[FOM] Thoughts on CH 2

[Harvey Friedman]

Concerning "there is a countably additive non atomic probability measure on
all sets of reals".

It has the advantage for set theorists in that it proves the existence of
an inner model of ZFC + "there exists a measurable cardinal".

It has the disadvantage for mathematicians in that the probability measure
cannot be translation invariant.

As we now know, mathematicians are content to leave the usual independent
set theoretic questions open - and they would rather leave them open than
consider new axioms. They have decided that for what they really care
about, they do not need to consider these set theoretic problems - and they
generally recognize that they are different. It would be interesting to
document just how mathematicians view set theoretic problems as different
than what they do. E.g., do they regard the difference as fundamental (as I
do), or "merely" sociological?

Harvey




On Sun, Aug 17, 2014 at 2:22 AM, Arnon Avron <aa at tau.ac.il> wrote:

>
> Quoting joeshipman at aol.com:
>
> "No detailed structure theory" is such a cop-out.
>
> They simply don't like the axiom and so give a vague standard that is
> neither described precisely, nor demonstrated for alternative axioms.
>
> Several times on this forum, I have challenged set theorists to explain
> what is wrong with the real-valued-measure axiom and I have never gotten
> a response. From the RVM axiom, all kinds of things can be proven about
> sets of reals, in fact very little of interest is left undecided by this
> axiom. It also has the consistency strength of a large cardinal axiom
> (measurable cardinal). What's not to like?"
>
>
> I am not an official set theorist, so I suppose my opinion does
> not count. Still I would like to point out two small problems that
> I see with this axiom:
>
> A) It is no more definite mathematical proposition than CH (personally
>    I find it totally meaningless).
>
> B) Even if it is meaningful, I see no convincing *mathematical* argument
>    for its truth.
>
>      Concerning the second point, I know that I am out-dated and
>    old-fashioned, but for me the truth of mathematical assertion is not
>    determined by voting or rating, but only by a full, rigorous
>    mathematical proof.
>
> Arnon Avron
>
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