WILLIAM BOOS

THE

WORLD,

THE

FLESH

FROM

AND

THE

ARGUMENT

DESIGN

• . . it must happen, in an eternal Duration, that every possible Order or Position must be try'd an infinite Number of times. This World, therefore, with all its Events, even the most minute, has before been produc'd and destroy'd, and will again be produc'd and destroy'd, without any Bounds and Limitations. Noone, who has a Conception of the Powers of Infinite, in comparison of finite, will ever scruple this Determination . . . . Suppose, (for we shall endeavor to vary the Expression) that Matter were thrown into any Position, by a blind, unguided Force; it is evident that this first Position must in all Probability be the most confus'd and most disorderly imaginable . . . . Thus the Universe goes on for many Ages in a continu'd succession of Chaos and Disorder. But is it not possible that it may settle at last . . . . so as to preserve an Uniformity of Appearance, amidst the continual Motion and Fluctuation of its Parts? Philo, in Hume's Dialogues, pp. 209 and 211 ABSTRACT. In the the passage just quoted from the Dialogues concerning Natural Religion, David Hume developed a thought-experiment that contravened his betterknown views about "chance" expressed in his Treatise and first Enquiry. For among other consequences of the 'eternal-recurrence" hypothesis Philo proposes in this passage, it may turn out that what the vulgar calt cause is nothing but a secret and concealed chance. (In this sentence, I have simply reversed "cause" and "'chance" in a well-known passage from Hume's Treatise, p. 130). In the first eight sections of this essay, I develop one topological and model-theoretic analogue of Hume's thought-experiment, in which 'most' ('A-generic') models rvl of a 'scientific' theory U are both 'eternally recurrent' and topologically random (in a sense which will be made precise), even though they are 'inductively' defined, via a step-bystep ('empirical'?) procedure that Hume might have been inclined to endorse. The last aspect of this model-theoretic thought-experiment also serves to distinguish it from simpler measure-theoretic prototypes that are known to follow from Kolmogorov's Zero-One Law (cf. the Introduction, 5.2, 6.1 and 6.7 below).

Synthese 101: 15-52, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands•

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WILLIAM BOOS

In the last three sections, I wilt argue more informally (1) that the metamathematical thought-experiments just mentioned do have a genuine metaphysical relevance, and that this relevance is predominantly skeptical in its implications; (2) that such 'nonstandard' instances of semantic underdetermination and 'pathology' seem to be t h e metatheoretic rule rather than the exception; and therefore, (3) that metamathematical and metatheoretic 'malign-genius' arguments are quite coherent, contrary (e.g.) to assertions such as that of Putnam (1980), pp, 7-8. In the essay's conclusion, finally, I assimilate (2) and (3) to the familiar datum that 'simplicity', rather than 'pathology', has more often than not turned out to be an anomalous 'special case' in the historical development of scientific and mathematical ontology.

INTRODUCTION

David Hume's thought-experiment, quoted above, is actually a rather curious conflation of stoic doctrine and "Epicurean Hypothesis". It is also, in my view at least, his most cogent formulation of an acute global form of the "problem of induction". The principal burden of this essay will be to provide Hume's speculation about 'eternal recurrence' ("that every possible Order or Position must be try'd an infinite Number of times" ) with a rigorous metamathematical analogue. In the Dialogues, of course, Hume's quasi-spokesperson Philo presents the idea to his interlocutors Demea and Cleanthes not as a speculation about empirical causation or 'induction', but as a confutation of traditional theological arguments from design. His thought-experiment may be an appropriate Leittext for this essay, however, if one considers claims made for assorted forms of 'metaphysical realism' and 'bootstrapping' techniques as more secular counterparts of such "arguments" (replacing the deists' 'god', if you like, with Einstein's "old man" ). It should perhaps also be noted that Philo's sharply-focused imagery has no counterpart in the better-known and more obviously 'metaphysical' Treatise and first Enquiry. This too may be understandable, however, if one recalls Hume's curiously Demea-like assertions in these better-known texts, that what the vulgar call chance is nothing but a secret and concealed cause (Treatise, p. 130), and that chance, when strictly examined, is a mere negative word, and means not any real power which has anywhere a being in nature. (Enquiries, p. 95; see also Dialogues, p. 215) Though there be no such thing as Chance in the world; our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion. (Enquiries, p. 56)

THE ARGUMENT

FROM DESION

17

In a sense, Philo's basic idea in the Dialogues is an inversion of the claim made in the last of the three passages about "chance" just cited. We might formulate the Dialogues' (and this essay's) 'inversion' in Hume's eighteenth-century diction, with only two changes of wording (italicized below), as follows: T h o u g h there be no such thing as Cause in the world; our ignorance of the real randomness of any event has the same influence on the understanding, and begets a like species of belief or opinion.

More precisely, we will explore some implications of a metalogicat framework in which 'most' models M of a 'scientific' theo~" U are topologically random (in a sense which will be made more precise), even though they are 'inductively' built-up by a step-by-step 'forcing' procedure. In effect, this procedure permits members of wide classes of infinitary sentences to hold in such 'inductive' models M (called 'Robinson A-generic below, for certain structures A) if and only if they are never countermanded by 'later evidence' about the structure of M. And this observation, in turn, suggests an underlying affinity between facts about spaces of models we will use, and topological analogues of results from Kolmogorov's probability theory (as might well be expected, in an argument which is supposed to be relevant to claims about 'induction'). For it is well-known to probabilists that a result known as (Kolmogoroy's) Zero-One Law provides one suggestive mathematical analogue for Philo's basic insight - that eternally-recurrent forms of local 'order' might be embedded in a measure-space of (sequences of) events characterized by complete global randomness (a space of infinite sequences of tosses of an unbiased coin, for example). It then follows as a straightforward corollary of the Zero-One, Law that almost all such sequences (all, that is, but a set of measure 0 in the usual product measure on the space 2 °,) do embed infinitely many repetitions of an arbitrary finite structure in this way. One could also derive a 'dual' topological analogue of this corollary, in which the set of all such 'eternally recurrent' sequences is comeager (the intersection of countably many dense open sets) in 20'. In effect, this essay's results about 'eternal recurrence' and other 'pathologies' provide a refinement of this analogue (in a 'Stone space' homeomorphic to 2'~). For they hold in a smaller class of 'inductively'

18

W I L L I A M BOOS

defined (A-generic) models that might appear at least initially to be better-'controlled'. More precisely, we will focus in the following sections primarily on a metamathematical framework, first explored by Abraham Robinson, in which we will be able to verify a model-theoretic refinement of Hume's recurrence-conjecture in all ('inductively' defined) A-generic models M of an arbitrary scientific theory U, where A is an arbitrary countable 'fragment' of infinitary logic. In Sections 1-4 below, we will explain what such a 'fragment' is, outline Robinson's definition of model-theoretic genericity, and discuss some sense(s) in which generic models are 'inductively defined'. In Sections 5-7, we will then show that the 'pathological' properties of such models do in fact include: 0.1. counterparts in all A-generic models M of U of Philo's geniusmalignus-like illusion - of endlessly recurrent local 'order' embedded in utter global disorder; and, further, as an additional consequence, 0.2. the validity in all such models M of G6delian rinternal contradictions ~ which rfollow~ (in the sense of the models) from inductively 'correct' finite extensions U' of U. (In plainer language, each A-generic model M of U in 0.2 will satisfy many conjunctions of the form '~ and rq~ is inconsistent with t~P,' so that the theory U' = U U {q~}, modeled in M, must appear ~wrong7 in M. 'Induction' does not take kindly to at least one natural attempt to 'formalize' ~itself~). In the straightforward topological senses alluded to above, moreover, 'most' models M of U then satisfy the properties requisite for 0.1 and 0.2 (that is, the class of those which do is comeager in an appropriate space St(U) of models of U). 0.3. To me at least, these observations suggest several tentative but rather obvious conclusions. 0.3.1. The first is that metamathematical and metatheoretic malign-genius arguments of this sort are quite coherent, contrary (e.g.) to assertions such as that of Putnam, 1980, pp. 7-8. 0.3.2. The second is that instances of such semantic underdetermination are in fact, in at least one relevant sense, the meta-inductive rule rather than exception. 0.3.3. The third is that metamathematical arguments of the sort we consider do have metaphysical relevance, and that the lines of argument they suggest are predominantly skeptical in their implications - or so,

THE ARGUMENT

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19

at least, I will argue in the essays concluding sections (8 through 10 below). 1.

FORMAL

PRELIMINARIES

1.1. In this section, we sketch the basic formal framework we will use to construe one version of David Hume's notorious 'problem of induction' (which I suggested earlier might well be called 'Hume's problem of design'). 1.2. For the remainder of the essay, we will consider a putatively 'universal' first-order theory U (or a higher-order one, reconstrued as a first-order theory of 'sorts'), and make the following technical assumptions. 1.3. The language L(U) of U is pieced together from 'abstract' and 'concrete' sublanguages L(T) and L(T,), for some terminal segment Z of positive and negative integers n, along with a separate lists of unary predicates I, (cf. 1.5 and 1.7 below), and unary function symbols fn for

n~Z. 1.4. The nonlogical vocabularies of the subtheories T and T, for n ~ Z are disjoint, except for the equality predicate =, and we assume that L(T) includes a fixed countable list of constant symbols C. 1.5. The axioms of U will include assertions that the 'universal' unary predicates In in L(U) are formally disjoint - that is, that -13x(L,(x) A I,~(x)), for m ~ n in Z. A preliminary 'Humean' rationale for these assumptions can quickly be given as follows: the In's serve as formal counterparts for extensions of discrete temporal stages in a model-theoretic 'world' M, and the T~'s provide 'experiential' theories about the 'evidence' collected at these stages). 1.6. The axioms of U will also include assertions which restrict the extension of the atomic predicates of each T,, to In, and guarantee that the I,,'s and corresponding atomic predicates of L(Tn) yield (trivial) faithful interpretations I and In of the 'abstract' and 'concrete' theories T and T,, in U, for each n ~ Z (cf., e.g., Shoenfield (1967), pp. 61-65, or Monk (1976), pp. 216-17). We will sometimes abuse the notational distinction between the interpretations I and In and their 'base sets' I and/,,. 1.8. We further assume that each of the theories T and T~ is recursively axiomatizable, and that Tincludes a recursively axiomatizable set-theor-

