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If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is incompatible with even (apparently) mild large powerset axioms like "$2^\kappa$ always strictly exceeds $\kappa^+.$"

This begs the question:

Is there a "large powerset axiom" $\varphi$ so extreme that $\mathrm{ZFC}+\varphi$ disproves the existence of strongly inaccessible cardinals? Let us also require that $\mathrm{ZFC}+\varphi$ does not prove $\neg \mathrm{Con}(ZFC + \varphi).$

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I think the statement, "There are no inaccessibles" is a large powerset axiom already: it asserts that, for every uncountable regular $\kappa$, there is some $\lambda<\kappa$ such that $2^\lambda\ge\kappa$; and we can make this even stronger by adding "$\mu<\nu\implies 2^\mu<2^\nu$, to demand a $\lambda<\kappa$ with $2^\lambda>\kappa$. In fact, these two axioms together imply that below a regular, we can find cardinals $\lambda$ with $2^\lambda$ arbitrarily large below the $\kappa$th cardinal above $\kappa$. To me, this is a large powerset axiom. –  Noah S May 22 '14 at 13:58
I think you mean to raise the question, rather than to beg it. begthequestion.info –  Joel David Hamkins May 22 '14 at 14:04
"While descriptivists and other such laissez-faire linguists are content to allow the misconception to fall into the vernacular, it cannot be denied that logic and philosophy stand to lose an important conceptual label should the meaning of BTQ become diluted to the point that we must constantly distinguish between the traditional usage and the erroneous "modern" usage. This is why we fight." @JoelDavidHamkins, an interesting statement. I'm almost convinced. –  goblin May 22 '14 at 14:11
Is there a way to construe (something like) the Inner Model Hypothesis as a large powerset axiom? I don't see it off the top of my head, but maybe it's equivalent to something along those lines. –  Noah S May 22 '14 at 14:16

1 Answer 1

up vote 10 down vote accepted

Foreman's maximality principle is as you have requested, though it is not yet known if it is consistent or not.

Foreman's maximality principle: Any non-trivial forcing notion either it adds a real or colapses some cardinals.

It follows from it that:

1) $GCH$ fails everywhere,

2) there are no inaccessible cardinals.

This principle is stated in the following paper:

Foreman, Magidor, Shelah, "$0^\sharp$ and some forcing principles", J. Symbolic Logic, 51 (1986) 39-46.

See also "Questions about $\aleph_1-$closed forcing notions".

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That's really nice! –  Noah S May 22 '14 at 14:15
So it's still open whether or not this is consistent with $\sf ZFC$? –  Asaf Karagila May 22 '14 at 17:35
Also, are there known consistent "limited" versions (e.g. "every proper forcing ..." or some other reasonable class of forcing notions)? –  Asaf Karagila May 22 '14 at 17:54
@AsafKaragila Martin's maximum. –  Andres Caicedo May 22 '14 at 18:10
@Andres: For which class of forcing notions? –  Asaf Karagila May 22 '14 at 20:43

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