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Groups > comp.lang.c++ > #67458 > unrolled thread
| Started by | peteolcott <Here@Home> |
|---|---|
| First post | 2019-08-08 12:15 -0500 |
| Last post | 2019-09-09 13:30 -0700 |
| Articles | 20 on this page of 100 — 16 participants |
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Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-08-08 12:15 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-08-08 21:28 +0100
Re: Provably unprovable eliminates incompleteness Dhu on Gate <campbell@neotext.ca> - 2019-09-08 06:06 +0000
Re: Provably unprovable eliminates incompleteness Stuart Redmann <DerTopper@web.de> - 2019-09-09 11:30 +0200
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 10:01 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibble@i42.removethisbit.co.uk> - 2019-09-09 21:38 +0100
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 18:05 -0500
Re: Provably unprovable eliminates incompleteness David Brown <david.brown@hesbynett.no> - 2019-09-10 09:01 +0200
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-10 10:38 -0500
Re: Provably unprovable eliminates incompleteness Reinhardt Behm <rbehm@hushmail.com> - 2019-09-11 15:31 +0800
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 12:04 -0500
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Reinhardt Behm <rbehm@hushmail.com> - 2019-09-12 10:20 +0800
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) peteolcott <Here@Home> - 2019-09-11 23:51 -0500
[OT] Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-12 04:20 -0700
Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) Keith Thompson <kst-u@mib.org> - 2019-09-12 01:22 -0700
[OT] Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-11 19:52 -0700
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-12 22:07 +0000
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-12 18:53 -0500
Re: Provably unprovable eliminates incompleteness Keith Thompson <kst-u@mib.org> - 2019-09-12 17:47 -0700
[OT] Re: Provably unprovable eliminates incompleteness James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-12 19:00 -0700
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-13 08:14 +0000
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-13 09:35 -0500
Re: Provably unprovable eliminates incompleteness Mr Flibble <flibble@i42.removethisbit.co.uk> - 2019-09-13 15:55 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 13:35 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-14 22:52 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 17:18 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-14 23:30 +0100
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] peteolcott <Here@Home> - 2019-09-14 17:42 -0500
Re: Provably unprovable eliminates incompleteness--- [ Cantor was wrong about cardinality ] Jeff Barnett <jbb@notatt.com> - 2019-09-14 18:12 -0600
Re: Provably unprovable eliminates incompleteness Juha Nieminen <nospam@thanks.invalid> - 2019-09-15 08:09 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 09:50 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 17:13 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 11:24 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 17:30 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 12:08 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 18:15 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 13:35 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 20:02 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 14:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-16 22:26 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 18:11 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-16 18:12 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] guinness.tony@gmail.com - 2019-09-17 00:16 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-16 17:54 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-17 09:10 -0400
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-17 08:11 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 11:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-17 21:10 +0200
Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 14:30 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-17 08:05 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 11:45 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-17 18:14 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 12:36 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 19:45 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-17 20:09 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 14:14 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 18:59 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Keith Thompson <kst-u@mib.org> - 2019-09-17 11:10 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-17 13:48 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Keith Thompson <kst-u@mib.org> - 2019-09-17 12:01 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-18 07:35 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Öö Tiib <ootiib@hot.ee> - 2019-09-18 04:22 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-18 15:41 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 10:02 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-18 22:25 +0100
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 16:41 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-18 22:56 +0100