20

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BOOS

etic subtheory Z in L(ZF) C_L(T), which is strong enough to do arithmetic and analysis, and accommodate whatever other mathematics may be needed for scientific work (one plausible candidate for 2 might be

Zermelo set theory, augmented by replacement axioms for all formulas provably equivalent in Zermelo set-theory to Y)a-formulas; a definition of the ~-hierarchy may be found in Jech (1978), p. 114 ff.). 1.9. Metatheoretically, we will also assume the existence of additional (not necessarily faithful) interpretations E,, of a fixed 'testable' subtheory S of the 'abstract scientific theory' T in the 'evidentiary' theories Tn for n E Z. These interpretations En (which are not expressed in the vocabulary of U) will remain fixed for the remainder of the essay. 1.10. By 1.6, the E.'s will induce 'composite' interpretations l~n = In ° En of S in U, again one for each n E Z. We sometimes blur the notational distinction between E,, and l~n. 1.11. Subsumed in the interpretations En, by definition (cf. Monk (1976), p. 216) are metatheoretic functions from the constants of T into the constants of T~ for each n ~ Z. We 'internalize' these components of the interpretations E,, with the aid of new object-theoretic functionsymbols f, in L(U), along with axioms that specify that the fn's have domain I and range included in I,~, for each n E Z, and that they coincide extensionally with the metatheoretical functions determined by En on the elements of C. 1.12. the axioms of U are exhausted by of those of T and the T,'s, along with the 'bookkeeping' axioms mentioned above in 1.5, 1.6 and 1.11. In effect, we have set up a theory U which relativizes the theories T and T, to the disjointly sorted 'temporal stages'/,, and ensures that U considers the 'scientific theory' T 'correct' at each such stage, via the interpretations E.. Assorted 'Humean' motivations for this will be developed more carefully in the next section. 1.13. For each n ~ Z, we further let Un be the deductive closure in the language L(U,) = L(T) U {Ii l i < n} U {f,- l i < n} U U i < , L(T,) of the axioms of U which are in L(U,). The intended interpretation for the Un's should by now be clear: they encode 'old' information (that the sun has risen up to time n, as it were). We close the section with a few last bits of indispensable boilerplate. 1.14. We introduce a countable list of new (Henkin) constatns D for U, and stipulate that D includes the disjoint union of countable sublists

THE ARGUMENT

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21

C, the new constants for T, and Cn, for n ~ Z, the new constants for the Tn'S.

1.15. For each k E Z, further, we let Dk be the union C with the Cn's for n < k, and form the languages M(T), M(S), M(T.), M(Un) and M(U), by adding C, C~, D~ and D to L(T), L(S) (which may be a proper subset of L(T)), L(T~), L(U~) and L(U), respectively. 1.16. We also write T, S, Tn, Un and U, for the theories, in the languages thus augmented, whose nonlogical axioms are identical with those of T, S, T,, U~, and U, respectively. 2,

AN

INTENDED

INTERPRETATION

2.1. An 'intended (metatheoretic) interpretation' for the last section's formal machinery of theories and interpretations may help make the stipulations of 1.2 and t.3 a bit clearer. 2.1.1. The Tn's and 'cumulative' Un's represent theories about 'the world' at successive discrete temporal stages tn, whose assertions are 'witnessed' by the Henkin 'names' (constants) provided by C, for the theories T~. 2.1.2. As such, therefore, the Tn's (U,,'s) are supposed to subsume whatever theoretical measurements, observations, identifications, assertions and reconstructions 'we' may be able make in some appropriate neighborhoods of the 'instants' tn ('instants' tk for k ~< n). 2.1.3. The En's. accordingly, witness that the 'testable' subtheory S of T is 'borne out' in the T,'s. We use (syntactical) interpreta6on rather than (semantic) satisfaction, since interpretation is a 'local,' proof-theoretic procedure: it does not require extensional canvass of 'all' the structure(s) of an infinite model M of U, and it preserves a measure of ontological neutrality about T and U. On the construal of 2.1.1-2.1.2, then, the T~'s simply represent linguistically representable 'evidence' and 'data' which become available in a given scientific research tradition at 'time' t,. We have made the provisional metatheoretic assumption that the T~'s are consistent, but otherwise we accept them (essentially) because they are all we have. Our use of syntactic interpretation and assumption that Tn is incomplete thus disclaim 'knowledge' about 'the' (?) 'world at time t~.' A model-theoretic realist would presumably identify such knowledge with ability to 'survey' the - epistemically inaccessible - submodels M, of

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a model M of U which are given by the faithful interpretations In of T, in U. But the possibility that such indefinitely long sequences of Mn's may have baffling surprises in store for us is precisely the point of this paper. Be that as it may, I will now ask the reader to hold in suspension, for the duration of the paper at least, any reservations we both may have about the tenability of the formalism of the last section, much less the ideality and epistemic accessibility of the 'instants' tn, the 'universality' of the L,'s, or the reliability or 'correctness' of the T,,'s. We consider this (re)construction only as a provisional and not entirely unreasonable framework, and the results of Sections 5-7 as a partial indication of the latitude such provisions will permit. Within some metatheory for the framework, at any rate, we can at least provide a clear rationale for the disjointness-assumptions about the languages L(T) and L(Tn), for n ~ Z. 2.2. For suppose we do wish to formulate and 'test' axioms for the 'bridging' theory U, which we hope will 'account' for any emerging relations between our 'abstract, scientific' theory T, and the more 'concrete' and 'factual' theories T,. Then our stipulations about the extensional disjointness of the In's and

metatheoretic disjointness of the nonlogical vocabularies of T and the T,,'s, mentioned above, simply reflect Hume's characteristic and wellknown doubts about persistence and identity of predicates through time. If any correlations between the nonlogical vocabularies of 'successive' T,'s (metatheoreticaly expressible as extensions of the En's to extensions T' of T in M(T)) should turn out to be satisfied in a model M of the inclusive theory U, therefore, they will have to emerge from

conditions on M's interpretation of the 'bridging' theory U. 2.3. In U, for example, we might stipulate of a given (k + 1)-ary predicate A in L(Tn), that A(m + 1 , . . . ) is the 'time-evolved' counterpart of A(m . . . . ) for all m in some finite range of U's integers, where the notion of 'time-evolution' involved would depend on the details of U, and its tenability on the En's. In classical physics, such 'evolution' might be given by a suitable Hamiltonian function on phase space; in quantum mechanics, it might be represented by the standard time-evolution operator associated with an instance of the Schr6dinger-operator. Mathematical vocabulary for all this, at any rate, is supposed to be provided by 2 in L(ZF), interpreted variants of which will presumably

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be common to the 'abstract' theory T and its 'concrete' counterparts 7",, (We would like at least to be able to say in L(T) that some things are members of assorted collections at the moment tn). 2.4. The problem for a 'Humean' scientist would then be to 'test' ~empirieal' assertions ¢ ~ L(T) to see whether they are 'satisfied,' 'borne out', or 'confirmed' by the 'interpretations' En or T, where we provisionally assume all this can be suitably interpreted and reinterpreted, from time to time?), via I and the In's, in U. For each metatheoretically finite set of integers K such that T~- (the numeral r~ is in this range) for n in K, for example, one might imagine that workers in some sort of idealized scientific tradition might carry out such 'tests' at the intervals specified by K. 2.5. In the same fashion, we can obviously express arbitrarily long finite fragments of 'lawlike,' 'deterministic' relation we might propose between assertions ,~n in u (using 'time'-parameters t~zin T to represent the different 'instants'), and 'test' these assertions over very long finite stretches of metatheoretic 'time' (the t,'s for n in K). 'Decisions' that arise from answers to these test-questions about propositions ~ which are undecidable in the T~'s might reflect aspects of an underlying metaphysical or metatheoreticat 'reality', in the form of some unknown (and presumably unknowable) background-model M of U. But I have already suggested that our 'internal' framework U should be ontologically neutral about such issues. Speaking with the vulgar and thinking with the learned, as Berkeley once put it, we merely interpret (finite fragments of) theories in (finite fragments of) other theories. We claim no extensional insight into this 'Vernunftidee' M. We consult 'experience' (?) as it emerges. It 'answers', and we write down the results. With the evident finitude in real time of (informal counterparts of) such assertions quite probably in mind, Hume simply posed (in much more judicious and elegant terms) an obvious irreverent question:

"So what?" This essay, in effect, is an attempt to offer one metamathernatical analogue for Hume's blunt question, and stay for a partial answer. Its terms will emerge in the model-theoretic interpretation of Hume's thought-experiment sketched below.

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WILLIAM

3.

