[FOM] Inconsistency of Inaccessibility
Joe Shipman
JoeShipman at aol.com
Tue Oct 25 19:06:07 EDT 2011
I disagree about the truth of measurable cardinals bring more easy to argue for than their consistency. The statement that Lebesgue measure can be extended to a countably additive measure on all subsets of R^n has plenty of intuitive plausibility but implies only the consistency of measurable cardinals, and any universe in which measurable cardinals exist can be cut down to one in which they don't exist (but are consistent) but in which all the same statements are true up to a very high level of statement complexity (all the arithmetical sentences, and all the analytical sentences up to something like sigma^1_4 if I recall correctly).
What is the logically and ontologically simplest meaningful statement whose truth depends on the existence of measurable cardinals and not simply on the arithmetical consequences of measurable cardinals such as their consistency?
-- JS
Sent from my iPhone
On Oct 25, 2011, at 9:09 AM, MartDowd at aol.com wrote:
>
>
> In a message dated 10/24/2011 5:01:50 P.M. Pacific Daylight Time, meskew at math.uci.edu writes:
> Are you claiming that giving a convincing philosophical argument for
> the addition of a mathematical axiom makes it likely that the axiom is
> consistent?
> 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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