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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Fri, 15 Nov 2024 18:24:32 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers Newsgroups: sci.math References: <6d9f3b10-47ad-459c-9536-098ce91f514b@att.net> <16028da0-456b-47ad-8baa-7982a7cbdf10@att.net> <1fca3a53-1cb4-4fd2-85b6-85e9b69ca23b@att.net> <97304048-24f5-4625-82a7-d17427f2f6e3@att.net> <65febd06-662b-4fa4-9aa8-f7353a79a110@att.net> <157a949d-6c19-4693-8cee-9e067268ae45@att.net> From: Ross Finlayson Date: Fri, 15 Nov 2024 10:24:38 -0800 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: 104 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-KlAw8kkMh8I1I5l1ubkkxvWJuW4J9+i45fr6RLDg1CeAOCGx6KI3m5dq7LeiLCca+1e4yDRQV31APtx!xOJ1fYinZk+O5LDDV+CjlE42s5x90pSiDvB4hNVZvcJf7DFsNovZDVk0XE+BKXbZBbDeAvTKhdTA 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: 5363 On 11/15/2024 10:05 AM, Ross Finlayson wrote: > On 11/15/2024 09:55 AM, Jim Burns wrote: >> On 11/15/2024 5:10 AM, WM wrote: >>> On 14.11.2024 19:31, Jim Burns wrote: >>>> On 11/14/2024 5:20 AM, WM wrote: >> >>>>> Therefore >>>>> a geometric representation let alone proof of >>>>> most of Cantor's bijections is impossible. >>>> >>>> Consider geometry. >> >>>> For two triangles △A′B′C′ and △A″B″C″ >>>> if >>>> △A′B′C′ and △A″B″C″ are similar triangles >> >>>> then >>>> corresponding sides are in the same ratio >> >>>>> Therefore >>>>> a geometric representation let alone proof of >>>>> most of Cantor's bijections is impossible. >>> >>> Your writing is unreadable >> >> A geometric representation of >> square.root, multiplication, and division exist. >> One representation uses similar triangles. >> >> Also, a geometric representation of >> addition, subtraction, and order exist. >> >> Cantor's bijection ⟨i,j⟩ ↦ k ↦ ⟨i,j⟩ >> ⎛ k = (i+j-1)⋅(i+j-2)/2+i >> ⎜ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2 >> ⎝ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k >> is composed of >> square.root, multiplication, division, addition, >> subtraction, and ⌈ceiling⌉ (order), >> for all of which geometric representations exist. >> >>> but that does not matter because >>> of course only a disproof is possible, >>> since there are no bijections. >> >> After all bijections are excluded, >> of course there are no bijections. >> >> On the other hand, >> ⎛ k = (i+j-1)⋅(i+j-2)/2+i >> ⎜ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2 >> ⎝ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k >> exists. >> >>>> Setting aside for a moment >>>> what you _think_ Cantor's bijection is, >>>> what part of _that_ >>>> is impossible to represent geometrically? >>> >>> It is impossible to cover the matrix >>> XOOO... >>> XOOO... >>> XOOO... >>> XOOO... >>> ... >>> by shuffling, shifting, reordering the X, >>> because they are not distinguishable. >> >> ⟨k,1⟩ ↦ ⟨i,j⟩ ↤ ⟨k,1⟩ >> >> ⎛ i = k-⌈(2⋅k+¼)¹ᐟ²-1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-3/2⌉/2 >> ⎜ j = ⌈(2⋅k+¼)¹ᐟ²+1/2⌉⋅⌈(2⋅k+¼)¹ᐟ²-1/2⌉/2-1-k >> ⎝ k = (i+j-1)⋅(i+j-2)/2+i >> >> Each ⟨k,1⟩ sends X to ⟨i,j⟩ >> Each ⟨i,j⟩ receives X from ⟨k,1⟩ >> >> According to geometry. >> Which I predict makes geometry wrong[WM], too. >> >> > > Non-standard models of integers exist. > > > Russell's retro-thesis "ordinary infinity" > is sort of a lie - if there's infinity > it's extra-ordinary - somebody like Dana Scott > had introduced "circle" and "box" modalities, > because he was into modal logic and relevance logic, > and "each" is not always "all". > > Of course it's well-known that any mere stipulation > is formally refutable, rather trivially. That > doesn't excuse absence of reason by any means. > > "Statistics" does not "predict", though > "guesses" I suppose may be said - > retro-troll. > > "A restriction of comprehension is not a truth."