;INDUCTION'

AND

BOOS

INFINITARY

PROPOSITIONAL

LOGIC

3.1. The problem, of course, is that 'design' is great, and we are small. One interpretation of 'great' is 'infinite', and of small, 'finite'. Another might assimilate "great' to 'nonstandard finite' and 'small' to ~standard finite', in the sense of 'nonstandard' models of number theory; or 'great' to 'nonrecursive', and 'small' to 'recursive' ('computable'), in the case of sets of integers. Still another can be elicited from other aspects of the elusive distinction(s) between object-theoretic 'syntax', and metatheoretic 'semantics'. 3.2. Consider once again, for example, the problem of trying to extend and revise our interpretations of 'abstract theory' T, in conformity with 'empirical evidence', provided by successive Tn's. If we could correlate 'time-evolution' of assertions in T with (countably) infinite conjunctions /x~<~ q~,, of their interpreted counterparts q~ = E,(~) in L(Tn), then we could obviously canvass 'all' the 'evidentiary' counterparts of universal temporal claims about 'causal chains' in 'the world' M ~ U. One cannot write infinite conjunctions and disjunctions in the firstorder languages we have introduced, however, and attempts to use syntactic, object-theoretic quantification to substitute for such infinitary propositional assertions encounter well-known metamathematical problems, most of which amount to versions of the 'great'/°small' dichotomy alluded to briefly above (and discussed at somewhat greater length in Section 9 below). 3.3. Suppose, then, that we focus our dissatisfaction for now on the 'first-orderness' of the language L(U) and M(U), and simply enlarge these languages to allow countable propositional conjunctions of the sort we require, and would like it to express (Quine be damned, as it were). Model-theoretic implications in background set-metatheories of such enlargements do turn out to be relevant to the 'pathological' models we will consider, and these implications have been well-studied, moreover (cf. Keisler, 1971, and Keisler, 1973, for example), and a rich theory of them has evolved - though it should perhaps be noted that the set-meta(...) theories in these monographs, and in studies of 'abstract model theory' in general (cf. Barwise and Feferman, 1985) are again first-order... (Quine may rise again.). 3.4. Essentially, such extensions add (theory-relative) expressive

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strength in the resulting logics, called L(T)~o~o~,M(U)o~o~, etc., in the cases at hand here. In return (there is always a metatheoretic price to pay for such object-theoretic 'gains'), one loses some of the standard metatheorems, known from the work of Lindstr/3m to characterize firstorder logic (the compactness-property, for example; cf., e.g., Monk (1976), p. 424). 3.5. It will also be noticed rather quickly, of course, that languages such as L(T)o~o~ have the power of the continuum, even though L(T) itself is countable. We can mitigate this unwanted surplus of cardinality, fragments however, by considering certain countable K = M(U, A) C M(U),o,o~ (etc.), as shown in Keisler (1971), pp. 17-18, and Keisler (1973), pp. 98-99. Roughly speaking, a fragment K is a countable collection of infinitary formulas closed under finita~ formula-formation and term-substitution, and also under breakdown into subformutas (cf. [Keisler 1973], p. 98) of given infinitary formulas ~ ~ K. 3.6. Certain fragments L(U, A) and M(U, A) arise naturally, as classes of infinitary formulas which can be formed from languages L(U) using sequences from countable transitive models A of weak set theories (much weaker than the theory S introduced above; cf. Keisler, 1971, pp. 35-41). In particular, we will refer later from time to time to fragments L(A, U) and M(U, A) associated with admissible sets A. The latter are models A of a particular weak set-theorj, whose definition may be found (e.g.) on Keisler, 1971, p. 37. 3.7. Can one use such countable fragments M(U, A) of infinitary logics M(U)o~lo, to provide reasonable representations for the desire to secure infinite conjunctions with finite information, which Hume (in effect) queried? More tellingly, perhaps, can we also use such representations to provide natural analogues of Philo's thought-experiment in the Dialogues, quoted above? 3.8. In both cases, I think, the answer is "yes". In order to explain how in somewhat more detail, we will develop in the next section some of the basic notions of model-theoretic forcing and genericity, which will then be central to the thought-experimental observations and results of Section 3. More precise details about these constructs may be found, e.g., in Barwise and Robinson (1970), Hirschfeld and Wheeler (1975), Keisler (1973), Lee and Nadel (1975), and Lee and Nadel (1977).

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4.

BOOS

MODEL-THEORETIC

FORCING

4.1. D E F I N I T I O N (of Robinson's model-theoretic forcing). 4.1.1. Fix a given countable fragment L(U, A) in what follows, and form M(U, A) from L(U, A) (etc.), as M(U) was from L(U). A forcing condition, state of affairs or matter of fact for U is a consistent finite collection p of sentences q~E M(U). If each element o f p is in M(T,,), we will call p state of affairs or matter of fact at time t~. Denote the set of conditions by P. Standing on the shoulders of Cohen and Robinson, we now (re)state Robinson's definition of the ('strong') forcing relation iF between conditions p E P and elements q~ of M(U, A)). This relation is written 4.1.2. p lY p, and read p (strongly) forces ~, p (strongly) establishes q~ or p provides ('strong') 'inductive' evidence for ~. The 'inductive' deftnition (Lewis Carroll might be pleased) proceeds as follows. 4.1.2.1. p IF*q~ for atomic ~ iff ~ p ; 4.1.2.2. p IV q~ for finite or countable disjunctions q~ = VnEK ~,, i f f p I~ tn for some n ~ K; 4.1.2.3. p t Y q~ for existential ~v = 3xO(x) i f f p lY O(d) for some d C D. Equivalently, p IF* Vdez) p(d). 4.1.2.4. p ~ p for negations ~p = - n ~ iff for no condition q such that q_Dpdoesq

IVy.

4.1.2.5. p IF*q~ for finite or countable conjunctions q~ = A,,~K p,, i f f p IP ~ VnEK-n q~n. Equivalently, for each conjunct q~n and condition q _Dp, there is another condition r __Dq such that r t~ q~n. 4.1.2.6. p IP q~ for universal q~ = Vx~x) iff p If ~ 3 x - 1 ~ x ) . Equivalently, for each c in D and condition q _Dp, there is another condition r _D q such that r iF O(c). In a sense which can be m a d e precise, strong forcing is a partly 'intuitionistic' notion. Its 'classical' counterpart is the following

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4.1.3. (Definition of 'weak' forcing), p IF-q~, read p (weakly) forces ~, p (weakly) establishes q~ or p provides ('weak') 'inductive' evidence for 4.2. Philosophers more familiar with the work of Kripke than that of Robinson will recognize close affinities in 4.1.2 with Kripke's inductive definition of the forcing relation for intuitionistic logic (with 'constant domains'). There may be something appropriate about these affinities, for Kripke's definition is often construed as a processive semantics for 'stages of knowledge', and we are attempting to consider a processive, stage-by-stage semantics of (putatively) 'inductive' evidence. The details are somewhat more complex, however, for the 'possible worlds' of Kripke's proof of the completeness-theorem for intuitionistic logic (cf. Bell and Machover, 1977, p. 441, and van Dalen, 1983, pp. 182-3) are not finite conditions of the sort considered in 4.1. They are the "prime theories" of van Dalen (1983) - which are, however, closely related to the 'generic models' we consider. Counterparts of the central results of Robinson's theory and the 'eternal recurrence' phenomena of the next section can in fact be developed for generic Kripke-structures (One version of this can be gleaned from a modal treatment given in Bowen (1979), and others can also be introduced). These counterparts would require that we devise 'natural' topologies to impose on spaces of prime theories and Kripkestructures, however, and we will not explore such arguments here. Historically, at any rate, the central motivation for Barwise and Robinson's study of model-theoretic forcing was the following modeltheoretic analogue of Paul Cohen's "generic sets" or "complete sequences of conditions", the staple objects of independence-proofs in set-theory. We will make heavy use of this notion, and therefore review it in some detail.

4.4. DEFINITION. A subset G of P is (Robinson-) A-generic iff 4.4.1. whenever conditions p and q are such that p ~ G and q C p, q is also an element of G; 4.4.2. whenever p and g are in G, so is p U q; and 4.4.3. for each sentence q~~ M(U, A), there is a p ~ G such that p decides q~- that is, p I~-q~o r p 1~-- ~ . The basic portmanteau-theorem of the subject is the following composite result of Robinson et alo We state it in a way which highlights

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its parallels with Henkin's formulation of the completeness theorem for theories in L(U) (ef. 5.5-5.9 below). 4.5. T H E O R E M 4.5.1. (Henkin's Completeness Theorem) ~ois a (classical) consequence of p in U iff M(V) ~ ~ for each model M(V) determined by a complete Henkin extension V of U such that p ~ V; 4.5.2. (Generic sets yield Henkin models; Robinson et al.) A-generic sets of forcing conditions G determine complete Henkin extensions (cf. Bell and Slomson (1969), p. 119-122) V(G), and therefore canonical A-generic models M(G) of U. 4.5.3. (Interrelation between i~- and genericity; Robinson et al.) Each forcing condition p weakly forces ~ ~ M(U, A) iff M(G) for each Ageneric G such that p ~ G. 4.5.4. (Characterization of all A-generic models as those which omit all

nonprincipal types in A; cf. Keisler, 1973, and Lee and Nadel, 1977) A Henkin model M is M(G) for some A-generic G iff M(G) 'omits every nonprincipal type in A', that is, M(G) satisfies every V v 3 - (read "for

all-or-exists"-) formula ~D =

VX

I . . . VXrr,~J(Xl,

3y,

. • . ,Xm)

^ 4,.j) ( x l . . . . .

=

x.,

VX

1 . . . VX m

....

Vn<::o~

3yi • • .

, y,,,)

which is nonprincipal, that is, which has the following property: 4.5.4.1. for each C l , . . . , C m from D and ~ / ( q , . . . , c m ) from M(U), there are n < w and d l , . . . , di, such that T~(Cl .....

Crn)

A (~.tni

" " " ~!lnj) X

× (Cl,. • -,c,,, dl . . . . ,dl) holds in some model N of U. 4.6. We will consider later the sense and wider implications of the somewhat abstruse and obscure-looking conditions in 4.5.3 (cf. 5.5-5.6 and 6.4-6.5 below). For now, simply note that the close analogies we have traced between the 4.5.3 and the Henkin completeness theorem 4.5.1 suggest that weak forcing plays the role of a 'logical consequence'relation, and A-generic sets~structures the role of 'models', which are related to this notion of 'consequence' by the 'completeness'-result 4.5.3. The nature of this parallel is further clarified by Lee and Nadel (1977)'s proof of the 'if'-direction of 4.5.3 (that (M omits every nonprincipal type nA) --~ (M is A-generic)), for their argument actually provides

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a countable list of V v 3-sentences (essentialy formalizing the definition of A-genericity), which taken together axiomatize the class of A-generic M's in (A, U). 4.7. In order to show in the next section that topologically 'ahnost all' models of U satisfy infinitary 'eternal recurrence laws', we will briefly review in the next section Stone's compact Hausdorff topology for complete Henkin extensions V D U. 5.

