FOM: Set-theoretic hypotheses

Karlis Podnieks podnieks at
Mon Feb 16 04:26:02 EST 1998

I am preparing (in HTML format) the English translation of my
book "Around the Goedel's theorem" (the 2nd edition was
published 1992 in Russian). See the current state of this work
on my home page. May I ask two questions?

All the so called "powerful" set-theoretic hypotheses H have -
to my restricted knowledge - the following property:
PA proves: Con(ZFC) -> Con(ZFC+ not-H),
but Con(ZFC) -> Con(ZFC+H) cannot be proved (sometimes, even in
ZFC+H). Or, the same property with ZF instead of ZFC.

Just two examples:

1. AC - the axiom of choice, AD - the axiom of determinateness.
Since  AC implies not-AD:
PA proves: Con(ZF) -> Con(ZFC) -> Con(ZF+ not-AD).

My 1st question: what is known about provability or
unprovability of Con(ZF) -> Con(ZF+AD)?

2. I - "there exists an inaccessible cardinal".
PA proves: Con(ZFC) -> Con(ZFC+ not-I).
But since in ZFC+I we can build models of ZFC,
ZFC+I proves: Con(ZFC), and hence, Con(ZFC) -> Con(ZFC+I) cannot
be proved even in ZFC+I.

My 2nd question: among the set-theoretic hypotheses considered
as interesting ones, is there some hypothesis H such that
Con(ZFC) -> Con(ZFC+H)
Con(ZFC) -> Con(ZFC+ not-H)
can be proved (in PA, in ZF etc. - as you please)? Or, the same
property with ZF instead of ZFC?

Best wishes,
podnieks at
University of Latvia
Institute of Mathematics and Computer Science

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