[OT] Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] James Kuyper <jameskuyper@alumni.caltech.edu> - 2019-09-18 19:59 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-18 21:34 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-18 16:42 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-19 17:54 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 13:26 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:06 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:37 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 16:07 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:04 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 17:32 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:10 +0000
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:33 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 16:05 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:48 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] peteolcott <Here@Home> - 2019-09-19 18:00 -0500
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-20 08:52 +0200
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-20 00:10 -0700
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] David Brown <david.brown@hesbynett.no> - 2019-09-20 10:11 +0200
Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ] Juha Nieminen <nospam@thanks.invalid> - 2019-09-20 08:25 +0000
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 13:13 -0700
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:54 -0700
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-19 15:09 -0500
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:45 -0700
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:34 -0700
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] peteolcott <Here@Home> - 2019-09-19 14:58 -0500
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 15:57 -0700
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] peteolcott <Here@Home> - 2019-09-19 18:10 -0500
Re: Provably unprovable eliminates incompleteness [ COMPLETE-2 ] Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> - 2019-09-20 13:18 +0100
Re: Provably unprovable eliminates incompleteness Louis Valence <lvalence@example.com> - 2019-09-09 17:43 -0300
Re: Provably unprovable eliminates incompleteness Siri Cruise <chine.bleu@yahoo.com> - 2019-09-19 12:28 -0700
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-19 15:04 -0500
Re: Provably unprovable eliminates incompleteness peteolcott <Here@Home> - 2019-09-09 09:57 -0500
Re: Provably unprovable eliminates incompleteness Keith Thompson <kst-u@mib.org> - 2019-09-09 13:30 -0700
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-08-08 12:15 -0500 |
| Subject | Provably unprovable eliminates incompleteness |
| Message-ID | <YdmdnSGX1cpdxdHAnZ2dnUU7-WXNnZ2d@giganews.com> |
"This sentence is unprovable" can be proven to be unprovable on the basis that its satisfaction derives a contradiction. Proof that Wittgenstein is correct about Gödel https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Mr Flibble <flibbleREMOVETHISBIT@i42.co.uk> |
|---|---|
| Date | 2019-08-08 21:28 +0100 |
| Message-ID | <3C%2F.5$3s7.3@fx01.fr7> |
| In reply to | #67458 |
On 08/08/2019 18:15, peteolcott wrote: > "This sentence is unprovable" can be proven to be unprovable > on the basis that its satisfaction derives a contradiction. > > Proof that Wittgenstein is correct about Gödel > https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel Modulo the Sausage Conjecture of course. /Flibble -- "Snakes didn't evolve, instead talking snakes with legs changed into snakes." - Rick C. Hodgin “You won’t burn in hell. But be nice anyway.” – Ricky Gervais “I see Atheists are fighting and killing each other again, over who doesn’t believe in any God the most. Oh, no..wait.. that never happens.” – Ricky Gervais "Suppose it's all true, and you walk up to the pearly gates, and are confronted by God," Bryne asked on his show The Meaning of Life. "What will Stephen Fry say to him, her, or it?" "I'd say, bone cancer in children? What's that about?" Fry replied. "How dare you? How dare you create a world to which there is such misery that is not our fault. It's not right, it's utterly, utterly evil." "Why should I respect a capricious, mean-minded, stupid God who creates a world that is so full of injustice and pain. That's what I would say."
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| From | Dhu on Gate <campbell@neotext.ca> |
|---|---|
| Date | 2019-09-08 06:06 +0000 |
| Message-ID | <ql25og$fnm$1@news.albasani.net> |
| In reply to | #67458 |
On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote: > "This sentence is unprovable" can be proven to be unprovable > on the basis that its satisfaction derives a contradiction. > > Proof that Wittgenstein is correct about Gödel > https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel Since this is unreadable by peons, I can only speculate, but provably unprovable sounds like a useless tautology. Dhu -- Je suis Canadien. Ce n'est pas Francais ou Anglaise. C'est une esp`ece de sauvage: ne obliviscaris, vix ea nostra voco;-) http://babayaga.neotext.ca/PublicKeys/Duncan_Patton_a_Campbell_pubkey.txt
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| From | Stuart Redmann <DerTopper@web.de> |
|---|---|
| Date | 2019-09-09 11:30 +0200 |
| Message-ID | <581781046.589713241.741596.DerTopper-web.de@news.eternal-september.org> |
| In reply to | #68279 |
