Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Sun, 9 Mar 2025 10:55:21 +0100 Organization: A noiseless patient Spider Lines: 121 Message-ID: References: <8e7b322e-1259-4563-b2d5-37983249a397@att.net> <4ae7b6d4-49a9-47ba-b2ac-c77238e93545@att.net> <60614d74-cf15-4e8e-8390-f8861bff44f9@att.net> <76d99693-1dcf-4049-98b9-a33edced2e83@att.net> <123fb080-4f72-482c-a6e9-aa525aa7150b@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sun, 09 Mar 2025 10:55:22 +0100 (CET) Injection-Info: dont-email.me; posting-host="7ac4cbb5476a43fc8115796013b9e2bc"; logging-data="718502"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/08X1IyQm/Gu1n5gZos35FxhmDePRBgBA=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:1ghZ70BX21ztFZF3mkhpBFiZ/KY= Content-Language: en-US In-Reply-To: On 08.03.2025 20:27, Jim Burns wrote: > On 3/8/2025 9:09 AM, WM wrote: >> On 08.03.2025 12:58, Jim Burns wrote: >>> On 3/8/2025 3:45 AM, WM wrote: > >>>> You need not the intersection however >>>> because >>>> Z₀ can also be defined by >>>> { } ∈ Z₀, and >>>> if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀ >>>> then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀. >> >>> Zermelo's Axiom of Infinity describes Z >>> Z ∋ {} ∧ ∀a: Z ∋ a ⇒ Z ∋ {a} >>> Multiple sets satisfy that unique description. >>> That's an indefinite description. >> >> My description is definite. > > Your description is only definite because > ℕ ∋ n is definite. No, my description is definite because every n can be obtained by addition of 1's (or of curly brackets). It is not necessary to have another definition of ℕ_def before. > > What ℕ is, > the interesting part of your description of Z₀, > must be found elsewhere. You are wrong. ℕ_def is the potentially infinite sequence of sums 1 1+1 1+1+1 .... ℕ is its completion, a set larger than all FISONs. > >>> If only >>> {{{...{{{ }}}...}}} with finitely.many curly brackets >>> each with immediate.predecessor ⋃{{{...{{{ }}}...}}} >>> and with {} being one of its priors >>> are in Z₀ >>> then, yes, that is a definite description. >>> >>> And, yes, that seems to be what you meant. >>> If you ever want to make your descriptions clearer, >>> you won't be getting complaints from me. >> >> That is exactly what I meant. >> And that's also the induction of my argument >> UF = ℕ  ==> Ø = ℕ. > > A proof by induction > (a proof by this.inductive.subset.is.the.whole.set) > is only reliable for > a set which is its.own.only.inductive subset. No. A proof by induction produces a set. If applied as I did above, the produced set is its own only inductive subset. > ℕ is its.own.only.inductive subset. > Z₀ is its.own.only.inductive subset. > ⋃{F} union of the set {F} of all FISONs, >  is its.own.only.inductive subset. Right. > > For any set W which > is its.own.only.inductive subset (ℕ,Z₀,⋃{F},...), > ∀j∈W:∃k∈W: j < k Yes, that is called potential infinity. A k larger than all j is not produced by induction. That is only produced by appointment. > {F} is its.own.only.inductive.subset. > No FISON in {F} is F(ω-1) Right. ω and ω-1 cannot be attained by induction. Therefore UF = ℕ ==> Ø = ℕ. >>> you should have said somewhere >>> what a finite ordinal is, >> >> That is not under discussion here. > > That has been under discussion for decades. > > I think that these decades of discussion > have been, in large part, you assigning > different meanings to 'finite', etc. > and matheologians (among whom I place myself) > trying to discern what your meanings are. > > Here's my best guess: > definableᵂᴹ  ==  finite Of course. We know also what we mean by ω, but it has no FISON. Therefore it is not a visible or definable number. > darkᵂᴹ  ==  finite  ==  big Look into the new thread "The truncated harmonic series diverges." to see that dark numbers are necessary. Having only definable denominators, the harmonic series could not diverge. >> n is usually denoting >> a natural number. > > Do we mean the same by 'natural number'? There are two different meanings: All positive integers having FISONs or all positive integers. > >> FISONs are finite by definition. > > Do we mean the same by 'finite'? A natural number n is finite. It is an integer between 0 and ω: 0 < n < ω. Sometimes 0 is included, never ω is included. Regards, WM