STONE

SPACE

OF

ULTRAFILTERS

AND

MODELS

Throughout this section, unless otherwise mentioned, U is an arbitrary consistent, recursively axiomatizable first-order theory without finite models in a countable language L(U), to which we adjoin Henkin constants D to form the enlarged language M(U), as we did above for the particular theory U introduced in Section 1. For each such theory U, we let 5.1. S(L0 be the Stone space of ultrafitters of the Lindenbaum algebra B = B0 of (equivalence-classes of sentences in the) the theory U in M(U), as defined in Bell and Machover (1977), pp. 141 ft. and 191 ff. (there are also Lindenbaum algebras Bn(U) of formulas in which the variables vo. . . . . v,,_, may occur free, for each n, with corresponding Stone spaces S~(U); cf. Bell and Machover, 1977, pp. 191-208). 5.1.1. The basic neighborhoods N ( ~ ) = [qq ={V_D U I ~ V} for E M(U) form a countable clopen base for a compact Hausdorfftopology o- on St(U), and a fixed recursive enumeration of the sentences of M(U) determines a homeomorphism rr from (S(U), or) to the product space 20~ with the Tychonoff product topology (cf. Bell and Slomson, 1969, pp. 28-29). 5.1.2. This homeomorphism, in turn, will allow us to carry over the (Lebesgue) product measure on 2 ~' to a measure pc on S ( U ) , in an obvious and essentially canonical way. In order to carry out several later proofs that certain sets in S(U) are null with respect to this measure, we sketch the construction of this h o m e o m o r p h i s m / x in somewhat more detail. 5.2. D E F I N I T I O N S 5.2.1. We fix first a recursive enumeration p = (q~ i n < ~o}of M(U), and define a tree of basic neighborhoods Ns for n < w and finite sequences s : n ~ z, by recursion on I h s = n, as follows.

30

WILLIAM

BOOS

Let 3e)~ for the empty sequence () be S(U). If N%is already defined, let ~"be the least sentence in the list p which is not decided by UU {As}, and let N, A1 (NsA0) be [4'] for ~ = As t, (As A ~ ) . 5.2.2. Using the argument of Bell and Slomson (1969), pp. 28-29, one can then show 5.2.2.1. that each element y of 2 `0 corresponds in a one-to-one way with a unique element M r of S(U), and 5.2.2.2. that this correspondence, - r : y ~ Mr, is actually a homeo-

morphism of 20, onto S(U). 5.3. The measure Ix on S(U) is then obtained by transferring the product measure of 2 °, to S(U) via r, so that the measure of each N, is 1/(2~hs). For simplicity of notation, we will sometimes write/x(p) for/x(N(q0). It should be clear that all of this is essentially canonical, once the enumeration p is fixed, as asserted above in 5.1.2. 5.4. In the event, we will not wish to work with arbitrary elements V of S(U), but rather with a subspace which we may identify with a space of (countable) Henkin models M for U, and which we will call St(U). We will give St(U) the relative topology, and identify by abuse of language the clopen neighborhoods N(qO = N ( ~ ) N S t ( U ) with {M ~ St(U) I M ~ q~}. In order to clarify the nature of this subspace St(U) in Booleanalgebraic terms, and show (among other things) that St(U) is a corneager subset of measure I in S(U), it will be helpful to retrace briefly in the next few paragraphs the model-theoretic rationale behind the 'type-omitting' language of 4.5 (cf. Bell and Machover, 1977, pp. 20312). 5.5. DEFINITIONS 5.5.1. An m-type X($) in L(U) is a consistent (satisfiable) collection of formulas (infinitely many, in the cases of interest) in the free variables -- ( x l . . . . .

xm).

E(x) is principal iff there is a single (satisfiable) formula o-($) which provably implies each element of E(Y) in U; otherwise li;(y) is nonprincipal. A model M realizes the type X(~) iff some m-tuple of elements ril of M satisfies each element of E(Y) in M. A model M which does not realize the type X(X) is said to omit E(Y;). 5.6. Consider now the VV3-formula ~ introduced above in 4.5.4. A few moments' inspection wilt then convince the reader 5.6.1 that

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the condition called 'nonprincipality' of the VV3-formula ~ in 4.5.4 above is indeed equivalent to nonprincipality of the m-type ~(2)= {o-,,(2) t n < o)} { ~ ( 3 y ~ . . . ~yin (t~n ! A " ' " A @,yn) ln % o9}; and 5.6.2 that a model M of U satisfies the VV3-formula ~ iff M omits the type X(x) = I n < o)}. Varying slightly ideas first developed in Rasiowa and Sikorski (1970), we can use this formalism to provide characterization of the subspace St(U) of S(U), and show that this subspace is a dense open subset of S(U) of measure 1. 5.7. PROPOSITION. If ~q~ is the equivalence-class of q~in the Lindenbaum algebra B of U in M(U), then

[[3x~(x)] = VdeD [[p(d)]] in B. Sketch of the proof. Using the theorem on constants (cf., e.g., Bell and Machover 1977, pp. 112-113), one can adapt almost word for word the argument for variables given in Bell and Machover (1977), pp. 193194. This permits us to (re)state the following 5.8. DEFINITION. An element V of S(U) is Henkin (cf. 4.5.t) iff V preserves' each of the suprema in 5.7, that is, ~3xq~(x)~ ~ V iff [[q~(d)~ ~ V for some d in D. As befits a notion which has been enormously useful since it was devised in the late 1940's, this property can be characterized Jn more than one equivalent way, and valiants of the following result are proven in every good textbook-introduction to mathematical logic (cf., e.g., Bell and Machover 1977, pp. 118-22 and 191-98). 5.9. THEOREM (Henkin; Rasiowa and Sikorski; cf 4~5). 5.9.1. Henkin ultrafilters exist in S(U); moreover, the following (5.9.25.9.4) are equivalent for each V St(U): 5.9.2. V in S(U) is Henkin; 5.9.3. V omits each (nonprincipat) type Z(x) of the form Z ( x ) = {¢(x) A ~¢(d) fd E D}. 5.9.4. V gives rise to a unique Henkin model M(V) of U in M(U), that is, a countable model M(V) of U such that 5.9.4.1. V is the set of equivalence-classes ~¢~ of sentences ~p in M(V) satisfied by M(V), and

32

WILLIAM

BOOS

5.9.4.2. M(V) satisfies a sentence 3xq~(x) of M(U) iff M(V) satisfies ~(d) for some d in D (D is a 'set of witnessing constants for U in M(V)). We have followed Rasiowa and Sikorski's (1970) emphasis on the equivalence between the 'witnessing' role of the constants D and preservation of the Lindenbaum-algebra suprema in 5.8, in order to show more easily that 'most' elements of S(U) are Henkin structures with respect to the given 'base'-set of constants D. We will briefly consider some elaborations of this theme at the beginning of the next section.

6.

GENERIC

AND

RANDOM

MODELS

6.1. P R O P O S I T I O N , The collection St(U) of Henkin elements V of S(U) is a comeager subset (intersection of countably many dense open subsets) of measure 1 in S(U). 6.1.1. Observe first that subsets X of S(U) exist which are comeager but have measure 0 (cf. the proof of 6.7 below, and the reference therein). The complements Y = S(U) - X of such subsets are therefore meager subsets of S(U) (countable unions of closed subsets with empty interior), but have measure 1. Separate arguments must therefore establish 6.1's two claims.

6.1.2. (Sketch of the proof that St(U) is comeager) Abusing from time to time the notational distinction between ~ and [~], we show that the set S C S(U) of Ws such that 3x?(x) ¢ V iff q~(d) ¢ V for some d ~ D is a dense open subset of S(U) for each sentence 3xq~(x) in M(U) and infinite subset D of D. 6.1.1 will then obviously follow. S is open, since it is the union of the clopen sets (3x¢(x) --+ ~(d)], for d ~ D. It is dense, for if there were an [~/] C the complement T of S for some sentence ~7, we would have U t-~l--*3x¢(x), and U~-~/---~-Tq~(d) for each d E D. But we could then choose a d such that d does not occur in ~/, and the theorem on constants would ensure that U t- r/--, Vx-7 ?(x); r/would therefore be inconsistent with U, and It/] the empty set. Note also that the argument given here for D works equally well for any infinite subset D' of D. 6.1.3. (Sketch of the proof that St(U) has measure 1) It will suffice to show that for each p(x) with exactly x free,

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0.

dED

We assume the measure ol of [3xq~(x)] is nonzero, and let ~(d) be (3x~(x)) A (-lq~(d)). Let (d~ In < a~) e n u m e r a t e D, and for increasing finite sequences o" of natural numbers of length lho-, let ~(o-) be 3x?(x)

A

qS(d~(0).

i
To prove 6.1.3, it will clearly suffice to show that there is a positive • < 1 such that for each n > 0, there exists a cr with lhcr = n such that the measure of [~(cr)] is less than ~ . Suppose not. T h e n for each positive • < 1, there is an n ~ w such that for all cr of length n, the measure of [~(o-)] ~> e~. Equivalently, for all positive 6 < 1, there is an n ~ o) such that for all o- of length n, /z([~(d~(0)) v • -. v q~(d~(~h~))]) < a" (choose e such that (l - g ) i / ~ < • < 1). But this implies that S N [3x~(x)] has measure 0, where S is as in the p r o o f of 6.1.2. But S is dense open in S(U), so its imersection with the clopen set [3x¢(x)] must have positive measure. This contradiction establishes 6.1.3. To clarify below why the set of Robinson A-generic models of U is also comeager for each countable admissible A, it will be helpful to have a further 6.2. D E F I N I T I O N . Elements p of M ( U ) code unique Borel subsets of S(U) in an obvious way, as follows: 6.2.1. each first-order ? codes the clopen set [?l = N ~ =

{v s(u) I

v};

6.2.t.2. if q~ codes X, -nq~ codes S(U) - X; 6.2.1.3. if q~ codes X and 4' codes Y, ~/x O codes X N Y and ~p v 4J codes X U Y; 6.2.1.4. if ~,, codes X,, for each n < ~o, then V,,<~, ~,, codes U ..... X , and/~,,<~o q~ codes O,,
34