Dhu on Gate <campbell@neotext.ca> wrote: > On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote: > >> "This sentence is unprovable" can be proven to be unprovable >> on the basis that its satisfaction derives a contradiction. >> >> Proof that Wittgenstein is correct about Gödel >> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel > > > Since this is unreadable by peons, I can only speculate, > but provably unprovable sounds like a useless tautology. > > Dhu > It isn't. It rather shows that - like the speed of light in physics - there are some rather unexpected limitations to what is possible in mathematics. For many years mathematicians believed that, given a set of axioms, every possible theorem in the given field can be proven to be either true or false based on logical deduction from these axioms. Goedel showed that this is not always the case (see the continuum hypothesis). This implies that there are things in mathematics that cannot be proven. This culminated in Goedels incompleteness theorem, which is considered by many mathematicians as a severe throwback in maths. Regards, Stuart
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-09 10:01 -0500 |
| Message-ID | <T9mdnX__WZbR9OvAnZ2dnUU7-R_NnZ2d@giganews.com> |
| In reply to | #68283 |
On 9/9/2019 4:30 AM, Stuart Redmann wrote: > Dhu on Gate <campbell@neotext.ca> wrote: >> On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote: >> >>> "This sentence is unprovable" can be proven to be unprovable >>> on the basis that its satisfaction derives a contradiction. >>> >>> Proof that Wittgenstein is correct about Gödel >>> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel >> >> >> Since this is unreadable by peons, I can only speculate, >> but provably unprovable sounds like a useless tautology. >> >> Dhu >> > > It isn't. It rather shows that - like the speed of light in physics - there > are some rather unexpected limitations to what is possible in mathematics. > For many years mathematicians believed that, given a set of axioms, every > possible theorem in the given field can be proven to be either true or > false based on logical deduction from these axioms. > > Goedel showed that this is not always the case (see the continuum > hypothesis). This implies that there are things in mathematics that cannot > be proven. This culminated in Goedels incompleteness theorem, which is > considered by many mathematicians as a severe throwback in maths. > > Regards, > Stuart > https://en.wikipedia.org/wiki/Continuum_hypothesis It turns out that the basis concept of cardinality is ill-formed. All infinities are the same size. The number of real numbers is the number of adjacent geometric points on a number line, countable using integers. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Mr Flibble <flibble@i42.removethisbit.co.uk> |
|---|---|
| Date | 2019-09-09 21:38 +0100 |
| Message-ID | <dLydF.116034$Gz5.16238@fx08.fr7> |
| In reply to | #68285 |
On 09/09/2019 16:01, peteolcott wrote: > All infinities are the same size. Nonsense. Infinity doesn't have a size by definition. /Flibble -- “You won’t burn in hell. But be nice anyway.” – Ricky Gervais “I see Atheists are fighting and killing each other again, over who doesn’t believe in any God the most. Oh, no..wait.. that never happens.” – Ricky Gervais "Suppose it's all true, and you walk up to the pearly gates, and are confronted by God," Bryne asked on his show The Meaning of Life. "What will Stephen Fry say to him, her, or it?" "I'd say, bone cancer in children? What's that about?" Fry replied. "How dare you? How dare you create a world to which there is such misery that is not our fault. It's not right, it's utterly, utterly evil." "Why should I respect a capricious, mean-minded, stupid God who creates a world that is so full of injustice and pain. That's what I would say."
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-09 18:05 -0500 |
| Message-ID | <MfCdnXtPEYlXR-vAnZ2dnUU7-UnNnZ2d@giganews.com> |
| In reply to | #68288 |
On 9/9/2019 3:38 PM, Mr Flibble wrote: > On 09/09/2019 16:01, peteolcott wrote: >> All infinities are the same size. > > Nonsense. Infinity doesn't have a size by definition. > > /Flibble > Congratulations your answer is more precisely accurate than mine. I did not provide the more precise answer because it would have been to much for the target audience to handle. So Cantor was not just wrong about differing cardinality he was dead wrong. One can count all of the (otherwise uncountable) real numbers by merely counting all of the adjacent geometric points on a number line. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | David Brown <david.brown@hesbynett.no> |
|---|---|
| Date | 2019-09-10 09:01 +0200 |
| Message-ID | <ql7hnn$52u$1@dont-email.me> |
| In reply to | #68285 |
On 09/09/2019 17:01, peteolcott wrote: > > https://en.wikipedia.org/wiki/Continuum_hypothesis > It turns out that the basis concept of cardinality is ill-formed. No, it is not. Cardinality is a mathematical concept with a mathematical definition. It turns out that we can do useful and interesting things depending on certain extra features that we can choose to define, or not to define, but that does not make the definition ill-formed in any sense. > All infinities are the same size. Absolute drivel. > > The number of real numbers is the number of adjacent geometric > points on a number line, countable using integers. > The cardinality of the set of real numbers is /easily/ shown to be bigger than the cardinality of the integers (countable infinity). (Well, /easily/ if you know that kind of mathematics.) Since you posted a link to the Wikipedia article on the Continuum Hypothesis, did you actually attempt to read it? The second line sums up the hypothesis: """ There is no set whose cardinality is strictly between that of the integers and the real numbers. """ The implication from this is that you have at least two different sizes of infinity, the cardinality of integers (aleph_0) and the cardinality of real numbers (2 ^ aleph_0). Whether you accept the continuum hypothesis or not, you can't get past that sentence and still think those two cardinalities are the same size.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-10 10:38 -0500 |
