FOM: Criteria for new axioms of set theory

Harvey Friedman friedman at
Thu Jan 13 19:26:33 EST 2000

>	                    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.

A special case of your Theorem is that your proposed axioms imply the
continuum hypothesis.

How does this work relate to other proposed axioms that "solve" the
continuum hypothesis. E.g.,

1. The existence of a nontrivial real valued measure on all sets of real
numbers implies the continuum hypothesis is false.

2. Axioms of Woodin that imply 2^aleph_0 = aleph_2.

I am under the impression that there is a new view among many full time
professional set theorists as to appropriate criteria for new axioms of set
theory. I gather that the view is rather subtle to explicate, and Steel
began an explanation for the FOM e-mail list some time ago in his postings
Fri, 19 Dec 1997 10:13, Fri, 30 Jan 1998 10:14. An important buzzword is
generic absoluteness. I am under the impression that axioms like 1 above
and presumably your axioms do not meet these new criteria.

It would be very valuable for FOM subscribers to see this new view of new
axoms for set theory explained in clear terms on the FOM.

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