FOM: GCH for some cardinal nos.
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Fri Dec 3 15:13:10 EST 1999
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).
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