[FOM] Harvey on invariant maximality

[Andrew Arana]

These results of Harvey are fantastic & show the stunning interpenetration of higher set theory into elementary mathematics. I look forward to seeing the magical work behind them, for not only are the results breathtaking but so, I am sure, are the methods.

I wanted to remark on the "naturalness" issue. Harvey talks about the "naturalness" of his theorems, or even their "inevitability", but I worry that this way of putting it suggests that the issue is sociological or even aesthetic, a matter of taste. Framing it in terms of elementarity seems more promising to me, where its elementarity is a matter of the concepts involved in the relevant theorems, i.e. Q, <, invariance, maximality, square, etc., & of the way those concepts are combined in the theorem's formulation (one might say its "form"). It would be good to get sharper in particular about the latter notion: perhaps it's just an issue of its syntax.

The purpose of "naturalness" observations is a regressive argument for the acceptance of large cardinals: the theorem is natural but its proof requires large cardinals, so you should accept large cardinals. In addition to the usual worries about how "requires" here should be understood, I wonder about the role of "naturalness" in the argument. I take it that the normative force of the argument depends on it, but how exactly? Does the argument work for weaker normative properties of theorems than "natural"? Does it work for "elementary" as I have sketched it above? (And what measure of weakness ought be used here?) Can we determine which normative properties of theorems enable the argument, & which do not? Resolving these would help me get more clarity about what exactly is at stake in debates concerning naturalness, & what has been accomplished.

Finally, a question for Harvey in particular: fixing the relational structure (Q,<), are there values of k,n,m for your templates for which the statements are provable in ZFC but not PA? In PA but not EFA? That is, if I tweak k, say, can I reduce the provability strength of the statements in this systematic way? If not, is such a development reasonable to expect, if not for (Q,<), then for some other relational structure?


Best,

Andy Arana