On the Gödel-Friedman Program (for the Conference in Honor of Harvey Friedman) Geoffrey Hellman

Transcription

1 On the Gödel-Friedman Program (for the Conference in Honor of Harvey Friedman) Geoffrey Hellman May, 2009

2 1 Description of Gaps: the first of two 1. How bridge gap between consistency statements pertaining to large-cardinal axioms and justifying those axioms themselves? 2. How bridge gap between literal truth of strong settheoretic axioms as standardly formulated and truth of more general (and strictly weaker) formulations not committed to a fixed universe of sets as abstract particulars? (We hope to do better than the paleontologist who, according to creationists, fills a gap only by creating two new ones!) Turning to 1, consider as a warm-up example, the case of Friedman s finitiization of Kruskal s Theorem.

3 Define a finite tree to be (infimum) embeddable into another 0 just in case there exists a 1-1 mapping : 0 such that for any x, y, ( ) = ( ) ( ) Kruskal s theorem can be stated as: There is no infinite set of pairwise non-embeddable finite trees. Equivalently, Kruskal s Theorem: For any sequence of finite trees, T, there exist i, j with i j such that is embeddable into This is not a finitary statement as it quantifies over inifinite sequences. However, Friedman discovered a finite form of Kruskal s Theorem, which is not predicatively provable. Friedman s Finite Form of Kruskal s Theorem ( FFFKT") : For any positive integer c there exists a positive integer n = n so large that if 1 2 is any finite sequence of finite trees with for all i

4 then there exist indices i,j such that i j and embeddable into Friedman proves (over weak base theory): 1-Con(ATR 0 ) FFFKT. Thus, FFFKT requires ordinal levels Γ 0 and beyond, exceeding a widely recognized bound on predicative mathematics. Indirectly, this implicates the uncountable: FFFKT transcends mathematics restricted to countable objects. A convinced "predicativist" can reply: "1-Con(ATR 0 )" is only a statement at the level of the natural numbers, so, strictly speaking, this provides no case for "indispensability of the uncountable". How can this kind of objection be answered? Another example bearing on large cardinals is the following from Friedman s more recent Boolean Relation Theory:

5 Let denote the union of A and B but implying that they are disjoint. A multivariate function f from N into N has expansive linear growth iff there are c, d, e 1 such that for all in dom(f ), if then c ( ) d, where is the maximum term of the tuple. For N we write for the set of all values of at arguments from. Now consider the following Proposition: For all multivariate from N into N of expansive linear growth, there exist infinite sets contained in N such that Now let MAH+ denote the theory ZFC + for all n there exists an n-mahlo cardinal. Friedman proves the following

6 Theorem: The Proposition is provable in MAH+ but not in ZFC (assuming ZFC is consistent). In fact, the Proposition is not even provable in ZFC + the scheme, there exists an n-mahlo cardinal, each n, as it proves the 1-consistency of the latter (over the subsystem of second-order arithmetic known as ACA ( arithmetic comprehension axiom ). Remarkably, if the pattern of letters, is altered at all in the statement of the Proposition, it becomes provable in RCA 0, the weakest of the main subsystems of analysis studied in reverse mathematics. Since the first sentence of the Theorem states only a sufficiency for proving the Proposition, again what is shown indispensable is not the statement of existence of the large cardinals per se but rather the 1-consistency of the indicated scheme for them, a statement at the level of natural numbers. This is symptomatic of a general method, made explicit by Feferman, of "defanging Gödel s doctrine" (the claim

7 that indefinite extendability of transfinite types constitutes "the true reason" for the incompleteness phenomena): wheneveraproofofasentence from a large cardinal hypothesis, LC, is given along with a demonstration that is unprovable from the original theory (if it is consistent), it suffices to replace LC with the proof-theoretic reflection principle, (R) prov ( ) for each sentence of the original language. (In Friedman s program, is typically of low set-theoretic rank.) This expresses confidence in the consistency of adding LC as an axiom, but R itself is not committed to anything of higher type than what is quantified over in itself. The challenge is to bridge the gap between R and LC. Remark: A similar situation arises in the link between large cardinals and descriptive set theory. Martin and Steele (in 1985) proved that the existence of infinitely many Woodin cardinals implies Projective Determinacy (PD), the statement that all projective sets of reals are

