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Keith Elliott Peterson (Aug 06 2021 at 18:59):One doesn't usually see mathematical arguments for God, but Christopher Menzel gives an argument for God from collections, though really it seems to be variations of arguments on large collections. From Alvin Plantinga's "only an infinite mind (which he calls then God), can collect up the infinite hierarchy of sets" to more and more complex versions of this argument.
I'm no expert on large structures or large categories, so I thought it might be better to ask actual experts that deal with large categories, large higher categories, and other large structures as part of their work.
https://www.researchgate.net/publication/308890413_The_Argument_from_Collections
Keith Elliott Peterson (Aug 06 2021 at 19:08):Also, can we just take a moment to appreciate how extremely large n-categories of (n-1)-categories and their infinite counterparts are! I mean, it hurts my head to think about Set and Cat, let alone 2-Cat, or any n-Cat, n>2, or (∞,1)-Cat...
John Baez (Aug 07 2021 at 00:50):It's pretty amusing for humans to start thinking about the infinite hierarchy of sets, then claim only god has what it takes to really understand this hierarchy, and then conclude that god must exist. But I like the chutzpah.
John Baez (Aug 07 2021 at 00:51):I guess you know Goedel's ontological argument for the existence of god.
Morgan Rogers (he/him) (Aug 07 2021 at 11:19):Keith Peterson said:
One doesn't usually see mathematical arguments for God, but Christopher Menzel gives an argument for God from collections, though really it seems to be variations of arguments on large collections. From Alvin Plantinga's "only an infinite mind (which he calls then God), can collect up the infinite hierarchy of sets" to more and more complex versions of this argument.
https://www.researchgate.net/publication/308890413_The_Argument_from_Collections
This was a fun read. It was nice to get a recap on ZF, and the modal interpretation of statements about set theory was an interesting new perspective for me. No detailed argument was provided for the convergence axiom, which I didn't understand (it seems plausible to me that there could be two incompatible extensions of a given universe of sets), but not too much hinged on that.
In light of another discussion happening here, it was amusing to see mention of "the first/smallest
inaccessible cardinal" around page 29, in spite of no mention having been made of the axiom of choice (or other principle required to demonstrate that there is a well-defined such cardinal).
Morgan Rogers (he/him) (Aug 07 2021 at 11:57):Regarding the eventual (surprisingly brief?) argument against realism on the basis of the modal interpretation, Menzel commits the strawman fallacy in a form which I've encountered on several occasions in my philosophy reading lately: his argument is based on the commitment of "the realist" to an overly rigid version of their position. To quote:
A set exists because its members do, and no further explanation is needed. But this explanation is utterly undercut if all sets are contingent; if a set exists in one world but not in another despite the existence of its members in , then its existence in is metaphysically capricious; since might not have existed even if its members had, the existence of its members does not after all explain its existence in .
The different "worlds" here correspond to different extensions of the axioms of ZF accommodating extensions of the cumulative hierarchy. Since any well-trained set-theoretic realist knows they cannot naively take arbitrary collections of things to be sets in a consistent way, the fact of which collections constitute sets is a contingent one; the first sentence in the quote cannot be mistaken to mean that the existence of the members of a collection is always sufficient for that collection to constitute a set. So Menzel is claiming that any realist must make a choice as to which constructions they believe produce real sets.
Menzel presents this as the "realist's impasse", essentially observing that any such choice is arbitrary (conveniently ignoring the finitist's choice, for example). In contrasting it with the argument of collections, he essentially observes that God would have the same problem, but claims that's okay because God can make a choice greater than any we could conceive of, and so any extension we lowly mortals reach will de re be realised.
But in the practice of using ZF (and extensions thereof), a set-theoretic realist is using the axiomatic framework to determine collections which they can be confident form sets based on their confidence in the individual constructions in the framework. In Menzel's own exposition, these are the constructions producing "safe" pluralities, and so to which we can assign the label "set" without problems. A set-theoretic realist never commits to saying that the smallest initial segment of the hierarchy closed under their chosen safe constructions (if indeed one believes the hierarchy to be ordered enough for there to be a smallest such) constitutes the entirety of the collection of sets.
Linnebo's modality can reasonably be interpreted purely formally by the realist, not as regarding what is possible regarding the reality of sets, but as what is possible regarding how ZF may be extended to capture that reality, so that the existence of sets is no longer contingent on the modality. Based on the quote that Menzel provides, I believe this is what Linnebo intended.
Morgan Rogers (he/him) (Aug 07 2021 at 14:04):I've defended the set-theoretic realist's position here, but I should add that I don't find either side particularly compelling. One doesn't need to commit to any metaphysical claims over mathematical objects such as sets in order to use and work with them, and any such metaphysical claim is susceptible to the same arguments as metaphysical claims regarding the existence of God (although perhaps the stakes are lower for mathematical objects than God in most social circles).
David Michael Roberts (Aug 08 2021 at 03:48):@Morgan Rogers (he/him)
(it seems plausible to me that there could be two incompatible extensions of a given universe of sets),
Yes, this happens for every countable transitive model, and the extensions can be gotten by adding a single new subset of the naturals. See Joel David Hamkins' https://arxiv.org/abs/1511.01074