| Message-ID | <pMqdnQFBSMfCXurAnZ2dnUU7-d3NnZ2d@giganews.com> |
| In reply to | #68293 |
On 9/10/2019 2:01 AM, David Brown wrote: > On 09/09/2019 17:01, peteolcott wrote: > >> >> https://en.wikipedia.org/wiki/Continuum_hypothesis >> It turns out that the basis concept of cardinality is ill-formed. > > No, it is not. Cardinality is a mathematical concept with a > mathematical definition. It turns out that we can do useful and > interesting things depending on certain extra features that we can > choose to define, or not to define, but that does not make the > definition ill-formed in any sense. > >> All infinities are the same size. > > Absolute drivel. > >> >> The number of real numbers is the number of adjacent geometric >> points on a number line, countable using integers. >> > > The cardinality of the set of real numbers is /easily/ shown to be > bigger than the cardinality of the integers (countable infinity). > (Well, /easily/ if you know that kind of mathematics.) > Yes I am aware of diagonal argument. You did not pay attention that I just refuted this: >> The number of real numbers is the number of adjacent geometric >> points on a number line, countable using integers. The next real number after 3.0 is the geometric point on the number line that is immediately adjacent to 3.0 on its right side. This geometric point is an infinitesimally larger than 3.0: encoded as the first point in the interval: (3,4] https://en.wikipedia.org/wiki/Bracket_(mathematics) > Since you posted a link to the Wikipedia article on the Continuum > Hypothesis, did you actually attempt to read it? The second line sums > up the hypothesis: If the concept of cardinality is bogus, then any theorem based on a bogus concept is also bogus. > > """ > There is no set whose cardinality is strictly between that of the > integers and the real numbers. > """ > > The implication from this is that you have at least two different sizes > of infinity, the cardinality of integers (aleph_0) and the cardinality > of real numbers (2 ^ aleph_0). Whether you accept the continuum > hypothesis or not, you can't get past that sentence and still think > those two cardinalities are the same size. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Reinhardt Behm <rbehm@hushmail.com> |
|---|---|
| Date | 2019-09-11 15:31 +0800 |
| Message-ID | <qla7sa$d23$1@dont-email.me> |
| In reply to | #68298 |
On 9/10/19 11:38 PM, peteolcott wrote: > The next real number after 3.0 is the geometric point on the number > line that is immediately adjacent to 3.0 on its right side. This geometric > point is an infinitesimally larger than 3.0: encoded as the first point in > the interval: (3,4] And between 3.0 and your "next" number (xnext) to the right is another real number: (3.0+xnext) / 2.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-11 12:04 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <er6dndJ1p8WItOTAnZ2dnUU7-d_NnZ2d@giganews.com> |
| In reply to | #68303 |
On 9/11/2019 2:31 AM, Reinhardt Behm wrote: > On 9/10/19 11:38 PM, peteolcott wrote: >> The next real number after 3.0 is the geometric point on the number >> line that is immediately adjacent to 3.0 on its right side. This geometric >> point is an infinitesimally larger than 3.0: encoded as the first point in >> the interval: (3,4] > > And between 3.0 and your "next" number (xnext) to the right is another real number: (3.0+xnext) / 2. > I specified a whole Infinitesimal number system: Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number of geometric points offset from the real part. 0.0[1,∞] specifies the entire set of geometric points with a value greater than zero on the number line which self-evidently corresponds to the entire set of positive reals. Notice that this entire set is countable using the set of positive integers. -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Reinhardt Behm <rbehm@hushmail.com> |
|---|---|
| Date | 2019-09-12 10:20 +0800 |
| Subject | Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <qlca11$5fp$1@dont-email.me> |
| In reply to | #68336 |
On 9/12/19 1:04 AM, peteolcott wrote: > On 9/11/2019 2:31 AM, Reinhardt Behm wrote: >> On 9/10/19 11:38 PM, peteolcott wrote: >>> The next real number after 3.0 is the geometric point on the number >>> line that is immediately adjacent to 3.0 on its right side. This >>> geometric >>> point is an infinitesimally larger than 3.0: encoded as the first >>> point in >>> the interval: (3,4] >> >> And between 3.0 and your "next" number (xnext) to the right is another >> real number: (3.0+xnext) / 2. >> > > > I specified a whole Infinitesimal number system: > Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number > of geometric points offset from the real part. > > 0.0[1,∞] specifies the entire set of geometric points with a value > greater than zero on the number line which self-evidently corresponds > to the entire set of positive reals. > > Notice that this entire set is countable using the set of positive > integers. If they are countable you can always list them in the counted order. With my above example I can always construct numbers that are not in your counted set. This disproves that you have counted them all. It disproves that they are countable at all.
