[FOM] Criteria for New Axioms

[Dmytro Taranovsky]

On 05/06/2015 05:03 PM, martdowd at aol.com wrote:
> The existence of Mahlo cardinals, which I consider has been strongly
> justified by ``iteration of collecting the universe'', settles independent
> questions of Harvey Friedman's Boolean relation theory.
Correction (or clarification): One needs n-Mahlo cardinals for every 
finite n and not just Mahlo cardinals.  (But arguments for Mahlo 
cardinals naturally lead to n-Mahlo cardinals.)

> >Pick an ordinal kappa_1 with sufficiently strong reflection 
> properties in L
> I'm not sure what this means.

Assuming 0^# (zero sharp), you would pick a Silver indiscernible. If, in 
building up sets, you have not reached 0^# yet, "sufficiently strong 
reflection properties" gives an intuitive criterion of what to pick.  
Notions of reflection properties form a directed system (for example, 
Sigma_2-correct + weakly compact + subtle --> Sigma_2-correct weakly 
compact subtle cardinal).  By taking a limit (for notions, such as kappa 
is regular in L, expressible in L with parameters in L_kappa), one gets 
reflective cardinals for L, and after iterating omega times, 0^#.

> There are reasons to suspect that the Mitchell order is bounded below
> $\kappa^{++}$.  Namely, it might be pathological that there are chains of
> ultrafilters longer than any chain of stationary sets (GCH). This 
> being so,
> supercompact cardinals don't exist.  I don't know about Woodin cardinals.
Woodin cardinals imply existence of cardinals with o(kappa)=kappa^++.  I 
do not see anything pathological about o(kappa)=kappa^++.  Stationary 
sets are subsets of kappa, while ultrafilters are subsets of P(kappa).  
The same subset of kappa can be present in multiple ultrafilters.

> >See my "Reflective Cardinals" paper
> I'll look at this ASAP.
Let me know once you have comments.  The paper should clarify the 
intuition of "sufficiently strong reflection properties".  By doing 
higher-order set theory inside V, it also clarifies arguments (for large 
cardinal axioms) that may otherwise look as handwaving about proper 
classes.  My previous paper "Extending the Language of Set Theory" may 
also be helpful for additional philosophical and less technical background.

Sincerely,
Dmytro Taranovsky