WILLIAM BOOS

6.3.1. an element V of S(U) is topologically A-generic or Cohen-Ageneric iff V is an element of every comeager subset of S(U) which is coded by an infinitary sentence p ~ M(U, A); and 6.3.2. an element V of S(U) is A-random iff V is an element of every Borel subset of S(U) of measure 1 which is coded by an infinitary sentence ~ ~ M(U, A). The precise relation between Robinson and Cohen A-genericity may or may not have been put to paper, but some aspects of it are not hard to establish. The following observation will turn out to be useful. 6.4. PROPOSITION. For each finite sequence d of elements of the set of 'new' constants D for U in M(U), let B(d) be the subalgebra of the Lindenaum algebra B (of sentences, not formulas) of M(U) involving only the 'new' constants in ran d. Then 6.4.1. (the completionof) B is isomorphic to the direct limit of (the completions of) the B(d)'s (cf., e.g., Jech, 1978, pp. 237-8 and 456-7); 6.4.2. each bijection/3 : v,, --~dn between the variables of M(U) and D yields an isomorphism between the Lindenbaum algebra of L(U)formulas ~07) in the variables v., for n in some finite subset s of ~o, and the Lindenbaum algebra of sentences B(d), where d = (dn I n ~ s); and 6.4.3. an ultrafilter V E St(U) is Robinson A-generic, for a given admissible A such that D C A, iff V A B(d) is Cohen-A-generic for each finite string d of elements of D; Sketch of the Proof(s) of 6.4.1 and 6.4.2. Essentially, 6.4.1 follows from the definition of direct limit, given in the pages cited above from Jech, and 6.4.2 from the definitions of the Lindenbaum algebras involved, along with appropriate instances of the theorem on constants. Sketch o_f 6.4.3 ( ~ ) . If V N B(d) is Cohen-A-generic for each finite sequence d of elements of D, V preserves each infimum of elements of the Lindenbaum algebra B(d) which is in A. If E(0) is a nonprincipal n-type, E(0) corresponds via the isomorphisms in 6.4.2 to collections of elements of various B(d)'s with infimnm 0. For each such d, Cohen A-genericity ensures that V satisfies -n~r(d) for some 0"07) E E(o), so that V omits the type E07). Sketch of 6.4.3 (--@ With the aid of the 'coding'-apparatus of 6.2, and analogues of standard arguments about set-theoretic Cohen-genericity (cf. Jech, 1978, pp. 139-140, 155 and 544), one can show that it is sufficient for 6.4.3 to verify the following:

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6.4.3.1. whenever/3 = ([ev,] I n E w) is an arbitrary 'descending chain' of elements [q~] of B(d) in A, where d = ( d o , . . . , d,,_~ ) is some fnite string of elements of D, and 6.4.3.2. /3 alsohas the property that A

[~,,1 =0,

6.4.3.3. V is not an element of [q~,,], for some n < to (i.e., some [~,~ is not in V). But note now that/3 corresponds once again, via one of the isomorphisms of 6.4.1 and 6.4.2, to a nonprincipal n-type E(O), and any ultrafilter V in S(U) which omits E(~) 'preserves the infimum' of/3 i.e., is such that -nq~(d) ~ V, for some e~(P) corresponding (via restoration of v's for d's) to ~n(d) in ran/3. Since/3 was arbitrary, 6.4.3.16.4.3.3 permit us to claim that we have verified 6.4.3. From 6.4 follows an obvious sharpening of 6.1 above. 6.5. PROPOSITION. For every countable admissible set A, 6.5.1. each Cohen-A-generic element V ~ S(U) is Robinson A-generic; and 6.5.2. the set of Robinson A-generic models is comeager in St(U). Sketch of the proof of 6.5.1. A Cohen-generic ultrafilter V on B restricts to a Cohen-generic ultrafilter on each B(d) C_B. Sketch of the proof of 6.5.2. Since the Lindenbaum sentence-algebra B of U in M(U, A) is the direct timit of the B(d)'s, Stone duality yields that its Stone space S(_U) is the topological inverse limit of the Stone spaces S,i(U) of the B(d)'s. By 6.4.3, an element V ~ S(U) is Robinson A-generic iff 6.5.2.1. V B(d) is Cohen A-generic on B(d) for each d from D iff 6.5.2.2. for each sequence (bn I n < w} of disjoint elements of B(d) such that V,,
36

WILLIAM

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ments of the Cantor space 2 °') have been thoroughly studied in the settheoretic literature (cf., e.g., Jech, 1978, pp. 540-5, and the more abstract treatment in Kunen, 1984). The papers of Keisler (1973), Lee and Nadel (1975) and Lee and Nadel (1977) also clarify fully the equivalence of Robinson A-genericity and omission of types in A, as we remarked earlier. Less well-studied is the relation between Robinson A-generic and A-random elements V of S(U). The authors of Lee and Nadel 1977, for example, apparently assert that each A-random V is Robinson A-generic (cf. p. 82). More correct is the following analogue of a complementarityresult which is known to hold between A-Cohen generic and A-random

reals. 6.7. PROPOSITION. There is an admissible set A ~ HF such that the collection of Robinson A-generic models of each recursively undecidable first-order theory U is /x-null in S(U). Therefore, no A-random model M of U is Robinson A-generic for this A. Sketch of the proof. There is a standard construction of a sequence of comeager sets X,~ for n < o9 such that/x(X~) = 1/2n for each n < o9 in R (cf. Oxtoby, 4-5), and this construction can be carried over to provide a similar sequence X, in SI(U), the Stone space Sa of the Lindenbaum sentence-algebra Bd, determined by U in the language with only one 'new' constant d from D. We can further assume 6.7.1 that each Xn is the disjoint union of clopen basis sets X~ for j < o9, such that the measure/x(Xnj) each Xnj, constructed as in Section 5 in Sa, is l/2~+J; 6.7.2 that the/Y on Sa is the restriction of the measure Ix constructed in Section 5 on S(U); and 6.7.3 that the sequence (Xnln < w) is in the smallest admissible set A such that A ~ HF. Clause 6.7 follows as a special case from the facts that that S(U) is isomorphic by duality to the inverse limit of the Stone spaces S,i of the B,i's, for finite sequences d from D, and that the measure /x is the corresponding inverse limit of its restrictions/zg on the 'factor' spaces S,i. By 6.4 above, then, every Robinson A-generic model M of U must preserve each supremum Vl<~oXnj = 1 in Ba, and therefore must lie in X = f'qn<,oXn. Since X has measure 0, this verifies 6.7. Essentially, this completes the technical review we will need to prove

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in the next section the claims we have made about 'eternal return'phenomena, in models of the 'scientific theory' U introduced above in Section 3. It should be observed, however, that despite results such as 6.7, there is a well-studied (partial) duality between the topological and measuretheoretic properties of spaces such as S(U), which translates in certain cases from 'meager' to 'null', for example, and from 'comeager' to 'measure 1'. An analogue of Kotmogorov's originaI version of the ZeroOne Law, for example, does hold in each case. An excellent and thoroughly readable survey of this theory may be found in Oxtoby (1970) (for the Zero-One Laws, see 82-5). For the simpler purposes of this essay, I will continue to emphasize in the next section two underlying consequences of the model-theoretic version of the theory we have sketched: 6.8.1. that the stone-space constructs of this section naturally yield two - essentially disparate - notions of "randomness': Robinson and Cohen A-genericity, in the topological case, and the measure-theoretic property called A-randomness in 6.3.2; and 6.8.2. that the 'inductive' definition of Robinson A-genericity, via the 'forcing'-clauses, will not obviate 'pathological' consequences of the topological notion of randomness we have sketched: it will actually ensure a special case of it. 7.

THE

UBIQUITY

OF

'ETERNAL

RECURRENCE'

7.1. In this section, we finally show, for sufficiently large admissible sets A, that appropriate forms of 'eternal recurrence', internal qnconsistency7 and other 'pathological' properties hold in all Robinson A-generic models M. It will follow that topologically 'almost all' the semantic structures to which 'internal' or 'metaphysical' realists commonly appeal, 'will deceive' us, for 'temporal' cons, about their 'true' nature (which is essentially random, in the topologicai sense explored in Section 6). 7.2. Recall first that the 'abstract' mathematical-'scientific' theory T has a subtheory S which is interpreted in each of the temporal 'crosssections' Tn, by (metatheoretic) interpretations En, which will be fixed in what follows. The language L(U) of the umbrella-theory U also includes function-symbols fn of 1.11 for n ~ Z, and among the axioms of U appear (for example) such assertions as Vx((Vy(f,~(x)=y))

38

WILLIAM

BOOS

~-~I(x)), and VxVy((f,(x) = y) --~ I,(y)), whose intended interpretation is that each fn maps all of I into I,. Using these function symbols and the En's, we can formulate the following definitions. 7.3. Let a property q~(x) E L(S) C_L(T) be given, and let z range over finite functions z: k ~ {0, 1} for k. A temporal pattern (based on ~ and z) is a conjunction q~n~k(x) of the form Ai<~Oi(x), where Oi(x) is En+i(qO(fi(x)) if z ( i ) = 1, and -nE.+~(qO(fi(x)) if z(i) = O. We wish to have a uniform way of generating the 'eternal recurrences' that we will show must hold in all A-generic models M of U for sufficiently large countable admissible sets A. The following construct provides one such scheme. 7.4. D E F I N I T I O N . Let ~p(x)EL(S)C_ L(T) be given, once again, and let z range over finite functions z: k ~ {0, 1} for k < w. 7.4.1. A property ¢(x) E L(S) is empirical in U iff for each m, k < o9, z: k --+ {0, 1} and c E C (the 'original' constants of the 'abstract' physical theory T), no consistent extension of U by sentences of pure equality theory (with constants from M(U)) decides ¢,¢k(c) for all n > m. 7.4.2. Given r: k--+{0, 1}, a property ¢(x) in L(S) is z-recurrent in a model M of U iff M satisfies q~,¢k(C) for arbitrarily large n < ~o and each

cEC. 7.4.3. q~(x) in L(S) is eternally recurrent in M iff it is r-recurrent in M for all k < w and all r: k ~ {0, 1}. The idea of 7.4.1 is that no assertion of 'pure logic,' or posited equivalence with such an assertion in U, will suffice to decide all the 'empirical' data given by a future temporal patterns ~,~k(c) based on q~ and z. 7.4.2 and 7.4.3 seem self-evident. With these notions in place, we can now prove the following simple application of the techniques of Robinson, Barwise, Keisler et al., introduced in earlier sections. To state it, let A be the smallest admissible set which properly contains HF, the collection of hereditarily finite sets. 7.5. T H E O R E M . Every empirical formula q~(x) E L(S) C_L(T) is eternally recurrent in every A-generic model of U. Proof. Let empirical ~(x) and z: k ~ {0, 1}, and m < 0 be fixed for the

