[FOM] Re: Choice of new axioms 1
Dmytro Taranovsky
dmytro at MIT.EDU
Sat Feb 11 14:52:37 EST 2006
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.
* 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.
* Harvey Friedman's arithmetical propositions are almost all decidable
from existence of zero sharp.
* Zero sharp even allows us to set up constructible higher order set
theory, making sense of constructs like the category of all groups. (An
indiscernible is like Ord, see "Extending the Language of Set Theory"
for details.)
* The incompleteness is resolved consistently and correctly.
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.
* If the incompleteness is unimportant, then there is no need for a
convention to resolve it.
* If the incompleteness is important, then it is important to resolve
it correctly, or at least reasonably, which rules out V=L.
I do not object to the study of L. The study is interesting on its own,
generalizes to higher inner models, and most importantly, is a source of
ideas and understanding. However, for any important result in L, one
should ask, "What assumptions does it use? Is it provable in ZFC? If
not, perhaps in ZFC + CH?"
> 18. Of course, PD doesn't even pretend to look like an axiom or like a
> convention. It is something that gets "justified" according to a rather
> elaborate story that starts off with taking complicated sets of reals and
> even arbitrary sets of reals as principal objects of mathematical study.
> This goes against the move towards concreteness in mathematics.
I have some results, which I am currently verifying, that strongly favor
accepting projective determinacy as an axiom. I will discuss projective
determinacy later.
I think that V=L is unacceptable even as a convention for general
mathematics.
Dmytro Taranovsky
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