8 determined. (This in turn yields a detailed structure theory of projective sets.) Moreover, work of Woodin, Martin, and Steele shows that this large cardinal hypothesis is minimal, in the sense that PD and the following statement are equivalent: For each Prov +"Therearen Woodin cards" ( ) for every sentence,, in the language of second-order arithmetic. As Steele puts it, "PD is the instrumentalist trace of Woodin cardinals in the language of second-order arithmetic.

9 2 Confirmation Short of Proof Gödel s prescient remarks: (quote from his "What Is Cantor s Continuum Problem?"). Key ingredients: fruitfulness, unification, and variety of evidence. Claim: The Bayesian apparatus can reasonably be applied. Even absent precise estimates of prior probabilities (as degress of rational belief), certain inequalities display the epistemic value of these ingredients. ( ) ( ( & ) Bayes Theorem: ( ( & ))=) ( ),where ( ) denotes the (conditional) probability of given, H is the target hypothesis, is given background knowledge, and is a body of evidence. (This is an elementary result in the standard probability calculus, independent of any interpretation of "probability".) In an epistemic, Bayesian application, it instructs

10 how to "update" degrees of rational belief when evidence is gained. (The first term on the right is the "prior" of the hypothesis, the second is called the "likelihood" of the evidence, and the denominator term is the "prior" of the evidence.) Typically, the hypothesis + background logically implies the evidence statement, so the second term = 1 and drops out. In assessing the denominator, we are nottousetheinformation(evenifwehaveit)thateactually has been found or established to be true, for then no known evidence could boost the posterior of H beyond its prior, i.e. there would be no incremental confirmation. This is known as "the problem of old evidence ". Thus, typically, the denominator is taken as strictly 1. The value of coming to know E is that it "licenses updating", i.e. setting ( ) = ( & ) As Gödel intended the term "fruitful", there is no sharp separation from the other properties indicated, "unifying", "[probabilifying] variety of evidence". At a minimum, we can take it as indicating a relative abundance of information in the role of. It could more explicitly be represented as a long conjunction of particular

11 statements, 1 & 2 & & Presumably, as grows, the denominator in Bayes Theorem shrinks, correspondingly boosting the posterior prob. of Handling "unification" in Bayesian terms is considerably more challenging, and we refer the reader to recent work of Wayne Myrvold. The Bayesian story regarding "variety of evidence" is short and sweet: the more varied the the smaller should be any positive difference ( ( & )) ( ) theformerofwhichappearinthedenominator of BT when written out for ;i.e. the less value "earlier" pieces of evidence have for predicting a new piece, at least in comparison with cases in which all the pieces are "of a piece", so to speak. (Here we re letting time do the ordering, but this is inessential.) Indeed, as Earman suggests, we can even let this behavior be our standard for "degree of variety".

12 Our main points are these: 1. These aspects of (thoroughly naturalistic) scientific method make sense, not only in empirical contexts, but also in purely mathematical ones. (Note how Gödel anticipates Quine in this regard.) 2. There isn t even a hint of ESP, i.e. all this stands independently of Gödel s controversial remarks (in the same essay) about mathematical faculties akin to sense perception. 3. Note an important difference between Gödel s vision and the situation of Friedman s and Woodin-Martin- Steele s results: G emphasized "explaining" low-level mathematical facts by simplifying and unifying their proofs without assumptions on LC, whereas F and WMS prove new theorems not derivable without new assumptions (e.g. of consistency or proof-theoretic