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-11 23:51 -0500 |
| Subject | Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <8bGdnYE-yO1FU-TAnZ2dnUU7-SXNnZ2d@giganews.com> |
| In reply to | #68347 |
On 9/11/2019 9:20 PM, Reinhardt Behm wrote: > On 9/12/19 1:04 AM, peteolcott wrote: >> On 9/11/2019 2:31 AM, Reinhardt Behm wrote: >>> On 9/10/19 11:38 PM, peteolcott wrote: >>>> The next real number after 3.0 is the geometric point on the number >>>> line that is immediately adjacent to 3.0 on its right side. This geometric >>>> point is an infinitesimally larger than 3.0: encoded as the first point in >>>> the interval: (3,4] >>> >>> And between 3.0 and your "next" number (xnext) to the right is another real number: (3.0+xnext) / 2. >>> >> >> >> I specified a whole Infinitesimal number system: >> Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number >> of geometric points offset from the real part. >> >> 0.0[1,∞] specifies the entire set of geometric points with a value >> greater than zero on the number line which self-evidently corresponds >> to the entire set of positive reals. >> >> Notice that this entire set is countable using the set of positive integers. > > If they are countable you can always list them in the counted order. With my above example I can always construct numbers that are not in your counted set. You can construct a number that is not on any point of the number line? In other words you can construct a number THAT IS NOT A NUMBER. That would be a (categorically impossible) neat trick. > This disproves that you have counted them all. It disproves that they are countable at all. > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | James Kuyper <jameskuyper@alumni.caltech.edu> |
|---|---|
| Date | 2019-09-12 04:20 -0700 |
| Subject | [OT] Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <2a2faf38-db00-4d90-b332-c7a866265549@googlegroups.com> |
| In reply to | #68350 |
On Thursday, September 12, 2019 at 12:51:50 AM UTC-4, peteolcott wrote: > On 9/11/2019 9:20 PM, Reinhardt Behm wrote: > > On 9/12/19 1:04 AM, peteolcott wrote: > >> On 9/11/2019 2:31 AM, Reinhardt Behm wrote: > >>> On 9/10/19 11:38 PM, peteolcott wrote: > >>>> The next real number after 3.0 is the geometric point on the number > >>>> line that is immediately adjacent to 3.0 on its right side. This geometric > >>>> point is an infinitesimally larger than 3.0: encoded as the first point in > >>>> the interval: (3,4] > >>> > >>> And between 3.0 and your "next" number (xnext) to the right is another real number: (3.0+xnext) / 2. > >>> > >> > >> > >> I specified a whole Infinitesimal number system: > >> Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number > >> of geometric points offset from the real part. > >> > >> 0.0[1,∞] specifies the entire set of geometric points with a value > >> greater than zero on the number line which self-evidently corresponds > >> to the entire set of positive reals. > >> > >> Notice that this entire set is countable using the set of positive integers. > > > > If they are countable you can always list them in the counted order. With my above example I can always construct numbers that are not in your counted set. > > You can construct a number that is not on any point of the number line? No, he can construct a real number that is on the number line, but is not a member of your set, proving that your set does not contain all of the real numbers on the number line. He only refers to the countable property of your set, which suggests that he might be thinking of Cantor's diagonal argument, but your description of this set implies that the set is ordered: if n < m, then x[n] < x[m]. Is that correct? If so, then constructing such a number is much simpler. Such a set cannot even contain all of the rationals, which is a countable set. If 0.0[1] is a positive real number distinct from 0.0, than (0.0 + 0.0[1])/2 is also a positive real number distinct from both 0.0 and 0.0[1]. If your set includes all positive real numbers, there must be a positive integral value of n such that 0.0[n] == (0.0 + 0.0[1])/2. What is that value?
