"$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-applied"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T10:14:56Z http://mathoverflow.net/feeds/question/15403 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15403/kappa-strongly-inaccessible-every-function-fv-kappa-to-v-kappa-can-b "$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-applied"? Adam 2010-02-16T01:58:27Z 2010-02-16T03:31:28Z <p>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.</p> <p>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:</p> <p><code>$$f(g) = \{\ z\ |\ \langle \langle x,y\rangle, z \rangle\in f\ \&amp;\ \langle x,y\rangle\in g\ \}$$</code></p> <p>Therefore one can say that if $\kappa$ is strongly inaccessible, then $V_\kappa$ is closed under self-application (as defined above) of functions $f:V_\kappa\to V_\kappa$.</p> <p>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.</p> <p>Question 0: does this make sense?</p> <p>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)?</p> <p>Question 2: if so, I'm sure this has come up before in the literature. In what sorts of directions does this investigation lead?</p> <p>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.</p> http://mathoverflow.net/questions/15403/kappa-strongly-inaccessible-every-function-fv-kappa-to-v-kappa-can-b/15410#15410 Answer by Joel David Hamkins for "$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-applied"? Joel David Hamkins 2010-02-16T02:56:33Z 2010-02-16T03:31:28Z <p>Unfortunately, your characterizations of the strongly inaccessible cardinals are not quite correct. The correct definition is that &kappa; is <em>strongly inaccessible</em> (also known as just plain <em>inaccessible</em>), if &kappa; is an uncountable regular strong limit cardinal. The cardinal &kappa; is <em>regular</em> if it is not the union of fewer than &kappa; many sets of size less than &kappa;. And &kappa; is a <em>strong limit cardinal</em> if whenver &beta; &lt; &kappa;, then the power set of &beta; also has size less than &kappa;. </p> <p>This is not equivalent to the assertion that V<sub>&kappa;</sub> is a model of ZFC. (Although, to be sure, this false assertion has appeared surprisingly often in print and I have even heard a famous proof theorist make this assertion to a very large audience of hundreds of logicians.) The reason is that if &kappa; is strongly inaccessible, then a Lowenheim-Skolem argument shows that there will be many &gamma; &lt; &kappa; for which V<sub>&gamma;</sub> is elementary in V<sub>&kappa;</sub>, and so these also will be models of ZFC. It is an exercise to show that the least &gamma; for which V<sub>&gamma;</sub> is a model of ZFC has cofinality &omega;, and so is definitely not inaccessible. </p> <p>Also, since ZFC is a first order theory in a countable language, if it has any models at all, then it has models in every infinite cardinality. So it is not correct to characterize inaccessible cardinals as the sizes of models of ZFC in that way either.</p> <p>It is also not equivalent to asserting that &kappa; is regular and not the size of a power set of a smaller set. The reason is that if, say, CH failed, then &omega;<sub>1</sub> would be regular and also not be the size of the power set of any smaller set (since 2<sup>&omega;</sup> would be already too large). But &omega;<sub>1</sub> is not an inaccessible cardinal. </p> <p>Your remark that V<sub>&kappa;</sub> is closed under pairs when &kappa; is inaccessible actually doesn't need any amount of inaccessibility. If x and y are sets in any V<sub>&alpha;</sub>, then the pair (x,y) appears just a few steps later (and actually, one can use flat pairing function that do not increase rank at all, for infinite rank sets), and so every V<sub>&lambda;</sub> is closed under pairing for any limit ordinal &lambda;. If one uses a flat pairing function (instead of the common Kuratowski pairing function), then every V<sub>&alpha;</sub> for every infinite ordinal &alpha; will be closed under pairing. </p> <p>Finally, yes, if V<sub>&lambda;</sub> is closed under pairing, then you can apply such functions to themselves, and this idea is used quite often when we have elementary embeddings defined on models of ZFC. For example, if j:V to M, then j(j) is a function defined on M, into some structure j(M), which will be the union of j(V<sub>&alpha;</sub><sup>M</sup>). This operation is called <em>application</em>.</p> <p>There is a famous result of Laver concerning the left distributive algebra of nontrivial elementary embeddings j:V<sub>&lambda;</sub> to V<sub>&lambda;</sub>. The first results characterizing normal forms in the free algebra with one generator used such embeddings, with the accompanying very large large cardinal hypothesis. For example, Laver produced a decision procedure, which was only known to work under these enormous large cardinal assumptions. Later, the large cardinal hypotheses were removed and the algebra became studied apart from the large cardinals, but the basic properties were definitely inspired and discovered by knowledge of what the large caridnals were like. The basic operation in this algebra is known as <em>application</em>, and is exactly the operation that you mention.</p>