Path: ...!weretis.net!feeder9.news.weretis.net!news.quux.org!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Sat, 22 Feb 2025 19:02:12 +0100 Organization: A noiseless patient Spider Lines: 13 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> <25e4529e6ea0cd575f92e6cfe481e9ed53d91850@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 22 Feb 2025 19:02:12 +0100 (CET) Injection-Info: dont-email.me; posting-host="e0db16360e7701d068d98e5964a441c0"; logging-data="78299"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18mr6IbGKD8vn0wG/ELMLKJU56z2BYSl2g=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:9W0No9rlo8M1rcJDbNz++CIi6M8= In-Reply-To: <25e4529e6ea0cd575f92e6cfe481e9ed53d91850@i2pn2.org> Content-Language: en-US Bytes: 2560 On 22.02.2025 15:03, Richard Damon wrote: > On 2/22/25 7:35 AM, WM wrote: >> The inductive step is proved in my example from |ℕ \ {1, 2, >> 3, ..., n}| = ℵo, i.e., by several axioms, and in Zermelo's example by >> a single axiom. > Which just proves that no specific FISON is individually REQURED to make > the set of Natural Numbers. If every FISON can be omitted, ten nothing remains for a sufficient set. Because if there was any sufficient set, it would have a first FISON. Regards, WM