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-12 01:22 -0700 |
| Subject | Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <lnwoedc4a0.fsf@kst-u.example.com> |
| In reply to | #68347 |
Reinhardt Behm <rbehm@hushmail.com> writes:
> On 9/12/19 1:04 AM, peteolcott wrote:
[...]
>> Notice that this entire set is countable using the set of positive
>> integers.
>
> If they are countable you can always list them in the counted order.
[...]
Reinhardt, let me urge you to take a look at comp.theory before
deciding whether it's worth your time to discuss anything with
Pete Olcott.
Either way, please drop comp.lang.c++ from the newsgroups list,
and consider whether the readers of comp.ai.philosophy want to read
about this.
--
Keith Thompson (The_Other_Keith) kst-u@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */
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| From | James Kuyper <jameskuyper@alumni.caltech.edu> |
|---|---|
| Date | 2019-09-11 19:52 -0700 |
| Subject | [OT] Re: Provably unprovable eliminates incompleteness (cardinality is incorrect) |
| Message-ID | <43f2f824-c1b8-43a4-a624-15f6bfba2f78@googlegroups.com> |
| In reply to | #68336 |
On Wednesday, September 11, 2019 at 1:04:36 PM UTC-4, peteolcott wrote: ... > I specified a whole Infinitesimal number system: I see no such specification in this thread. The term "whole infinitesimal number system" is made up of words that I understand very well, but in combination, they don't mean anything to me. They don't mean anything to Wikipedia or Google, either. The only hit I got with either search engine was your use of that term in this very discussion. Congratulations: you used a phrase that was a googlenope until you used it. > Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number > of geometric points offset from the real part. That notation is unfamiliar to me as well, and that definition seems meaningless as well. There's an infinite number of geometric points on the number line between any two distinct real numbers. But I presume that you could define an alternative mathematics where that isn't true, and perhaps that notation is meant to be interpreted within the context of some such alternative system? > 0.0[1,∞] specifies the entire set of geometric points with a value > greater than zero on the number line which self-evidently corresponds > to the entire set of positive reals. It's not particularly self-evident to me. That the set you're referring to can be described in that fashion seems self-evidently to imply that the set you're referring to is NOT the entire set of positive reals. However, it's entirely possible that I've misunderstood something, so I'll ask you a few questions about your notation. 1. Does 0.0[n] denote a real number for all positive integral values of n? 2. What is the precise numerical value of 0.0[1]? Please use some more conventional notation, such as a decimal expansion or an algebraic expression to answer that question. 3. Just in case the answer to 2 is too esoteric for me to understand, is it true that 0.0 < 0.0[1]? Assuming that the answers to 1 and 3 are "yes", is there any non-negative integral value of n such that 0.0[n] = (0.0 + 0.0[1])/2? In particular, is true for either n=0 or n=1? If 0.0[1] denotes a real number distinct from 0.0, then (0.0 + 0.0[1])/2 must denote another real number, distinct from both 0 and 0.0[1]. I can't imagine how you could disagree with that statement, but I also don't understand how you could make the claims you've made in that message unless you did disagree with it. If you do disagree, please explain. > Notice that this entire set is countable using the set of positive integers. Regardless of what your notation actually means, by construction the set you describe is countable. My only objection is that it doesn't seem to possess an important characteristic of the set of real numbers, one that is shared by the set of rational numbers, which is countable: between any pair of distinct points in the set, there's an infinite number of other distinct points that are also in the set.