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remainder of the proof, and recall the definition of the nth temporal

pattern pn~k(x) for n > m, given in 7.4.1 above. If M is an arbitrary A-generic model of U, we will actually show 7.5.1: the infinitary formula rl~L(A,U) CM(A,U), given by r / = Vx Vn>m~,,~(X), holds in M. To do this, in turn, it will suffice to verify that the V V 3-formula 77 is nonprincipal in M(A, U). Suppose not. Then there is a single (satisfiable) formula ,/(x) in M(U) such that U ~-x(T(x) --~ -q ~n,k(x)) for all m > m. Let ga be the least integer > m such that all the vocabulary in 7(x) comes from M(u,~) (cf. 1.13), and choose an arbitrary c ¢ C. Then there are finite sets of sentences A C U rq M(U,~), and Hn _CM m = U 71 (Ur~a M(Tr)) such that 7.5.2. A U H,, ~ 7(c) --+ -~ ~r,,k(c), for all n > r~. Then 7.5.3. A U {,/(x)} t- ~Hn --+ -7 ~n,k(c), where the right-hand side is in M ~a, so that 7.5.4. ~-( A (A U {7(c)})) --~ ( A H n -+ -7 ~ , k ( c ) ) . The antecedent of this implication is in M(U~), and the consequent is in M ~a. But recall that these languages are disjoint, except for = and the 'new' constants (my formal gloss of 'Hume's hypothesis' in 1.5 and 2.2). By the Craig Interpolation Theorem, then, there must be an interpotant (in that intersection, such that 7.5.5. ~-( A (A U {7(c)})) ~ ~', and 7.5.6. ~-~--~ (/~H~ --+ -7 ~n,k(c)), so that 7.5.7. U}-(--~q~,,k(c), for all n > r ~ . But this contradicts the hypothesis that the ~ is empirical. There can be no such consistent y(x), therefore, so that the V V 3 - f o r m u l a r/is nonprincipal. Since it is also in M(U, A), it must hold in every Ageneric model M of U, and we are done. We now turn to the prevalence of G/3delian phenomena in each Robinson A-generic M, for sufficiently large admissible sets A. 7.6. To simplify and fix background assumptions, we adopt the framework and notation of Smorynski (1977) for the rest of this section, and assume in particular that the syntax of M(U) can be enumerated in a primitive recursive way, that a coding-apparatus (Smorynski, 1977, pp.

40

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823-40) and RE provability-predicate Pru (Smorynski, 1977, pp. 8378) have been provided for the language M(U) and theory U in U, and that these support the 'derivability-conditions' D I - D 3 of Smorynski (1977), p. 827. 7.6.1. We also relax somewhat Smorynski's "inessential assumption" (p. 830) that only numerical variables and function symbols occur and construe constants d in D as 0-ary function symbols d,., coded for each i < ~o by <2, ((), i)) (cf. Smorynski (1977), pp. 830-1). This latitude will require that we relativize the predicates and terms used to code syntax in Smorynski (1977), pp. 835-8 to U's interpretation of the natural numbers or hereditarily finite sets; this can be achieved with appropriate modifications for the predicates and default-clauses for the terms, and we assume that this has been done in what follows (and therefore do not repeat it in 7.7 below). 7.6.2. We will also need a variant of Feferman's dot-notation which applies to the 'new' constants d~, rather than numerals (cf. Smorynski, 1977, p. 837). Something like this framework is presumably implicit in Smorynski's brief sketch of the Hilbert-Bernays Completeness Theorem, pp. 860-1. This notation may be introduced as follows, using the notation of Smorynski (1977), p. 837: 7.7. DEFINITION. Let the term s°(u, v) = sub(u, <2, >). 7.7.1. We usually write rp@)~ for s°(rq~(x) ~, y). We then have the straightforward 7.8. PROPOSITION. 7.8.1. s°(~p(x) 7, t ) = r0n, where t) is equivalent to q~(&,), for each closed term t which is provably equal to d~ in U; 7.8.2. each term s°(rq~(x)~, v) has the variable v free, though it formally encodes a sentence of M(U); and finally, 7.8.3. the result of substituting the constant d,~ in ~q~@)~ is provably equal to %~(dn)~ in U. Proof (of 7.8.3) rq~@)7(dn) = s(~(x) ~, dn)= sub(~(x) ~, <2, ~, 0})) = sub(~q~(x)~, ~d~~) = ~q~(dn)Z The following is similarly unproblematic.

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7.9. DEFINITION. For q~M(U), Con(UU{p}) is the sentence P r u ( ~ -,, 0 = 1~) in M(U). Witlh these notational conventions and propositions in place, the following observation now follows directly from G6del's work. 7.9. T H E O R E M . Let A be the smallest admissible set which properly contains HF, the set of hereditarily finite sets. Then for each Robinson A-generic model M of U, there are infinitely many sentences p of M(U) such that p and --7 Con(T U {q~}) both hold in M. Proof. For each n < co, and each dense set of formulas A(x) in Bn, form the n-type. 7.9.1. F(x) = {6(x) ~ ~Pru(rS(,~:) -+ 0 = ln) j 8(x) ~ A(x)}. By our metatheoretical assumptions about the 'mathematical' part Z of U, there are models of U whose substructures which model Z are standard. Since each such structure will 'accurately' reflect the metatheoretic consistency of each sentence p(dn) for d,, in D, we may assert that each such F(x) is finitely satisfiable. It is our aim to consider models which omit F(x), and we therefore undertake to prove 7.9.2. Each such F(x) is nonprincipal. Proof of 7.9.2. Suppose there were a formula ~(x) such that 3xw(x) were consistent with U, and U ~- ~(x) --+ ,/(x) for each 7(x) ~ F(x). By the density of zX(x), we can assume that ~(x) ~ 2~(x). Let dn in D be such that ~q(d~) is consistent with U, and write d for dn. Then g ~-rl(d) --> (-1Pru(r~(d) --+ 0 = 1~)), i.e., U ~-~q(d) -~

Con(U u {n(d)}). But this would imply (metatheoretically) that ~q(d) is refutable in U, by G6del's Second Theorem (cf. Smorynski, 1977, p. 828) for the theory U U {~(d,,)}. So there is no such r/(x ). 7.9.3. Since each F(x) is nonprincipal, each Robinson A-generic model M of U omits F(x), by 4.5.4, so long as A includes the n-type F(x). But omission of F(x) in M means that there is a d~ in D such that 6(d~) A --1 Con(U U {6(dn)}) holds in M. Since dense sets /X(x) of the sort we have considered do exist in B,, we have shown that there is at least one such sentence with this property in M(U) for sufficiently large A.

In order to verify that the set of (equivalence classes of) such sentences is actually infiniw, we need only note that 7.9.4. whenever a model M of U satisfies q~ A ~ Con(U U {p}), M satis-

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ties -nCon(UU {~0}) for each tp such that ~0 also holds in M, and ~On-~ ~q~, i.e., such that M~ ~, and UF- ~--~ q~. This follows immediately from the derivability-conditions, for if U ~ ~---~ q~and Pru(r~ ~ 0 = 17), D1 of (Smorynski, 1977, p. 827) yields that U~-Prc~(rq~---~p-) and D3 then that Pru(~0--~0 = P), so M ~--n Con(U U {0}) as well as ~. 8.

IS ; O N T O L O G I C A L E C O N O M Y ' A S O U R C E OF * P A T H O L O G Y '

8.1. We turn now to some extended arguments on behalf of the allegedly 'inductive' nature and 'ontological economy' of A-generic models M, in which the 'pathological' assertions discussed in the last section hold. 8.2. To see the rationale for such arguments, observe first that models N of U provably exist in ZFC whose properties are diametrically opposed, in a sense, to those of A-generic models M. These are the socalled A.saturated models, which realize every type in A which is consistent with U. The very language suggests that A-saturated models satisfy a kind of plenitude principle, and they do indeed exhibit a kind of ontological profusion, in that an), type in A that could be realized in N is. One can also show that such N's also admit many automorphisms (isomorphisms from N onto itself). Such models are even rather easy to 'define' in ZFC, but they remain the topological exception rather than the rule, in the senses alluded to in previous sections. 8.3. By contrast, 4.5.4 yields that an A-generic M only satisfies types ~;(x) in A which are principal. Since every element m of M determines a 1-type Ern = {o'(x)]M ~ o-[m]}, this yields an obvious dichotomy for such :Sin's in M: either 8.3.1. the set of formulas E,~ is not in A; or 8.3.2. there is a formula O-m which 'defines' the type or set of'properties' Era, in the sense that o-re(x) holds at m in M, and every other property ~(x) definable in the language L ( U ) which holds at m is a universal consequence of o'm(x) in M. 8.4. Suppose, for example, that A includes all recursive (computable) V V 3-formulas ~ (this will already be the case, for example, if A is the smallest admissible set such that o) ~ A, mentioned earlier). Then either

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an m in M is 'definable' in M, in the sense of the last paragraph, or the set of properties satisfied by m in M cannot be computed at all. 8.5. As A grows, moreover, this dichotomy grows more and more acute. Should the countable admissible A be a model of a stronger theory such as ZFC, for example, A-generic sets G D U E M will provably exist whose associated models M(G) are about as ontologically restricted as they can get, from the point of view of the 'universe' of A. 8.6. In this sense, then, we can recapitulate the basic observations of the previous sections in the following terms: 8.6.1. A-generic models M may be regarded 'inductive hulls' of the forcing conditions which define them as structures for M(U, A); 8.6.2. the larger the countable fragments A, the broader the scope of 2. l's 'inductive' forcing-definition, and the ontologically sparser the Ageneric models M; 8.6.3. topologically 'almost all' (Henkin) models M of U are Robinson A-generic, no matter how large the countable fragment A; and finally, 8.6.4. that the 'ontological sparseness' of such models does not ensure that their structures ('causal' or otherwise) are uniform or otherwise 'smoothed' or 'normalized' to conform to our expectations of them. On the contrary: such models seem to provide some of the more rigorous venues one could hope to find for the activities of that venerable Cartesian archimago, the genius malignus or mauvais gdnie, who neatly (and dialectically) inverted the role traditionally assigned by arguments from design to 'god'. 8.7. As another example of the dilemmas one encounters when one tries to anticipate 'gdnies' decisions - malign or benign - one might consider the following 'secularized' argument from design.: 8.7.1. ('Mill's Circle') John Stuart Mill's ingenious suggestion (cf. e.g., Mill, 1974, pp. 569-575, and Losee, 1980, pp. 156-158) that one might provide a subtly '(recta)inductive' 'proof of (his variety of) induction. (In recent years, Richard Boyd has offered a comparably self-referential rationale for abduction, and Kripke's analysis of Wittgenstein sometimes seems to apply implicitly similar lines of reasoning to the puzzle about following a rule). 8.8. In crude and reductive paraphrase, Mill's argument (originally formulated as a 'justification' of a 'law of universal causation') went roughly as follows. Enumerative induction - with all the cautions, reservations, provisos

