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Strongly inaccessible cardinals are usually introduced either as (a) cardinalities of models of ZFC or (b) cardinals which are not the power set of a smaller cardinal nor the supremum of a set with hereditarily lesser cardinality. These seem to represent the model-theoretic and set-theoretic perspectives on strong inaccessibility.

Recently I learned that if $\kappa$ is a strongly inaccessible cardinal, then $(V_\kappa)^2\subseteq V_\kappa$, so any function $f:V_\kappa\to V_\kappa$ is a set of pairs whose coordinates are members of $V_\kappa$, and so any such function can be "applied" to any other such function:

$$f(g) = {\{\ z\ |\ \langle \langle x,y\rangle, z \rangle\in f\ \&\ \langle x,y\rangle\in g\ }$$\}$$ Therefore one can say that if \kappa is strongly inaccessible, then V_\kappa is closed under self-application (as defined above) of functions f:V_\kappa\to V_\kappa. This seems to be sort of a "recursion-theoretic" characterization of strong inaccessibility: it identifies a definable operation under which all strongly inaccessible cardinals are closed. Question 0: does this make sense? Question 1: is the converse true, making this a complete characterization? (if V_\kappa closed under self-application of functions then \kappa is strongly inaccessible)? Question 2: if so, I'm sure this has come up before in the literature. In what sorts of directions does this investigation lead? This is one of the more-vague questions I've asked so far. I guess I'm sort of fishing for enlightenment here; it took me a long time to understand the point of inaccessibility, and I suspect that I might have caught on more quickly if this motivation (closure under self-application) had been introduced early on. 1 # "\kappa strongly inaccessible" = "every function f:V_\kappa\to V_\kappa can be self-applied"? Strongly inaccessible cardinals are usually introduced either as (a) cardinalities of models of ZFC or (b) cardinals which are not the power set of a smaller cardinal nor the supremum of a set with hereditarily lesser cardinality. These seem to represent the model-theoretic and set-theoretic perspectives on strong inaccessibility. Recently I learned that if \kappa is a strongly inaccessible cardinal, then (V_\kappa)^2\subseteq V_\kappa, so any function f:V_\kappa\to V_\kappa is a set of pairs whose coordinates are members of V_\kappa, and so any such function can be "applied" to any other such function:$$f(g) = {\ z\ |\ \langle \langle x,y\rangle, z \rangle\in f\ \&\ \langle x,y\rangle\in g\ }

Therefore one can say that if $\kappa$ is strongly inaccessible, then $V_\kappa$ is closed under self-application (as defined above) of functions $f:V_\kappa\to V_\kappa$.

This seems to be sort of a "recursion-theoretic" characterization of strong inaccessibility: it identifies a definable operation under which all strongly inaccessible cardinals are closed.

Question 0: does this make sense?

Question 1: is the converse true, making this a complete characterization? (if $V_\kappa$ closed under self-application of functions then $\kappa$ is strongly inaccessible)?

Question 2: if so, I'm sure this has come up before in the literature. In what sorts of directions does this investigation lead?

This is one of the more-vague questions I've asked so far. I guess I'm sort of fishing for enlightenment here; it took me a long time to understand the point of inaccessibility, and I suspect that I might have caught on more quickly if this motivation (closure under self-application) had been introduced early on.