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| From | Juha Nieminen <nospam@thanks.invalid> |
|---|---|
| Date | 2019-09-12 22:07 +0000 |
| Message-ID | <qlefjm$300c$1@adenine.netfront.net> |
| In reply to | #68298 |
In comp.lang.c++ peteolcott <Here@home> wrote: > The next real number after 3.0 is the geometric point on the number > line that is immediately adjacent to 3.0 on its right side. This geometric > point is an infinitesimally larger than 3.0: encoded as the first point in > the interval: (3,4] That would be "the smallest real number larger than 3". Which does not exist. Infinitesimals (no matter how you want to define them) are not going to help you. That number still does not exist, even if you add infinitesimals to your number system. It doesn't matter how you try to twist it, it's not going to work. You are not going to make that number exist. Not with infinitesimals, not with anything. The set of real numbers is genuinely larger than the set of natural numbers because there is no bijection between the two sets. (This is in contrast with, for example, the set of rational numbers, or the set of algebraic numbers, which are equally large as the set of natural numbers because there is a bijection between them.)
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| From | peteolcott <Here@Home> |
|---|---|
| Date | 2019-09-12 18:53 -0500 |
| Message-ID | <oNudndY5IKcOR-fAnZ2dnUU7-T_NnZ2d@giganews.com> |
| In reply to | #68372 |
On 9/12/2019 5:07 PM, Juha Nieminen wrote: > In comp.lang.c++ peteolcott <Here@home> wrote: >> The next real number after 3.0 is the geometric point on the number >> line that is immediately adjacent to 3.0 on its right side. This geometric >> point is an infinitesimally larger than 3.0: encoded as the first point in >> the interval: (3,4] > > That would be "the smallest real number larger than 3". Which does not > exist. > > Infinitesimals (no matter how you want to define them) are not going > to help you. That number still does not exist, even if you add > infinitesimals to your number system. It doesn't matter how you try > to twist it, it's not going to work. You are not going to make that > number exist. Not with infinitesimals, not with anything. > There is a contiguous set of points between (3.0, 4.0]. There is a first point in that interval, let's call it X. Every point on the number line represents some number therefore the first point X in the above interval represents a number, it might not be a real number, or any other conventional named type of number. X is a number on the number line. > The set of real numbers is genuinely larger than the set of natural > numbers because there is no bijection between the two sets. (This > is in contrast with, for example, the set of rational numbers, or > the set of algebraic numbers, which are equally large as the set of > natural numbers because there is a bijection between them.) > -- Copyright 2019 Pete Olcott All rights reserved "Great spirits have always encountered violent opposition from mediocre minds." Albert Einstein
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| From | Keith Thompson <kst-u@mib.org> |
|---|---|
| Date | 2019-09-12 17:47 -0700 |
| Message-ID | <lno8zpaupi.fsf@kst-u.example.com> |
| In reply to | #68373 |
peteolcott <Here@Home> writes:
[SNIP]
Pete, will you please consider *not* posting this to comp.lang.c++
and other newsgroups where it's off-topic? You've already taken
over comp.theory, and anyone who's interested can follow you there.
(I've cross-posted and redirected followups to comp.theory.)
Note to other readers: This is my first *and last* attempt to ask
Pete not to post to off-topic newsgroups.
--
Keith Thompson (The_Other_Keith) kst-u@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */
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| From | James Kuyper <jameskuyper@alumni.caltech.edu> |
|---|---|
| Date | 2019-09-12 19:00 -0700 |
| Subject | [OT] Re: Provably unprovable eliminates incompleteness |
| Message-ID | <4c5fbccb-0a4b-4970-9339-922a19ccb6f6@googlegroups.com> |
| In reply to | #68373 |
On Thursday, September 12, 2019 at 7:54:09 PM UTC-4, peteolcott wrote: ... > There is a contiguous set of points between (3.0, 4.0]. > There is a first point in that interval, let's call it X. The concept that you can identify a first point in that interval inherently leads to a contradiction. If X is in that interval, then 3.0 < X. Therefore,(3.0+X)/2.0 has a value that is manifestly greater than 3.0, so it should be in that interval, but is manifestly less than X, so it should occur earlier in that interval than X. Unless and until you say something to address that argument, I'll pay no further attention to you - I've already paid more attention than I should.
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