Timeline for Is platonism regarding arithmetic consistent with the multiverse view in set theory?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 2 at 12:33 | comment | added | Juan Atacama | I mentioned fictionalism in mathematics. More here. | |
| Sep 2 at 12:23 | comment | added | Juan Atacama | There are many philosophical approaches to mathematics. For example, some fictionalists may believe there are no mathematical objects and mathematics is based on a successful consistent pretense that we are looking into some (fictional) mathematical worlds. | |
| Sep 2 at 12:22 | comment | added | Juan Atacama | The topic of the post is the compatibility of two philosophical views regarding mathematics. Mr. Hamkins has some philosophical views, and so do I. They are determined by our philosophical intuitions (mine are reinforced by the intuitive notion of a finite series of interconnected infinitely accelerating Turing machines, with the output tape of one machine serving as the input tape for the next machine, with an arithmetic formula given on the start tape). | |
| Sep 2 at 12:21 | comment | added | Juan Atacama | The original post has two parts. The first talks about a truth platonist for arithmetic who believes in the unambiguous truth of the statements of arithmetic. There is no mention of von Neumann's set model of the natural numbers. The second talks about Hamkins' philosophical theory of the set multiverse, and different "arithmetics" in the various ZFC models. It looks like the author of the post is asking whether belief in the existence of the unique TA is compatible with belief in Hamkins' set multiverse. | |
| Sep 2 at 8:55 | comment | added | Mikhail Katz | @Juan, you wrote: "I don't see that explicitness in the original post". I am not sure what you mean. If you re-read the second paragraph of the question, you will find that the 16th word there is "ZFC". You can't be more explicit than that. You are certainly free to believe in true arithmetic, Santa Claus, or whatever you like, but that's not the subject of the present question. The subject is whether Platonism is compatible with the multiverse of Hamkins. | |
| Sep 1 at 22:11 | comment | added | Juan Atacama | In addition to the true arithmetic, I also believe in various finitely anchored realms of mathematical objects, such as some monoid of words over a finite alphabet, or the (true) universe of hereditarily finite sets containing the (true) von Neumann natural numbers that can serve as a model for TA (I know that non-standard ZF models can "lie" about the natural numbers). | |
| Sep 1 at 22:11 | comment | added | Juan Atacama | I don't see that explicitness in the original post, and I fear that many who read it will see it the same way I do. The post talks about truth platonist for arithmetic who believe in the unambiguous truth of the statements of arithmetic. I am also one of them, and I consider the truth of arithmetic statements to be independent from ZF(C). (The truth of some arithmetic formula could in principle be answered by a finite sequence of infinitely accelerating Turing machines, with the output tape of one machine serving as the input to the next machine, with the formula given on the first tape.) | |
| Sep 1 at 15:56 | comment | added | Mikhail Katz | The Original Post. @Juan P.S. I should mention that I didn't downvote your answer. | |
| Sep 1 at 15:52 | comment | added | Juan Atacama | What do you mean by "the OP"? | |
| Sep 1 at 13:36 | comment | added | Mikhail Katz | The OP explicitly spoke of ZFC. In the context of ZFC, the (von Neumann) natural numbers are just a type of set. The burden is on those who argue for a distinction between numbers and sets (of the type discussed in this question as well as the answers) to explain why some types of sets should be "fixed" and others "variable" as one passes from universe to universe within Hamkins' multiverse. I am not saying htat an explanation is impossible; only that one needs to be given. @Juan | |
| Aug 30 at 22:59 | comment | added | Juan Atacama | I'm a platonist about TA. (I believe there is the set of all natural numbers and all definable sets and relations in it, as well as the set of all true propositions about that realm.) But I am very skeptical about the existence of uncountable sets. Therefore, I am not a proponent of the view that there is both the realm of natural numbers and a full-blooded set multiverse also inhabited by uncountable sets, but this is not because I find such a view to be inconsistent. | |
| Aug 30 at 0:54 | comment | added | Juan Atacama | We are talking about the consistency of a philosophical theory. How are you going to bring into contradiction the view that there are two immaterially separate realms, one of which is inhabited by natural numbers and the other only by sets, if you have consistent descriptions of one and the other? Why can't they be peacefully next to each other? | |
| Aug 29 at 16:55 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Aug 29 at 16:38 | history | answered | Mikhail Katz | CC BY-SA 4.0 |