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It is easy to overlook the fact that the existence of a given large cardinal provides us with a true arithmetical statement that would otherwise be false if the large cardinal notion were not consistent with ZFC (See On statements independent of ZFC + V=L ). The arithmetical statement that I'm referring to here is CON(ZFC + large cardinal notion). This is an example of where we use a fact that is external to the set of Natural numbers, the existence of the large cardinal, to prove a result that's internal to $\mathbb{N}$. If we want a less esoteric statement of looking externally (not using a large cardinal notion) to prove an internal result about the Natural numbers, we can consider Goodstein's Theorem. Goodstein's Theorem states that a certain infinite collection of sequences, almost all of which grow very quickly on some initial segment, eventually descend to 0. The amazing result is that while this fact is not provable in Peano arithmetic, we can give a very simple proof of it in ZFC. In this case, we consider sequences of infinite ordinals to prove a true arithmetical statement representable in PA.

Let me now switch gears a little to say something about large cardinals. Assuming increasingly strong large cardinal axioms opens up the possibility for increasingly complex transitive set models of ZFC and then increasingly complex definable inner models of ZFC. Nevertheless, even if we assume a sufficiently strong large cardinal hypothesis, say the existence of a weakly compact cardinal $\kappa$, the elementary embeddings that arise fix all "small" elements (hereditary size less than $\kappa$ in the domain). I would therefore like to consider nonstandard models of PA that will not be fixed by such embeddings in an effort to make use of the large cardinal assumptions in a meaningful way for revealing internal truth by external examination. In order to avoid asking a rather vague question, let me pose it as follows:

Has anyone considered models of PA of large cardinal size?

As an example of what I have in mind, assume that we have a nonstandard model $M$ of PA of size $\kappa$ containing an unbounded well-order ( See Uncountable nonstandard models of PA ) for $\kappa$ supercompact. Can we use the fact that $\kappa$ is supercompact to reveal any interesting number-theoretic properties true in $M$?

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    $\begingroup$ I'm not sure what results along these lines one could hope to get; by the Lowenheim-Skolem theorem, any first-order phenomenon that can happen in a model of PA happens in some countable model of PA. Moreover, if $L$ is any logic (second-order, infinitary, etc.) such that the class of $L$-sentences in the language of PA is a set, then we can easily show that there is some cardinal $\kappa$ such that any phenomenon describable by $L$ that can happen to a model of PA happens to one of cardinality $\le\kappa$; and such $\kappa$ won't be a large cardinal. So I don't know what we can hope for here. $\endgroup$
    – Noah Schweber
    Dec 20, 2010 at 4:13
  • $\begingroup$ @Noah S: I don't think that the fact that we can move to countable models of a first-order theory prevents us from learning something new. For example, we prove many relative consistency results by assuming the existence of large cardinals. If a transitive set model of ZFC witnesses this large cardinal property, then we can get a countable transitive set model $M$ of ZFC witnessing this large cardinal property. But that doesn't change the fact that the countable ordinal having the large cardinal property in $M$ is still uncountable from the viewpoint of $M$. $\endgroup$
    – Jason
    Dec 20, 2010 at 5:32
  • $\begingroup$ To support Jason, the statement (in the language of ZF) that the model $M$ has size $\kappa$ certainly entails the fact that merely as a set $M$ has size $\kappa$, and with $\kappa$ a large cardinal, the resulting consistency statements constitute facts of arithmetic. On the down side, I don't see the added value of knowing that $M$ models PA. If you have any set of (large) size $\kappa$ and any model of PA, ultrapowers will give you a large model of PA. Thus large sets and large models have the same consistency strength. $\endgroup$
    – David Feldman
    Dec 20, 2010 at 6:12
  • $\begingroup$ @David: I was concerned about the same observation that you made. But I'm thinking that if $\kappa$ is say measurable and you have an ultrapower embedding $j: V \rightarrow N$ with critical point $\kappa$ for some (transitive) inner model $N$ and a nonstandard PA model $M$ of size $\kappa$, then maybe considering $j(M)$ in $N$ can tell you something about $M$. I'm not sure what to expect though since the strength of large cardinal embeddings often comes from closure conditions and $j(M)$ cannot even be countably closed since it has a nonstandard $\omega$. $\endgroup$
    – Jason
    Dec 20, 2010 at 6:36

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See Section 4.2 of Harvey Friedman's book draft for the details of a combinatorial, non-metamathematical application of n-Mahlo-sized extensions of structures of the form $(\omega,<,0,1,+,f,g)$.

In this example, $f$ and $g$ are not allowed to grow fast enough for either to be chosen to be the multiplication function. More generally, I am not aware of any "interesting" number-theoretic applications of large cardinals. That said, perhaps you should ask Friedman about large-cardinal-sized models of PA.

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