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and auxiliary hypotheses one may wish - has (usually) 'worked' rather well, has it not? Implicitly at least, we do also seem to take this curious efficacy as a kind of de facto meta-inductive warrant, to continue applying object-theoretic inductive arguments, do we not? And if so, does this not mean that there 'must' be something universally correct, or at least adequate, about such appeals to meta-inductive arguments (which Mill formulates as assumptions about 'causal' regularity) to support our (thereby warranted) use of object-theoretic induction - an 'inductive' version of 'inference to the best explanation,' as it were? 8.9. Yes and no. 8.9.1. The burden of this essay has essentially been to suggest - first that if one tries to formalize something very like Mill's argument in model-theoretic terms, one is likely to consider something very like the semantics of the 'generic' models mentioned above, in which we only admit in to the semantics of our models M what finite bits of information 'force' us to admit. 8.9.2. Indeed, the language of 'model-theoretic forcing' is built into the jargon used to define generic models: the properties which hold in such models of the theory U are exactly those which are are ('weakly') 'forced' by pieces of the elementary diagram of models of U, and the clauses for such 'forcing' closely resemble a more sophisticated metalogical counterpart of enumerative induction. 8.9.3. Yet the models which result from such ontologically economical 'forcing'-techniques, against 'Millian' intuition, turn out to be exactly the kinds of structures which bear out most strikingly some of the reservations expressed in Hume's thought-experiments - and in which, moreover, the G6delian problems of 7.10.2 abound. The metalogical results which provide a basis for these reconstructions are well-established in the literature, but they seem to have been little noticed by philosophers. 8.10. In the essay's final two sections, I will briefly suggest 8.10.1. that one might interpret these and other, deeper metamathematical results as formal miniatures of complex phenomena which have emerged in the ontologies of nineteenth- and twentieth-century mathematics, metamathematics, physics and metaphysics, and argue 8.10.2. that such 'formal miniatures' may have wider implications for metaphysics, in some broader senses of this very broad word.

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9.1. The notorious "problem of induction" is actually only one of several subtly interrelated dilemmas we have arguably created for ourselves, in our persistent (but presumably finite) formal attempts to canvass 'the' infinite. Indeed, I would argue (and have argued, in [Boos]) that such attempts may be constitutive of Kant's "fate of reason", mentioned in the opening pages of the First Critique. Other (more or less familiar) metamathematical reflections of this "fate" would include phenomena such as: 9.2. o3-incompleteness (in which a theory T proves q~(~) for each metatheoretic natural number n, but does not prove (for every natural number n) ~(n)); 9.3. theory-relativity and semantic underdetermination, of a cluster of cognate metatheoretical notions by formalized object-theoretic counterparts. One might call these notions liminal, in that they seem to hover and reappear at the threshhold of theories in which we might think we have definitively expressed them (or "captured" them, to use an irrelevant macho-jargon which many mathematicians and logicians seem to like in this context). The most notorious of these notions, of course, is 'truth' (for which few of us will stay long enough for an answer). But 'definability' is an equally elusive idea, and a more philosophically relevant one, perhaps, in a century whose 'analytic' metaphysicians have tried again and again to 'solve' or dismiss perennial dilemmas with the aid of a curious hybrid discipline which has come to be called "philosophy of language". In this connection, it may also be of some philosophical interest that Boolos and Vesley have recently explored a new version of G6del's fundamental results, which exploits the Berry paradox - the 'definition' of the least integer not 'definable' in less than so-and-so-many syllables rather than the classical paradox of the Liar. Other, somewhat parallel lines of inquiry about 'complexity' have also been pursued with great acuity by Chaitin (cf. Chaitin, 1988, and the popular exposition in Rucker, 1987, pp. 286ff.). Be that as it may, it should also be observed that the w-incompleteness of 9.2 may in fact be regarded as an instance of the 'theory-relativity' mentioned in 9.3 - namely, that of (apparently unproblematic) theoretical notions about "'the' natural numbers".

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Other liminal conceptions have turned out to straddle theory and metatheory in similarly characteristic ways. These notions - some of which may initially seem unproblematic - include 9.4. 'finiteness' (theories of the sort we consider, for example, cannot 'really' say of a given object-theoretic 'natural number' that it has an indefinitely large but metatheoretically finite collection of predecessors); 9.5. 'well-foundedness' of binary relations R (the assertion that there are no 'descending chains' r l R . . . R r n R . . . ; equivalently, that every subset of the field of the relation has an 'R-minimal' element), which would permit us to define and prove things by metatheoretic 'induction and 'recursion' over R (one cannot say 'internally' of such R's that they are 'really' well-founded); and 9.6. 'the' range of applicability given axioms and schemes of 'mathematical induction' (another variant, in effect, of ideas implicit in 9.4 and 9.5). The classical semantic paradox of the 'heaper' ('sorites') provides further informal evidence that the range(s) referred to in 9.6 are in some way contextual and theocv-marginal, or at least not conclusively 'definable' (whatever this may mean; cf., once again, 9.3). 9.7. More 'mathematical' counterparts of the sorites, of course, arise immediately when one attempts, quite naturally, to introduce 'the' (?) class of 'infeaslbly large' natural numbers. 9.7.1. For 0 is not 'infeasibly large'. And if n is not 'infeasibly large,' neither is n + 1. There are therefore, no 'infeasibly large' natural numbers (?). 9.8. Correlatively (?), one can consider 'the' (?) class of nonstandard natural numbers. In this case, it is provable (in plausibly relevant metatheories) that the class of 'infinitely large' numbers in a given nonstandard model M of number theory has no least element. 9.8.1. What, then, prevents us from arguing - incorrectly, but exactly as in 9.7.1 - with "infeasibly large" replaced by "nonstandard"? ("For 0 is not 'nonstandard'. And if n is not 'nonstandard', neither is n + 1. There are, therefore, no 'nonstandard' natural numbers". ) It is the theory-reIativity/metatheoretic contextuality of 'nonstandardness' - a 'constructive' application (or so we might now see it) of the 'relativism' of 9.3, in this case to 'intended' notions of the sort introduced in 9.4 and 9.5. 9.9. The basic outlines of such metamathematical states of affairs, of

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course, are familiar to every trained mathematical logician. But their potential implications for the aporiae of classical metaphysics seem to me not yet fully explored. 9.10. Probative, 'dogmatic' arguments in philosophy, for example, have often 'deduced' assertions of great rhetorical power as internal counterparts of begged metatheoretical 'principles' - c o n s i d e r , e.g., the socalled Cartesian circle; or Kant's (presumably synthetic a priori) 'transcendental deduction' of the existence of synthetic a priori judgment(s). 9.11. Reductive, 'skeptical' arguments, by contrast, have often denied object-level counterparts of metatheoretic 'principles,' and tacitly exempted these 'principles' from the precise scrutiny they otherwise enjoyed - consider, for instance, Berkeley's (thoroughly 'abstract' and nonconstructive) critique of the use of "general abstract ideas"; or Hume's application of reductive principles' (such as the 'fork') to everything but rthemselvesT. 9.12. Both of these classical patterns - the reductive 'razor' which does not shave itself, and the ambitious 'proof' which does - reappear often in the writings of twentieth-century 'analytic' and non-'analytic' philosophers. Several of Wittgenstein's more notorious conundrums, for example (both early and late), closely parallel failed attempts by contemporary logicians to resolve or dismiss problems caused by self-referential dilemmas and semantic paradoxes. 9.13. Roughly speaking, I approach these and some other venerable dilemmas of classical metaphysics with several working hypotheses, the irst of which is perhaps most relevant to the arguments offered in this essay. These hypotheses are: 9.13.1. that metalogical arguments do not 'solve' classical problems; but they do often reflect them, and in potentially useful and suggestive analogical ways; 9.13.2. that many of the deepest and most puzzling Classical arguments were ingenious attempts, in effect, to 'solve' informal counterparts of semantic dilemmas and paradoxes, with the aid of begged metatheoretic 'maxims' and 'principles'; 9.13.3. that the metalogical miniatures of these 'maxims' and 'principles' are characteristically 'lirninal', in the sense sketched above, as well as independent or undecidable, in the formal venues in which they are most naturally stated; and, finally, 9.13.3. that informal self-referential problems associated with these

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semantic dilemmas and paradoxes are not trivial or idly logocentric, as some literary theorists have glibly averred, but are regulative of our conscious inquiry and attempts to understand. In that sense, they may even be 'transcendental', or at least ineluctable predicaments of our lives as sentient beings . . . . Wittgenstein's Fliegenglas, in particular, may be a direct descendent of Plato's cave. 10.

IS ' P A T H O L O G Y '

GENERIC?

