The question is essentially what is asked in the title. I split it into two parts

(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense of the existence of large cardinals, and to what extend these arguments work (I mean if these arguments work for very large cardinals, say like $I_0$, ..., or they are limited in the hierarchy of large cardinals).

(B) (Arguments against the existence of large cardinals) What are the main philosophical arguments against the existence of large cardinals, and to what extend these arguments work (I mean if these arguments work even for small large cardinals like inaccessible cardinals, or they are essentially against some very large cardinals).

I am mainly interested in the arguments given by those people who have some experiences in set theory (they have important results in set theory).

A somehow related question is Arguments against large cardinals.


The book Frank R. Drake, Set Theory. An Introduction to Large Cardinals lists arguments for the existence of inaccessible, Mahlo, and hierarchy indescribable cardinals. As a consequence, weakly compact cardinals should also exist since they are precisely the $\Pi_{1}^{1}$-indescribable cardinals. The argument set forth by Drake states that the set of all ordinals should have no conceivable bound. Said differently, every property that $V$ satisfies should also hold for some $V_{\alpha}$ since one should not stop with the. For example, the axioms of replacement and the power set axiom allude to the fact that the class of all ordinals $\mathbf{On}$ should be an inaccessible cardinal. There should therefore be some cardinal $\kappa$ which is an inaccessible cardinal. In fact, a cardinal $\kappa$ is inaccessible if and only if $V_{\kappa}\models ZFC^{2}$ where $ZFC^{2}$ is the second order version of the $ZFC$ axioms where the axiom schema of replacement is replaced with a second order sentence. Similar arguments holds for the existence of Mahlo cardinals and indescribable cardinals. However, I am unaware of any similar arguments that hold for any of the large cardinal axioms beyond the indescribable cardinals since for example measurable cardinals seem to have no limit to the amount of indescribability that holds below them. Unless I am simply unfamiliar with a way to generalize this argument, this argument is limited to the lower regions of the large cardinal hierarchy.

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    $\begingroup$ The considerations described here make me fairly confident about the existence of indescribable cardinals. As I mentioned in my answer to the earlier question linked by the OP, that confidence does not extend even to subtle cardinals, because their definition seems to be not merely a matter of being "large" or "resembling the whole universe" but rather another sort of combinatorics for which I don't see a philosophical justification. $\endgroup$ – Andreas Blass Apr 2 '15 at 11:53
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    $\begingroup$ I am fascinated by large cardinals, but I've always looked upon those arguments as fundamentally weak. They remind me of the Ontological argument, attempting to pull the existence of increasingly perfect objects out of thin air. $\endgroup$ – Joel David Hamkins Apr 2 '15 at 12:57

You might want to take a look at the papers of Paul Corazza (of Wholeness Axiom fame--he, of course, was the one to first propose it) found on his homepage. The paper I would suggest one reads is his paper "The Spectrum of Elementary Embeddings j:$V$$\rightarrow$$V$" (found also on his homepage). I leave it to the individual reader to decide whether the Hindu Vedas (his inspiration for the Wholeness Axiom and the belief in the existence of large cardinals below I3--a consequence of assuming the Wholeness Axiom) are a reasonable philosophical basis for one's belief in their existence.


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