Path: nntp.eternal-september.org!news.eternal-september.org!eternal-september.org!feeder3.eternal-september.org!usenet.network!news.neodome.net!rocksolid2!i2pn2.org!.POSTED!not-for-mail From: joes Newsgroups: sci.logic Subject: Re: Simple enough for every reader? Date: Thu, 3 Jul 2025 14:12:29 -0000 (UTC) Organization: i2pn2 (i2pn.org) Message-ID: <979729b53f91bd779898f5241ca32cbb609f8ac5@i2pn2.org> References: <100a8ah$ekoh$1@dont-email.me> <102rfal$2am0a$1@dont-email.me> <102rhlt$2b85u$1@dont-email.me> <102u1gk$32100$1@dont-email.me> <102ug2t$35ekh$1@dont-email.me> <1030dft$3obvv$1@dont-email.me> <10315us$3tqn8$2@dont-email.me> <1033805$lvet$1@dont-email.me> <1033ks9$1075$1@dont-email.me> <10360hf$10lrl$1@dont-email.me> <10365va$11afj$3@dont-email.me> <1038it5$epe7$1@dont-email.me> <1039873$jtod$1@dont-email.me> <103b13q$14dr9$1@dont-email.me> <103bc1r$17360$2@dont-email.me> <103dqb3$1u2kv$1@dont-email.me> <103engv$25bv0$1@dont-email.me> <103g9t2$2l4am$1@dont-email.me> <103hkv3$2voqr$1@dont-email.me> <103j7qu$3dl3j$1@dont-email.me> <103jgq9$3fje0$1@dont-email.me> <103lhgp$11qu$1@dont-email.me> <103mrsa$b011$1@dont-email.me> <103oe8v$ppfi$1@dont-email.me> <103osb9$sphe$1@dont-email.me> <103r4a7$1fl13$1@dont-email.me> <103ukik$2ahp0$1@dont-email.me> <1042o2k$3d5cj$1@dont-email.me> <1043dg5$3hor7$1@dont-email.me> <1045itl$3le8$1@dont-email.me> <1045vc8$5pd6$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Injection-Date: Thu, 3 Jul 2025 14:12:29 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="3190503"; mail-complaints-to="usenet@i2pn2.org"; posting-account="nS1KMHaUuWOnF/ukOJzx6Ssd8y16q9UPs1GZ+I3D0CM"; User-Agent: Pan/0.145 (Duplicitous mercenary valetism; d7e168a git.gnome.org/pan2) X-Spam-Checker-Version: SpamAssassin 4.0.0 Am Thu, 03 Jul 2025 15:08:25 +0200 schrieb WM: > On 03.07.2025 11:35, Mikko wrote: >> On 2025-07-02 13:51:01 +0000, WM said: >>> The function is injective, or one-to-one, if each element of the >>> codomain is mapped to by at most one element of the domain, >>> The function is surjective, or onto, if each element of the codomain >>> is mapped to by at least one element of the domain; Wikipedia >>> Bijection = injection and surjection. >>> Note that no element must be missing. That means completeness. >> >> It does not mean that the bijection is completely known. > > It means that every element of the domain and of the codomain is > involved. > The domain must be complete by the definition of mapping, and the > codomain must be complete by the definition of surjectivity Which are the case for Cantor's function. > The rule of subset proves that every proper subset has fewer elements No such rule for infinite sets. > than its superset. So there are more natural numbers than prime numbers, No, you can number the primes. > The rule of construction yields the number of integers |Z| = 2|N| + 1 > and the number of fractions |Q| = 2|N|^2 + 1. Those numbers are equal. >>> "The arguments using infinity, including the Differential Calculus of >>> Newton and Leibniz, do not require the use of infinite sets." [T. >>> Jech: "Set theory", Stanford Encyclopedia of Philosophy (2002)] >> Differential calculus does not require sets at all. > But it needs potential infinity. Therefore your "the distinction between > complete and incomplete is not mathematical." is wrong. It doesn't need "actual infinities". >>> "Numerals constitute a potential infinity. Given any numeral, we can >>> construct a new numeral by prefixing it with S." [E. Nelson: >>> "Hilbert's mistake" (2007) p. 3] >> >> That is a possible way to view them. >> But a different view does not lead to different mathematical conclusion >> as they are irrelevant to inferences from axioms and postulates. > > Potential infinity is based upon other axioms than actual infinity and > has other results. Uh, no? >> That N has an order and can be given other orders is irrelevant. > Not for bijections. The enumeration of the rational numbers is > impossible in the natural order by size for instance. That's a different function. -- Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math: It is not guaranteed that n+1 exists for every n.