[FOM] Intrinsic justifications for large-cardinal axioms

Rupert McCallum rupertmccallum at yahoo.com
Sat Dec 9 04:33:18 EST 2017

I have recently uploaded the following pre-print to arxiv.org.


Note in particular the first two paragraphs of p. 9 and the second and third paragraphs of p. 14.

Thus we end up with two different possible accounts of what might be "intrinsically justified", one which extends up to 
an \omega-reflective cardinal, which is intermediate in strength between a totally ineffable cardinal and an 
\omega-Erd\H{o}s cardinal, and one which extends up to n-extendible cardinals for any positive integer n. The former 
notion of intrinsic justification can be plausibly be taken up to the level of remarkable cardinals or even virtually 
n-huge* cardinals, (see Victoria Gitman's recent paper about virtual large cardinals for the definition of this notion), 
whereas the latter notion can plausibly be taken up to the level of supercompact cardinals or even n-huge cardinals.

I do not know whether to say that only the first, weaker notion of intrinsic justification, which vindicates 
Peter Koellner's suggestion that intrinsic justifications do not give us as much consistency strength as an 
\omega-Erd\H{o}s cardinal, should be accepted, or whether the second stronger notion should be accepted, which 
plausibly justifies everything short of an I_3 cardinal, or whether I should say that this is a bifurcation in the 
very concept of intrinsic justification. I would welcome any thoughts about this.
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