[FOM] Re: Choice of new axioms 1

[Harvey Friedman]

On 2/11/06 2:52 PM, "Dmytro Taranovsky" <dmytro at MIT.EDU> wrote:

> Harvey Friedman wrote:
>> 17. Clearly V = L looks like an axiom - and perhaps even more convincing for
>> some, it looks like a *convention* that we will only consider mathematical
>> objects that we can generate.
> 
> If we accept V=L as a convention to avoid incompleteness in V, then
> perhaps we should also accept existence of zero sharp, at least
> instrumentally, to avoid incompleteness in L.

0# is completely different. V = L as a convention simply asks that we
restrict attention to sets that we already know exist on the basis of ZFC -
the present gold standard. As you well know, 0# is not something that can be
proved to exist in ZFC.

As a convention, V = L is guaranteed consistency (relative to ZFC). 0# is
not guaranteed consistency (relative to ZFC).

> *  Many general Sigma-2 statements, including a number of partition
> properties, are undecidable in ZF + V=L.  They are generally resolved
> with the help of zero sharp.

What does Sigma-2 mean, and what examples are you talking about? Also which
of them are of concern for the working mathematician?

> *  Harvey Friedman's arithmetical propositions are almost all decidable
> from existence of zero sharp.

This is only temporarily true. Also, I remind everybody, the "1" at the end
of the subject line means that I have not yet taken into account any
eventual effect of the kind of work I am doing now.
> 
> Anyway, there is not much interest in using V=L as a convention:
> *  Conventions are used to restrict to well-behaved sets, like those
> that are measurable.  This does not work with V=L.

This is a very special kind of problem: expand the axioms of set theory IF
AND WHEN the mathematical community changes it current mind and decides that
it wants to do away with these large number of independent statements about
arbitrary (or logically complicated) sets of reals (subsets of complete
separable metric spaces). And it has a very special kind of solution: L by
convention.

> *  If the incompleteness is unimportant, then there is no need for a
> convention to resolve it.

I wrote about the hypothetical that the incompleteness is deemed important
enough by the mathematical community for arbitrary (or complicated) subsets
of complete separable metric spaces.

> *  If the incompleteness is important, then it is important to resolve
> it correctly, or at least reasonably, which rules out V=L.

They are resolved correctly in ZFC + V = L - as statements about L.

This doesn't rule out V = L as a convention. There is no doubt that V = L
decides these statements correctly as statements about L.

E.g., the statement

"every field element is algebraic"

is false in lots of fields, but true in

"the subfield of algebraic points".
> 
> I think that V=L is unacceptable even as a convention for general
> mathematics.
> 
I see no reason to object to V = L as a convention. I have no doubt that if
the mathematical community were to get into the mood of wanting to rid
itself of the incompleteness regarding arbitrary (and complicated) subsets
of complete separable metric spaces - and this is a big if - then it would
turn to ZFC + V = L.

The situation is more problematic if and when things that I am doing with
Pi01 sentences - and extrapolations that are much more impressive, but not
done yet - sink in. Even here, it is not clear to me what path the
mathematical community will choose. But certainly the situation changes if
only for the reason that

*NO CONVENTION will clean out the Pi01 and Pi00 independence results.*

This is in sharp contrast to the present situation were the "offending"
incompleteness is completely cleaned out by L, and ZFC + V = L.

Harvey Friedman