[FOM] Re: Projective Determinacy and other topics

[Dmytro Taranovsky]

Harvey Friedman wrote,
> Alternatively, where is your proof that there are no new intuitively clear
> and evident axioms to be found?

I actually believe that the most important set theoretical
independences will be solved by intuitively clear and evident axioms.  
It is just the new axioms will be less evident than 2+2=4 and they,
like projective determinacy, will become completely evident only after
careful study.

>>  One can get only so far with simple existence axioms;
> Do you have a proof of this?

No, in fact it is interesting just how simply one can express set
theoretical independences.  Your work shows that there is a five
quantifier sentence that expresses the existence of a subtle cardinal: 
There is a subtle cardinal iff every sufficiently large transitive set
contains x and y such that x is a proper subset of y and x is neither 
{} nor {{}}.  I take this as evidence that subtle cardinals exist.


>Neither PD or V = L answer all consistency statements, although PD
>answers more of them than ZFC + V = L.

V=L does not answer any consistency statements.

>We don't know that 0# exists. Creates more trouble than it's worth??
What kind of trouble does 0# create?  Its existence resolves important
concrete questions and the inner model L[0#] is as canonical as L.

>Also, ZFC + V = L does so much already.
V=L does not answer whether inaccessible cardinals exist.


All uses of V=L must be explicitly noted in the statements of results. 
It is inappropriate to use the phrase "all real numbers" to mean all
constructible real numbers, absent unambiguous statement that
the nonstandard notation is used.


Sincerely,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm