Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: sci.logic Subject: Re: Simple enough for every reader? Date: Fri, 30 May 2025 12:36:39 +0300 Organization: - Lines: 90 Message-ID: <101bu77$dqtr$1@dont-email.me> References: <100a8ah$ekoh$1@dont-email.me> <100ccrl$upk6$1@dont-email.me> <100cjat$vtec$1@dont-email.me> <100fdbr$1laaq$1@dont-email.me> <100funo$1ous9$1@dont-email.me> <100haco$24mti$1@dont-email.me> <100hocb$2768i$1@dont-email.me> <100mpm7$3csuv$1@dont-email.me> <100much$3drk8$1@dont-email.me> <100p8v7$k2$1@dont-email.me> <100pbot$dmi$1@dont-email.me> <100rv2t$jpca$1@dont-email.me> <100sajh$lkp7$2@dont-email.me> <100us6p$1b4q2$1@dont-email.me> <100uvfe$1b4dh$3@dont-email.me> <1011fkf$1v4kb$1@dont-email.me> <1011qrn$20v83$1@dont-email.me> <10149ic$2jtva$1@dont-email.me> <1014kja$2l9jj$4@dont-email.me> <1016h9p$35it9$1@dont-email.me> <101797i$39rdb$1@dont-email.me> <1019bki$3qe6q$1@dont-email.me> <1019s2k$3sv8u$3@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Fri, 30 May 2025 11:36:39 +0200 (CEST) Injection-Info: dont-email.me; posting-host="30194b9ca93057da22fe329aa8d4529a"; logging-data="453563"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+YbsuYsimK0/cdigmQUMek" User-Agent: Unison/2.2 Cancel-Lock: sha1:oKvujHgzBZKvGOSsu2bO8i7nhuI= On 2025-05-29 14:47:49 +0000, WM said: > On 29.05.2025 12:07, Mikko wrote: >> On 2025-05-28 15:13:54 +0000, WM said: > >>>> There is no induction in plain logic. >>> >>> But it is in the mathematics we apply. >> >> It is in certain mathematical structures but not in all. > > Anyhow a reader in sci.logic should understand it. Everybody should understand at least arithmetic induction and its limitations. But everybody doesn't. >>> I have said: {1} has infinitely many (ℵo) successors. >> >> But you navn't proven that this infinity is not begger than some other >> infinity. > I am assuming Cantor's infinity. This is expressed by ℵo. Cantor did not use ℵo for infinity in general but only for a particular kind of infinity. He also used other symbols for other kinds of infinity. There are also kinds of infinity Cantor did not discuss at all. >>>> and P[n] -> P[n+1] before it can infer >>> >>> I did not expect that you need this explanation: >>> If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, >>> 3, ..., n, n+1} has infinitely many (ℵo) successors because here the >>> number of successors has been reduced by 1, and ℵo - 1 = ℵo. There is >>> no way to avoid this conclusion if ℵo natural numbers are assumed to >>> exist. And that is the theory that I use. >> >> To me this does not look like P[n] -> P[n+1]. > > P[n]: {1, 2, 3, ..., n} has infinitely many (ℵo) successors. > P[n+1]: {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors. But P[n] -> P[n+1] is not there. > Do you doubt ℵo - 1 = ℵo? That can't be said in Peano arithmetic. > It is basic mathematics as you learn it in the first semester. No, it is not that basic. There are no infinities there, and no induction, either. >> As I said the theory must be specified. >> >> In Peano arithmetic the induction axiom is applicable to everything. >> If you want something else you must specify some other theory, perhaps >> some set theory. > > Induction is applied to every natural number of the Peano set. The > proof shows that it cannot be applied to every natural number of the > Cantor set. You have shown no proof that shows that. >>>>> The set of finite initial segments of natural numbers is potentially >>>>> infinite but not actually infinite. >>>> >>>> There is nothing potential in a set. >>> >>> Then call it a collection. >> >> Things get soon complicated if we allow other than objects, first order >> functions and first order predicates. > > Here nothing gets complicated, but all remains very simple. Which "here"? With or without non-set collections? > All Cantor's natural numbers can be manipulated collectively, for > instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have > disappeared. > Could all Cantor's natural numbers be distinguished, then this > subtraction could also happen but, caused by the well-order, a last one > would disappear. Contradiction. It is not a contradiction that at least one disappears when all disappear. -- Mikko