[FOM] Criteria for New Axioms

[martdowd at aol.com]

In reply to Dmytro Taranovsky

>I am not aware of significant consequences of inaccessible cardinals
>(when added to ZFC)

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.

>Pick an ordinal kappa_1 with sufficiently strong reflection properties in L

I'm not sure what this means.

>There are intuitive arguments for strong large cardinal axioms, but they
>get progressively weaker as the axioms are strengthened.                       

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.

>Arguments based on "collecting the universe", which are most clear
>using reflective cardinals, lead to indescribable cardinals.                   

You seem to mean something different by "collecting the universe" than I do.
By my notion, it's a topic of current research whether weakly compact
cardinals exist.

>See my "Reflective Cardinals" paper

I'll look at this ASAP.

Regards,
Martin Dowd

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