In this brief final section, I will try to paraphrase ('inductively' generalize?) some of the (apparent) implications of the model-theoretic phenomena we have considered, in the following brief skeptical maxim, which could surely be cast as some sort of metamathematical descendent of the 'tropes' and 'modes' of classical pyrhhonism: 10.1. "In mathematical and mathematical-physical ontology, it may well

be 'simplicity' which is exceptional, and 'pathology' which is generic". In its most straightforward historical interpretation, this tentative "maxim" simply records the observation that patterns and structures initially considered 'normal' or 'desirable' in the history of mathematics have often turned out to be rare, sparse, meager or otherwise 'abnormal', in contexts 'naturally' devised for their description. 10.2. A nominally non-logical example of this (alleged) phenomenon may clarify a bit what I have in mind. A basic line of argument which underlies several of the metamathematical arguments of this essay is the topological Baire category theory (cf. Oxtoby, 1970, pp. 2, 41), according to which (e.g.) the union of nowhere dense subsets of a 'polish' space (X, ~-) (a space which is homeomorphic to a complete separable metric space) has dense complement in X. Among many other things, this theorem implies that 'most' continuous functions f from the real line R into itself are nowhere differentiable. Karl Weierstraf3 offered the first prescription for obtaining such functions, and the 'genericity' of them in an appropriate Banach space of continuous functions is now well-understood. Many nineteenth-century mathematicians found the first examples of such 'anomalous' functions deeply disturbing, however - abhorrent violations, as it were, of regulative ideals of mathematical unicity and simplicity ("Will no one free us from this lamentable plague of functions which have no derivatives?"). 10.3. In retrospect, many mathematicans would now find the responses

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expressed in such remarks rather quaint. Ultimately, there seems to be no particular reason why space-filling curves, Brownian paths, chaotic orbits and the like should be more "lamentable" than their simpler (and now quainter-sounding) ancestors - the 'surds', 'irrational' and 'imaginary' numbers - whose initially puzzling properties remained fossilized in some of the names we still give them. Be that as it may, each of these 'strange' and 'pathological' phenomena has in fact been shown to be generic (comeager) in appropriate topological venues. 10.4. I have already emphasized at some length two obvious ways in which the principal observations in Section 7 above seem to bear out the 'maxim' of 10.1. These are: 10.4.1. that models M of the theory U which are 'inductive'/Robinson A-generic with respect to all sentences in M(A, U) satisfy 'eternal recurrence laws' of the sort considered in 7.1; and 10.4.2. that such models also support G6delian ambiguity about the rinternal7 tenablity of many assertions ~ in M(A, U) which actually hold in M. 10.5. Two other observations in Sections 6 and 7 may also be arguable instances of 10.1, though I have given less emphasis to them so far. These are: 10.5.1. that 'most' models of U in the measure-theoretic sense introduced in 5.3 are not °inductive/Robinson-generic with respect to alt formulas in M(U, A), as remarked in 6.7; and 10.5.2. that the G6delian 'pathology' which holds in all A-generic models in 6.4.2 may not be so 'pathological' after all - or may, at least, reflect 'pathological' lines of epistemic argument which most of us would find useful, even indispensable. 10.6. The implications of 10.5.l seems reasonably clear: measure-theoretically, 'most' elements of S(U) are A-random, and do not, therefore, satisfy the requirements of 'ontological economy' associated above with Robinson A-genericity by 6.6. 10.7. The point of 10.5.2 is more involved than I can efficiently sketch here; I will try to adumbrate it briefly as follows. Suppose we construe the predicate I called Con(U U {~}) above not as consistency of U U {q~}, but as "UU{q~} has an interpretation", or "UU{p} is conceivable", on the (reasonable) assumption that U is strong enough to prove the Henkin Completeness Theorem. 10.8. Then if p E M(U) is existential, for example, one can naturally

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construe ~ A ~ Con(U U {~}) as an object-theoretic nonconstructive existence assertion, a "formal miniature" in U of assertions such as the one Berkeley rejected, in Section 23 of A Treatise Concerning the Principles of Human Knowledge: (cf. Berkeley, 1975, p. 83). 10.8.1. that there are "trees, for instance, in a park, or books in a closet,

and nobody by to" 'conceive' them. I have developed some implications of these ideas as partial formal miniatures of Berkeleyan metaphysics in another essay. (Another aspect of this formal reconstruction would assimilate some of the properties Berkeley associated with 'perception' to definability, and interpret the semantic paradoxes of Richard and Berry as formal miniatures of Berkeley's "master argument".) 10.9. In a wider and admittedly somewhat vaguer sense, finally, the "'Pathology' is generic"-maxim 10.1 above may also suggest another, rather oddly skeptical and processive variant of Lovejoy's plenitudeprinciple. By this I mean the following: 10.9.1. Historically, 'strange' new additions to traditional mathematical ontology have often arisen as second-intentional 'physical' objects. Striking examples of this phenomenon in this century, for example, have been provided by the theory of Schwartz distributions, and by von Neumann's synthesis of quantum mechanics and the theory of selfadjoint operators in Hilbert space. In each case, the 'physical' objects which served as models for 'new' mathematical creations initially seemed much more complex and 'paradoxical' than simpler or more mechanistic 'world-pictures' which preceded them (and which themselves may have scandalized a good many intellectual bourgeois in their turn, a few centuries earlier). 10.10. Ontological enlargements of 'mathematics' along such lines also suggest: 10.10.1. that contemporary 'pythagoreanism', and the process of "Mathematisierung der Natur" Husserl once decried in his Krisis der Europiiischen Wissenschaften (cf. Hussert, 1954, Sections 8 and 9) are not fixed world-views, but aspects of an interactive and processive research program, whose further ramifications we cannot predict; and 10.10.2. that the structure(s) which interpret(s) our theories about 'the world' may continue to evolve in ontologically richer and more complex ways than we initially lead ourselves to expect. 10.11. If so, a mathematical version of one of Einstein's wry remarks that the most incomprehensible thing about 'the world' is that it is

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(mathematically) comprehensible - may be a subtle sort of complex

'synthetic a priori judgment'. 10.11.1. By this I mean that 'the' ontology of 'mathematics' may be, by definition, whatever, we (must?) devise to formulate our successive (over)simplifications of 'what there is'. And if this were so, in turn, Einstein's remark would turn out to be even more accurate than he knew: for it would then be regulative, in some Kantian sense, both of 'mathematics' and of 'the world'. 10.12. A final consequence of 10.1's (tentative) 'maxim' might be that twentieth-century logicians, metaphysicians and philosophers of language who dismiss 'pathological' semantic interpretations for 'our' theories (as Einstein sometimes seemed to do in his later role as an opponent of quantum 'incompleteness', and Wittgenstein clearly did with great rhetorical force in Uber Gewissheit), may be appealing to tacit ontological background assumptions as simplistic and misguided as expectations of past mathematicians that 'all' numbers should be 'real', or 'all' functions smooth. 10.13. 'Most' interpretations for our modest physical (and metaphysical) theories, in other words, might also (and in ways Hamlet never intended) be undreamt of in our philosophies.

REFERENCES Barwise, K. and S. Feferman: 1985, Model-theoretic Logics, Springer, Berlin, Barwise, K. and A. Robinson: 1970, 'Completing Theories by Forcing', Annals of Mathematical Logic 2(2), 119-42. Bell, J. and M. Machover: 1977, A Course in Mathematical Logic, North Holland, Amsterdam. Belt, J. and A. Slomson, A.: 1969, Models and Ultraproducts, North Holland, Amsterdam. Berkeley, G.: 1975, Philosophical Works, M. Ayers (ed.), Dent, London. BooMs, G.: 1989, 'A New Proof of the G6del Incompleteness Theorem', Notices of the American Mathematical Society 36, 388-90. Boos, W.: 1983, 'Limits of Inquiry', Erkenntnis 20, 157-94. Bowen, K.: 1979, Model Theory for Modal Logic, D. Reidel, Dordrecht. Chaitin, G.: 1988, Algorithmic Information Theory, Cambridge University Press, Cambridge. Hirschfetd, J. and W. Wheeler: 1975, Forcing, Arithmetic, Division Rings, Springer, Berlin. Hume, D.: 1976, Dialogues Concerning Natural Religion, J. Price (ed.), Clarendon, Oxford.

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Hume, D.: 1975, Enquiries Concerning Human Understanding and Concerning the Principles of Morals, P. Nidditch (ed.), Clarendon, Oxford. Hume, D.: 1978, A Treatise of Human Nature, P. Nidditeh (ed.), Clarendon, Oxford. Husserl, E.: 1954, Die Krisis der Europiiischen Wissenschaften und die Transzendentale Phiinomenologie, Nijhoff, Haag. Jech, T.: 1954, Set Theory, Academic Press, New York. Keisler, H.: 1971, Model Theory for lnfinitary Logic, North Holland, Amsterdam. Keisler, H.: 1973, 'Forcing and the Omitting Types Theorem', in M. Morley (ed.), Studies in Model Theory, Mathematical Association of America, Buffalo, pp. 96-133. Kunen, K.: 1984, 'Random and Cohen Reals', in K. Kunen and J. Vaughan (eds.), Handbook of Set-Theoretic Topology, North Holland, Amsterdam, pp. 887-911. Lee, V. and M. Nadel: 1975, 'On the Number of Generic Models', Fundamenta Mathematicae 90, 105-14. Lee, V. and M. Nadel: 1977, 'Remarks on Generic Models', Fundamenta Mathematicae 95, 73-84. Losee, J.: 1980, A Historical Introduction to the Philosophy of Science, Oxford University Press, Oxford. Mill, J.: 1974, A System of Logic Ratiocinative and Inductive, Toronto University Press, Toronto. Monk, J.: 1976, Mathematical Logic, Springer, Berlin. Oxtoby, J.: 1970, Measure and Category, Springer, Berlin. Putnam, H.: 1980, Reason, Truth and History, Cambridge University Press, Cambridge. Rasiowa, H. and R. Sikorski: 1970, The Mathematics of Metamathematics, third edition, Polish Scientific, Warszawa. Rucker, R.: 1987, Mind Tools, Houghton Mifflin, Boston, Shoenfield, J.: 1967, Mathematical Logic, Addison/Wesley, Reading. Smorynski, C.: 1977, 'The Incompleteness Theorems', in J. Barwise (ed.), Handbook of Mathematical Logic, North Holland, Amsterdam, pp. 821-65. van Dalen, D.: 1983, Logic and Structure, second edition, second corrected printing, Springer, Berlin. Vesley, R.: 1989, Letter, Notices of the American Mathematical Society 36, i352. HR 1 Box 568A Greenwood Lake, NY 10925 U.S.A.