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Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM <wolfgang.mueckenheim@tha.de> Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Thu, 27 Feb 2025 11:45:41 +0100 Organization: A noiseless patient Spider Lines: 94 Message-ID: <vppfol$3280b$1@dont-email.me> References: <vmo1bs$1rnl$1@dont-email.me> <vp6rg8$2o8sd$7@dont-email.me> <9a88665f-211f-4260-b585-97c72c7b6d1b@att.net> <vp7qea$2tofq$5@dont-email.me> <8bed122d8b355eff96158e6f5cb76cffcc42925c@i2pn2.org> <vp9hl1$3afuk$6@dont-email.me> <7a26856916099747e76314a2b4c79693e14426fd@i2pn2.org> <vpacfc$3fush$1@dont-email.me> <b3d7070e-1381-41e3-9ace-0f21bc052d0b@att.net> <vpc885$3t9g8$1@dont-email.me> <e19b21b4-07f1-46ac-94f0-dac6cd114754@att.net> <vpetjf$eusv$3@dont-email.me> <98baf83e-820e-4e1b-be2c-e5ea4802683d@att.net> <vpfc74$hq5c$2@dont-email.me> <0876c2b9-2144-44c1-a26b-20176f5e2127@att.net> <vpft4t$kdg0$4@dont-email.me> <067f772a-4f4c-4c27-8042-3f605f814876@att.net> <vpi1g6$14ivq$7@dont-email.me> <vpi7eu$17stc$1@dont-email.me> <vpi9jo$18qai$2@dont-email.me> <fa7bb863-570e-4602-b932-277b01133bba@att.net> <vpk0nn$1s04m$1@dont-email.me> <dd62224a-579b-4032-be2c-04c305247753@att.net> <vpmvg3$2i1ev$1@dont-email.me> <558a879a-4130-476a-8b5d-d53cd371919b@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 27 Feb 2025 11:45:42 +0100 (CET) Injection-Info: dont-email.me; posting-host="c186eda793122898af663c392b76d5f9"; logging-data="3219467"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/HoglixdKER0XCmWdXpFdv4lm93Ei1+gQ=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:poO+gRQPd+/VWpHf4sETXGGmXG8= Content-Language: en-US In-Reply-To: <558a879a-4130-476a-8b5d-d53cd371919b@att.net> Bytes: 5304 On 26.02.2025 23:17, Jim Burns wrote: > On 2/26/2025 6:55 AM, WM wrote: >> On 25.02.2025 19:39, Jim Burns wrote: > >>> >> >> Zermelo for instance made such claims: >> Um aber die Existenz "unendlicher" Mengen zu sichern, >> bedürfen wir noch des folgenden, seinem wesentlichen Inhalte >> von Herrn Dedekind herrührenden Axioms. >> ... Der Bereich enthält mindestens eine Menge Z, welche die Nullmenge >> als Element enthält und >> so beschaffen ist, daß jedem ihrer Elemente a >> ein weiteres Element der Form {a} entspricht >> ... Die Menge Z_0 enthält die Elemente 0, {0}, {{0}}, usw. >> und möge als "Zahlenreihe" bezeichnet werden, >> ... Sie bildet das einfachste Beispiel einer >> "abzählbar unendlichen" Menge. >> [E. Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, >> Mathematische Annalen (1908), S. 266] > > ⎛ But in order to ensure the existence of "infinite" sets, > ⎜ we still need the following axiom, > ⎜ the essential content of which comes from Mr. Dedekind. > ⎜ ... The domain contains at least one set Z, > ⎜ which contains the zero set as an element and > ⎜ is such that each of its elements a > ⎜ corresponds to another element of the form {a} > ⎜ ... The set Z_0 contains the elements 0, {0}, {{0}}, etc. > ⎜ and may be referred to as a "number series", ... > ⎜ It forms the simplest example of a "countably infinite" set. > ⎜ [E. Zermelo: Investigations on the Foundations of Set Theory I, > ⎜ Mathematische Annalen (1908), p. 266] > ⎝ > -- google Good. Much better than 10 years ago. > > The domain contains at least one set Z. > 0∈Z ∧ ∀a:a∈Z⇒{a}∈Z > > It is indefinite which set Z refers to, > apart from that claim definitely being true of Z. The set containing 1 and with every x also x+1 may be ℚ, ℝ, or ℂ. Important is that it contains ℕ. > > That set Z has a subset Z₀ which might not be Z > (what is Z, after all?) > which may be referred to as a number series. > > The number series Z₀ holds no extra numbers. > Suppose set W and Y are candidates for Z > 0∈W ∧ ∀a:a∈W⇒{a}∈W > 0∈Y ∧ ∀a:a∈Y⇒{a}∈Y > and > some elements are in W but not in Y. > Those elements in W\Y are extra, and not.in Z₀ The specific form of Z is irrelevant. Z₀ is the Zahlenreihe, the set {0, 1, 2, ...} > > Perhaps, outside the material you quote, > Zermelo has said that > Z₀ holds no extra numbers > or some version of that. > I haven't investigated what Zermelo has said. > Whatever he's said, the claims I've made here are true. Yes. > Because Z₀ holds no extra numbers, > Z₀ is definite, even though superset Z is indefinite. > That is, in my opinion, what makes all this work. Z₀ is ensured by induction. That is the point. > > This next bit you (WM) might like, for a change. > It looks like the pseudo.induction.rule which > you have been trying to use. It is induction. > Z₀ being the unique intersection of inductive subsets > makes > Z₀ its.own.only.inductive.subset, > which makes > Z₀ the only correct set to use in a proof by induction. It is proved to exist by induction, i.e., by the rule that 0 exists and with a also a'. By the same induction I prove UF = ℕ ==> Ø = ℕ. Regards, WM