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*To*: fom@math.psu.edu*Subject*: FOM: GCH for some cardinal nos.*From*: Jan Mycielski <jmyciel@euclid.Colorado.EDU>*Date*: Fri, 3 Dec 1999 13:13:10 -0700 (MST)*Sender*: owner-fom@math.psu.edu

RESEARCH ANNOUNCEMENT GCH FOR FINITE INDICES FOLLOWS FROM SOME NATURAL AXIOMS Jan Mycielski I will state here three new set theoretic axioms: LAD (which is an axiom of determinacy for long ordinal definable games), DE (which is an axiom asserting the definability of certain ordinals in certain models), and DI (which is an axiom asserting the distinguishability of certain ordinals in certain models). And I will try to explain why I think that those axioms are natural. A theorem motivating those axioms is the following. THEOREM. The theory ZFC + LAD + (DE or DI) yields: If gamma = 0 or if aleph_gamma is a strongly inaccessible cardinal, then 2^aleph_(gamma + n) = aleph_(gamma + n + 1), for n < omega. The proof will appear elsewhere. (I have submitted to the JSL a paper containing this proof.) In order to state LAD we need the following concepts. Let alpha be an ordinal number and X be a set of sequences of 0,s and 1's of length alpha. We consider a game of perfect information of length alpha in which two players I and II choose alternatively the consecutive terms of such a sequence, I makes the choices at all limit steps and all even steps and II makes the choices at all odd steps. If the resulting sequence is in X then I wins; if it is not in X then II wins. LAD. For any alpha and X as above, if X is ordinal definable then one of the players has a winning strategy. {DISCUSSION. LAD is based on the same idea as AD (the Axionm of Determinacy). Namely, it is true for alpha < omega, and we do not know any definable X for which LAD would fail. Since, assuming the existence of appropriate large cardinals yields AD in L[R], perhaps LAD for all X in L also follows from some large cardinal axiom (which I will call LC). Then it may be also the case that the theory ZFC + (L = OD) + LC is consistent. Then LAD would be a theorem of this theory and so it would be strongly justified.} DE. For every cardinal number alpha and every ordinal beta of cardinality alpha there exists an ordinal gamma such that each ordinal less than beta is definable in the model V_gamma by a unary formula with ordinal parameters less than alpha. {DISCUSSION. If M is a (well-founded) model of the theory ZF, then there exists a (well-founded) model N elementarily equivalent to M such that all ordinal numbers of N are definable in N by unary formulas. (This is a theorem of J. Paris. It follows easily from the Omitting Types Theorem.) Hence, if we get rid of Platonic prejudices and we obey Ockham's principle of economy, we should accept in metamathematics that each ordinal number is definable in the language of ZF. How much of that can be known to the model N? I claim that at least as much as is expressed in DE. (An axiom still stronger than DE is proposed in my paper containing the proof of the THEOREM.) Thus, it appears that, if the theory ZFC + LAD is consistent, then ZFC + LAD + DE should be also consistent, and DE is well motivated. But the following criticism of DE was raised by A. Blass and R. Laver. Since an appropriate large cardinal axiom implies that AD holds in L[R], it should be the case (by analogy), that AD holds in OD[R]. And the latter is inconsistent with DE. I feel the this analogy should not be followed. Indeed, the statement "AD holds in OD[R]" implies that the language of ZF augmented with ordinal parameters as in DE (or with real parameters), is weak. Such a weakness suggests a kind of independence between the powerset operation and the structure of ordinal numbers. This weakness points toward the independence of CH. Thus "AD holds in OD[R]" does not appear to be an interesting axiom.} DI. For every cardinal alpha, there exists an ordinal beta, such that for every pair of distinct ordinals gamma and delta both of cardinality less that 2^alpha, gamma is distinguishable from delta in the model V_beta by a unary formula with ordinal parameters less than alpha. {DISCUSSION. The discussion is similar to that od DE. Once again DI assumes that certain models N (of Paris) know certain interesting things about themselwes, and we apply also the additional idea (of Leibniz) that distinct objects must have distinct properties.} OPEN PROBLEM. Evaluate 2^aleph_omega (using some reasonable axioms)! GENERAL REMARK ABOUT THE UNDERLYING PHILOSOPHY. Although the author rejects Platonism (see above) he does not want to be called a formalist. Indeed "formalism" is a misnomer with a pejorative significance attached (apparently by Brouwer?) to some ideas of Hilbert and Poincare. But, the latter were plain rationalists believing that mathematics is a human construction and not a description of an ideal world independent of humanity. A construction which is physical (electrochemical processes in brains, computer computations, and notes on paper) and is as real as other physical objects made by people and machines. Thus, in a real enough sense, mathematicians are no more formalists than engineers, architects, painters or sculptors. Hence it is misleading and inviting spurious discussions to use the name formalism for that philosophy of Hilbert and Poincare (which is my basis). Since many years, our growing knowledge (physiology and anatomy of the brain, mathematical logic, the ideas of universal Turing machines and learning machines, and dynamic systems theory) shows in a stronger and stronger way that this philosophy suffices to explain the phenomena of human intelligence and of mathematics. Since it is also the most economic theory (one which assumes the least), it is the only one which reason can accept today. So its proper name is not formalism but rationalism. This work develops some ideas in my paper in JSL 60 (1995) ,191 - 198, and a paper with weaker axioms which imply CH (to appear).

**Follow-Ups**:**Jan Mycielski**- FOM: Re: GCH for some cardinal nos.**Vaughan Pratt**- Re: FOM: GCH for some cardinal nos.**Jan Mycielski**- FOM: Re: GCH for some cardinal nos.

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