Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 28 Jul 2024 23:32:40 +0000 Subject: Re: Replacement of Cardinality Newsgroups: sci.logic,sci.math References: <881fc1a1-2e55-4f13-8beb-94d1f941b5af@att.net> <85194aeb-1b24-4486-8bcc-4dcd43b4fd2f@att.net> From: Ross Finlayson Date: Sun, 28 Jul 2024 16:32:58 -0700 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 111 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-nqs2aNKtA76D7cpHX+fKS/V/i2hPXYH7GhD9xzLXo0mcqTxp7HWV3I3wOZtdDfipUIFXytPdWTLVoQs!0dtAnjktcBnO3MY5pu0XcdXob6it33SFV8H1m654ueDoQmpL2zReFAWUscgaEJNWaSxIVxqppxI= X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 4510 On 07/28/2024 04:25 PM, Ross Finlayson wrote: > On 07/28/2024 11:17 AM, Jim Burns wrote: >> On 7/28/2024 8:17 AM, WM wrote: >>> Le 27/07/2024 à 19:34, Jim Burns a écrit : >>>> On 7/26/2024 12:31 PM, WM wrote: >> >>>>> _The rule of subset_ proves that >>>>> every proper subset has less elements than its superset. >> >>>> If ℕ has fewer elements than ℕ∪{ℕ} >>>> then >>>> |ℕ| ∈ ℕ >>> >>> |ℕ| = ω-1 ∈ ℕ >> >> ⎛ Each non.{}.set A of ordinals holds min.A >> ⎜ >> ⎜ Ordinal j = {i:i> ⎜ >> ⎜ Finite ordinal j has fewer elements than j∪{j} >> ⎜ >> ⎝ ℕⁿᵒᵗᐧᵂᴹ is the set of ALL finite ordinals. >> >> No finite.ordinal is last.finite, >> no visibleᵂᴹ finite.ordinal, >> no darkᵂᴹ finite.ordinal. >> In particular, no finite.ordinal is ω-1 >> >> Also, no before.first infinite.ordinal is >> before the first infinite.ordinal ω >> In particular, no infinite.ordinal is ω-1 >> >> ---- >> Consider ordinals i j k such that >> i∪{i} = j and j∪{j} = k >> >> Obviously, their order is i < j < k >> >> Either they're all finite >> |i| < |j| < |k| >> or they're all infinite >> |i| = |j| = |k| >> >> No finite.to.infinite step exists. >> no visibleᵂᴹ finite.to.infinite step, >> no darkᵂᴹ finite.to.infinite step. >> >> Defining declares the meaning of one's words. >> 'Defining into existence' that which doesn't exist >> makes nonsense of whatever meaning one's words have. >> >> ⎛ if >> ⎜ g: j∪{j}→i∪{i}: 1.to.1 >> ⎜ then >> ⎜ f(x) := (g(x)=i ? g(j) : g(x)) >> ⎜ (Perl ternary conditional operator) >> ⎜ f: j→i: 1.to.1 >> ⎜ >> ⎜ if >> ⎜ f: j→i: 1.to.1 >> ⎜ then >> ⎜ g(x) := (x=j ? i : f(x)) >> ⎝ g: j∪{j}→i∪{i}: 1.to.1 >> >> Therefore, >> i has fewer than j iff j has fewer than k >> >>>> ℕ has fewer elements than ℕ >>> >>> ℕ has ω-1 elements. >> >> ℕⁿᵒᵗᐧᵂᴹ holds all finite ordinals. >> >> Finite doesn't need to be small. >> ℕⁿᵒᵗᐧᵂᴹ holds ordinals which >> are big compared to Avogadroᴬᵛᵒᵍᵃᵈʳᵒ, >> but those big ordinals have an immediate predecessor, >> and each non.0.ordinal before them has >> an immediate predecessor. >> That makes them finite, but not necessarily small. >> >>>> Because ℕ does not have fewer elements than ℕ >>>> ℕ does not have fewer elements than ℕ∪{ℕ} >>>> and the rule of subsets is broken. >>> >>> ℕ = {1, 2, 3, ..., ω-1} = {1, 2, 3, ..., |ℕ|} >> >> ∀j ∈ ℕⁿᵒᵗᐧᵂᴹ: >> ∃k ∈ ℕⁿᵒᵗᐧᵂᴹ\{0}: >> k = j+1 ∧ ¬∃kₓ≠k: kₓ=j+1 >> >> '+1': ℕⁿᵒᵗᐧᵂᴹ→ℕⁿᵒᵗᐧᵂᴹ\{0}: 1.to.1 >> and the rule of subset is broken. >> >> > > That's, ..., nice and all, yet, are you, > "preaching to the choir", or, > "reaching to the higher", the higher ground. > > I.e., here it's not saying much. > > Where's the "extra"-ordinary. > > It's a matter of deductive inference there is one, > while the naive nicely arrives at it directly. > > Foundations is more than a field.