[FOM] Inconsistency of Inaccessibility
Monroe Eskew
meskew at math.uci.edu
Tue Oct 25 19:21:02 EDT 2011
Naive set theory formalized by Frege is inconsistent, but intuitively
appealing. Without being aware of Russell's paradox a mathematician
might be inclined to say it is a correct theory of sets that captures
the principles implicit in his own use of sets. Isn't this analogous
to the situation with inaccessible cardinals?
My point is that appeals to intuition are not, in all cases,
sufficient arguments for consistency. When a formal inconsistency is
claimed, it must be confronted head-on. Intuitions are no match for
proof.
Disclaimer: I am not endorsing Kiselev's argument.
On Tue, Oct 25, 2011 at 6:09 AM, <MartDowd at aol.com> wrote:
>
> It adds to the evidence. Statements which are independent of ZFC can only
> be accepted by agreement that they are true. At this point, such agreement
> is being argued for, but is by no means inevitable. A candidate for
> acceptance should already enjoy substantial likelihood that it is
> consistent.
>
> Existence of inaccessible cardinals seems consistent. They have appeared in
> logic, in settings such as Grothendieck universes, monster sets, etc. Most
> mathematicians would probably agree that consistency holds. Claiming that
> truth holds is a bolder leap, although various mathematicians have been
> inclined to make it. Actually, it seems that likelihood of consistency is a
> prerequisite to possibility of truth.
>
> The existence of measurable cardinals provides another example. Many set
> theorists are inclined to accept the truth of this (I think the opposite
> view should be well-considered, though). So far, attempts to prove
> inconsistency have failed; but to me at least, there are few if any
> arguments for consistency, other than that inconsistency has not been
> proved. Truth, in fact, for inaccessible cardinals, on the other hand, can
> be given various arguments.
>
> I might add that some of the arguments in my paper are mathematical, for
> example that certain axioms imply that Ord is Mahlo. This could be seen as
> "a posteriori" evidence, as so highly favored by the advocates of measurable
> cardinal existence.
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