Tag Archives: Large cardinal consistency implies truth

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I guess I should respond to your question as well.

On Oct 31, 2014, at 3:30 AM, Sy David Friedman wrote:

My point is that Hugh considers large cardinal existence to be part of set-theoretic truth. Why?

Let me clarify my position, or at least that part of it which concerns my (frankly extreme) skepticism about your anti-large cardinal principles.

(I am assuming LC axioms persists under small forcing and that is all in the discussion below)

Suppose there is a proper class of Woodin cardinals. Suppose M is a ctm and M has an iteration strategy \mathcal I at its least Woodin cardinal such that \mathcal I is in L(A,\mathbb R) for some univ. Baire set A.

Suppose some LC axiom holds in M above the least Woodin cardinal of M.Then in V, every V_{\alpha} has a vertical extension in which the LC axiom holds above \alpha.

The existence of such an M for the LC axiom is a natural form of consistency of the LC axiom (closely related to the consistency in \Omega-logic).

Thus for any LC axiom (such as extendible etc.), it is compelling (modulo consistency) that every V_{\alpha} has a vertical extension in which LC axiom holds above \alpha.

But then any claim that the LC axiom does not hold in V, is in general an extraordinary claim in need of extraordinary evidence.

The maximality principles you have proposed do not (for me anyway) meet this standard.

Just to be clear. I am not saying that any LC axiom which is consistent in the sense described above, must be true. I do not believe this (there are adhoc LC axioms for which it is false).

I am just saying that the declaration, the LC axiom does not hold in V, in general requires extraordinary evidence, particularly in the case of LC axioms such as the LC axiom: there is an extendible cardinal.