13 reflection). One reason for G s emphasis emerges on Bayesian analysis above: updating beliefs depends on invoking the relevant evidence; if you don t have independent access to it, you can t invoke it. 4. Thus, a major challenge for the F and WMS program(s) in addition to that of accumulating more andvariedresultsunified by the new axioms is to find independent justification for believing those results. Remark: All is not lost. Experience with axiom systems can support rational credibility of their consistency. Sometimes, a constructive or quasiconstructive model-theoretic argument can also be given. (For instance, the standard intuitionistic argument that the Dedekind-Peano axioms of arithmentic are true of our conception of "finitely constructed numbers", combined with Gödel s double-negation translation, does support the claim that classical DPA 2 is consistent.) More needs to be done on this topic.

14 3 The Second Gap: Alternatives to a fixed background universe of sets. Of special concern to me (unsurprisingly) is the effect of introducing modality into a (an otherwise) nominalistic framework for mathematics, employing the powerful combination of mereology with logic of plurals (achieving the effect of full, polyadic second-order logic). Main features relevant here: 1. As with the number systems, one recovers the expressive power of 2d-order axiom systems for set theory, esp. ZFC 2, along with Zermelo s [1930] results: (a) Given any two models with bijectively equivalent urelement bases, one is isomorphic to an initial segment of the other.

15 (b) The height of any such model is a strongly inaccessible cardinal. 2. All quantification over sets is relativized to a model. Quantification over models is achieved via plural quantification, e.g. "There are (better: could be) some things related thus-and-so..." Indeed, one replaces systematically by quantification w/r/to a 2-pl. relation variable. Moreover, set in its usual absolute sense is eliminated entirely, in favor of quantification over (first-order) items of arbitrary or stipulated models. 3. Proper classes are avoided in favor of unrestricted extendability: For any ZFC 2 model there might be, it might have a proper extension. (Note our reliance on modal operators; there is no literal quantification over "worlds".) Set-comprehension occurs only inside a model; necessarily, there is no collection of "all possible models or ordinals of models, etc." There are no ultimate infinities.

16 4. Small large cardinals, including the Mahlo s, are motivated, invoking the extendability principle, much as inaccessibles are. (This replaces top-down reflection on "the universe".) Question: What happens w/r/to confirmation of large cardinal hypotheses when "mathematical existence" is construed as "logico-mathematical possibility"? In particular, how can mere possibility explain (indeed, serve as "best explanation" of) relevant consistency or reflection phenomena? By way of an answer: Unlike causal explanation, actuality really plays no role in purely mathematical explanation due to the (S-5) necessity involved. If sentences Γ are satisfied in a structure, necessarily no contradiction is implied by Γ Itmakesnodifference if we say, "could be

17 satisfied". The modal-structural construal of mathematical existence, under translation, respects the inference from existence to necessary existence: e.g. we have [Φ( )] [Φ( )] (S-5 axiom). And the (platonist) inference from possible existence to actuality is also respected (under translation): [Φ( )] [Φ( )] (S-4 axiom) Also, natural scientific explanations not uncommonly appeal to possibilities, especially when statistics are involved. Thus, thermodynamical behavior of complex systems is regularly explained by appeal to relative "sizes" of collections of exact microstates compatible with given macroscopic constraints, etc.; observed frequencies may be explained by citing rules of,say,fermi-dirac(asopposedtobose-einstein) statistics in which only certain configurations of a

18 system are "allowable" or physically possible. In general, we have explanations of the nature and behavior of quantum systems via their pure states, and these are irreducibly dispositional, specifying physical probabilities of outcomes for any possible experiment for an observable pertaining to the system. Many such explanations are also not really "causal"; rather they invoke complex properties of systems pertaining to "potentialities", giving information about the kind of system we re dealing with. (We may call this "explanation? " as contrasted with "explanation? ", which has dominated discusions in philosophy of science.) We conclude: A modal-structural approach to large cardinals is not worse off than the standard, face-value platonist reading of set theory.