FOM: Re: determinate truth values, coherent pragmatism

[Harvey Friedman]

Reply to Shipman 10:48AM 10/5/00:

Friedman wrote:

>4. I also see the possibility that the general math community may be
>more
> comfortable with "there is an atomless probability measure on all
>subsets
> of the unit interval" than with large cardinal axioms, in that this
> involves objects that are far less abnormal than a large cardinal... Of
>course, a byproduct of this
> probability measure is that CH is very badly false. But, when viewed as
>an
> approach to CH, this is seriously at odds with the approach currently
>taken
> by the set theorists... However, from the coherent
> pragmatism that I see in the math community, they may well accept this
>when
> shown how to use this in an easy and coherent and effective way for a
> sufficient body of attractive normal mathematics. This would be where
>the
> set theorists' approach  via some notion of truth and the
>mathematicians
> approach via coherent pragmatism would clash.

Shipman replied:

...it seems to me that the set theorists are also
>using
>"coherent pragmatism" (though they may like to talk in terms of truth).
>I have certainly not seen any satisfactory arguments from set theorists
>why the
>axiom of an atomless measure on the continuum is FALSE; though I have
>seen
>arguments that alternatives to this axiom (such as Martin's axiom) are
>USEFUL, I have not seen any to persuade me that those alternatives are
>"true".

Quoting some further postings on the FOM, Shipman concludes:

>Therefore, I draw the conclusion that Harvey is wrong, and that set
>theorists do NOT differ from "ordinary mathematicians" in their approach
>to new axioms, because if they WERE in fact motivated "via some notion
>of truth" rather than "the mathematicians approach via coherent
>pragmatism", someone would have been able to explain their rejection of
>RVM by discussing reasons it might actually be false.

You are making the hidden assumption that the set theorists regard the
determination of the truth value of

"there exists an atomless probability measure on all sets of reals"

as a reasonably high priority. They don't. They seek to determine the truth
value of sentences of set theory in order of logical complexity as measured
by the Pi-n-m and Sigma-n-m hierarchy of prenex formulas. Here n denotes
the highest order of the quantifiers, and m denotes the number of
quanitifiers of order n. The Pi means that the leading n-th order
quantifier is universal, and the Sigma means that the leading n-th order
quantifier is existential. Here 0-th order means over the natural numbers,
1-th order means over the power set of the natural numbers, etcetera.

Actually, set theorists are very sensitive to an additional consideration.
Specifically, quantification over all subsets of omega-1. This is weaker
than 3-rd order quantifiers but stronger than 2-nd order quantifiers.

RIght now, having been very satisfied with their "determination" of the
truth value of sentences in Pi-1-m, for any m, they consider the
"determination" of the truth value of sentences quantifying over all
subsets of omega_1 as of the highest priority, with 1 or 2 such quantifiers
(and any first or second order quantifiers as well).

This list of priorities is of course at odds with other lists of
priorities. Specifically, probability measures on all sets of reals are
closer to current mathematical practice, and more readily appreciated by
normal mathematicians, than statements formulated in terms of subsets of
omega_1. (Of course both are still of some considerable distance from the
current normal mathematics). Set theorists have their ways of trying to
"determine" truth values in set theory, and for them, they find that their
list of priorities according to these logical hierarchies is the way to do
their research.