Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Fri, 27 Dec 2024 11:14:37 +0100 Organization: A noiseless patient Spider Lines: 101 Message-ID: References: <4051acc5-d00a-40d2-8ef7-cf2b91ae75b6@att.net> <8d69d6cd-76bc-4dc1-894e-709d044e68a1@att.net> <7356267c-491b-45c2-b86a-d40c45dfa40c@att.net> <4bf8a77e-4b2a-471f-9075-0b063098153f@att.net> <31180d7e-1c2b-4e2b-b8d6-e3e62f05da43@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Fri, 27 Dec 2024 11:14:37 +0100 (CET) Injection-Info: dont-email.me; posting-host="a6bf3ec35b348782241589a6c89f5a4d"; logging-data="3733213"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19+j5FMtBqML7xqbiFFCjlnj3iG/yeuZo4=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:EoVp2z7sBNTLNg7WPxxqR770WQM= In-Reply-To: Content-Language: en-US Bytes: 5490 On 26.12.2024 19:41, Jim Burns wrote: > On 12/22/2024 6:32 AM, WM wrote: > I (JB) think that it may be that > 'almost.all' '(∀)' refers concisely to > the differences in definition at > the center of our discussion. > > For each finite.cardinal, > almost.all finite.cardinals are larger. > ∀j ∈ ℕⁿᵒᵗᐧᵂᴹ: (∀)k ∈ ℕⁿᵒᵗᐧᵂᴹ: j < k That is true in potential infinity. It is wrong in actual infinity because there ℕ \ {1, 2, 3, ...} = { } shows that all finite cardinals can be manipulated such that none is larger. > > I think that you (WM) would deny that. > You would say, instead, > ᵂᴹ⎛ for each definable finite.cardinal > ᵂᴹ⎜ almost.all finite cardinals are larger. Right. > For sequence ⟨Sₙ⟩ₙ᳹₌₀ of sets > Sₗᵢₘ is a limit.set of ⟨Sₙ⟩ₙ᳹₌₀ > if > each x ∈ Sₗᵢₘ is ∈ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀  and > each y ∉ Sₗᵢₘ is ∉ almost.all Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀ > ⟨Sₙ⟩ₙ᳹₌₀ ⟶ Sₗᵢₘ  ⇐ > ⎛ x ∈ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  x ∈ Sₖ > ⎝ y ∉ Sₗᵢₘ  ⇐  (∀)Sₖ ∈ ⟨Sₙ⟩ₙ᳹₌₀:  y ∉ Sₖ > A limit is a set S​͚ such that nothing fits between it and all sets of the sequence. > --- > The notation a​͚  or S​͚  for aₗᵢₘ or Sₗᵢₘ > is tempting, but > it gives the unfortunate impression that > a​͚  and S​͚  are the infinitieth entries of > their respective infinite.sequences. > They aren't infinitieth entries. > They are defined differently. The last natural number is finite, and therefore objectively belongs to a finite set. But like all dark numbers it has no FISON and therefore the dark realm appears like an infinite set. > >>> E(n+2) is >>> the set of all finite.cardinals > n+2 >>> E(n+1)  =  E(n+2)∪{n+2} >>> E(n+2)∪{n+2} isn't larger.than E(n+2) >> >> Wrong. > > Almost all finite.cardinals are larger than > finite.cardinal n+1 > {n+2} isn't large enough to change that. > Almost all finite.cardinals are larger than > finite cardinal n+2. That is true for visible n. > >>> #E(n+2) isn't any of the finite.cardinals in ℕ >> >> It is an infinite number but >> even infinite numbers differ like |ℕ| =/= |ℕ| + 1. > > Infiniteᵂᴹ numbers which differ like |ℕ| ≠ |ℕ| + 1. > are finiteⁿᵒᵗᐧᵂᴹ numbers. No. They are invariable numbers like ω and ω+1. The alephs differ only because they count potentially infinite sets which always can be bijected as far as is desired. > > The concept of 'limit' is a cornerstone of > calculus and analysis and topology. A limit is a number or set such that nothing fits between it and all numbers or sets of the sequence. > > That cornerstone rests upon 'almost.all'. > Each finite.cardinal has infinitely.more > finite.cardinals after it than before it. For each finite cardinal that can be defined this is true. Dark cardinals never have been considered. > If one re.defines things away from that, > it is only an odd coincidence if, after that, > any part of calculus or analysis or topology > continues to make sense. > Without dark cardinals set theory does not make sense. ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0 would contradict each other because more than all n are not in {1, 2, 3, ....}. Regards, WM