Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: FromTheRafters Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Sat, 22 Feb 2025 07:13:36 -0500 Organization: Peripheral Visions Lines: 52 Message-ID: References: <865ffa4c4c1091981c5b3b93ddf3dba690cd5ad2@i2pn2.org> <559d228c01ea290aec13e735ec85036862578165@i2pn2.org> <5220af0cb7d579f20d58809659d8dcb8d7ba046c@i2pn2.org> <685c1274-e22f-409d-b39c-c3a5430c2f57@att.net> <69f56ce0-08a2-4614-b102-e333175c643d@att.net> <9a88665f-211f-4260-b585-97c72c7b6d1b@att.net> <8bed122d8b355eff96158e6f5cb76cffcc42925c@i2pn2.org> <4909c3c162f948930321987279d0e04864f68f9d@i2pn2.org> Reply-To: erratic.howard@gmail.com MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-15"; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 22 Feb 2025 13:13:46 +0100 (CET) Injection-Info: dont-email.me; posting-host="7c26a6f785c58bda4533875532e14e1b"; logging-data="4141940"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/0tQW9qkCi56N+bkGkk6ssZq68QwYL138=" Cancel-Lock: sha1:Gf2U212rwf/82wWW8HZK+Ra2Ns8= X-ICQ: 1701145376 X-Newsreader: MesNews/1.08.06.00-gb WM formulated on Saturday : > On 22.02.2025 12:14, FromTheRafters wrote: >> WM explained on 2/22/2025 : >>> On 22.02.2025 02:05, Richard Damon wrote: > >>>> Note, as I understand it, that initial Zermelo Set Theory didn't even >>>> HAVE "induction", >>> >>> You don't understand it. >> >> He understands it better than you do apparently. Zermelo's theory didn't >> have transfinite induction. > > Who claimed that? Certainly not you. Induction is a theorem there, and not extended to the infinite sets. >>>> but its last axiom was the Axiom of Infinity that say that there exists >>>> in this domain the set Z that contains the null set as an element and is >>>> so constituted that to each of its element a, there corresponds a further >>>> element of the form {a}, in other words with each of its elements a, it >>>> also contains the corresponding set {a} as an element. >>> >>> That is induction. >> >> That is the successor function, it doesn't extend to the transfinite. > > Induction covers the infinite set of elements. Transfinity is not included. > First element and the successor function is induction. First element and successor function is 'inductive hypothesis' not induction. https://www.cs.cornell.edu/courses/cs2800/2011fa/Lectures/induction.pdf >>>> Note, this is NOT the statement of induction. >> >> AI Overview >> >> No, Zermelo's set theory by itself does not explicitly include the >> principle of finite induction; > > It does. Not explicitly, it is a theorem. >> Key points: > > AI is not suitable for mathematics. It is okay when it is correct though.