tag:blogger.com,1999:blog-3891434218564545511.post5654666557537290790..comments2024-03-28T13:23:50.623-05:00Comments on Alexander Pruss's Blog: Might all infinities be the same size?Alexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-3891434218564545511.post-15120307587552901322016-09-02T13:13:26.048-05:002016-09-02T13:13:26.048-05:00Here's a clearer way to run my argument, in hi...Here's a clearer way to run my argument, in hindsight. Assume PPP with "pairings" being taken to be metaphysical pairings. Assume ZFC and the Standard Model Hypothesis (which is true for aught that we know). Now either the hypothesis that all infinite sets are the same size (as defined by PPP) is true or not. If it is true, we're done.<br /><br />Suppose it's not true. Then it follows from the Standard Model Hypothesis that there is a minimal standard model U of set theory which is a set, in fact a countable set. Not all "true" sets are members of U, since there are more than countably many "true" sets. U is a standard model, meaning that its membership relation is just a restriction of the "true" membership relation and if a set is in it so are all the "true" sets that are "truly" members of it. Call the members of U the U-sets. Any two U-sets that have a mathematical pairing that is itself in U will be said to be U-same-size. But any two U-sets that have a mathematical pairing among the "true" sets (which can go beyond U) will be said to be "truly" same-size. Then, infinite U-sets are in general not U-same-size, but they are all "truly" same-size.<br /><br />Now take this story, which is mathematically coherent given the Standard Model Hypothesis, and drop from it all the "true" sets. But for each "true" set that was a pairing, add to the story a Platonic relation such that it is possible for its extension to correspond to that "true" set. So the only sets now are what we used to call the U-sets. The infinite sets now differ in mathematical cardinality, but any two of them can still be paired metaphysically by some Platonic relation, and that makes them be metaphysically the same size. And, the claim is, for aught we know (at least assuming Platonism and the like), this is how things are and must be.<br /><br />I should add one caveat. My stories depend on the denial of a plausible extension of the separability axiom, namely that for any set S of n-tuples and any *metaphysical* n-ary relation R, there is a subset of S consisting of all the tuples whose elements stand in R. (Separability is basically the restriction of this to relations R that can be expressed in the language of set theory.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.com