Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ]

From	David Brown <david.brown@hesbynett.no>
Newsgroups	comp.lang.c++, comp.ai.philosophy, comp.lang.prolog, comp.lang.lisp
Subject	Re: Provably unprovable eliminates incompleteness [ All infinite sets have the same cardinality ]
Date	2019-09-17 21:10 +0200
Organization	A noiseless patient Spider
Message-ID	<qlrb3p$ipv$1@dont-email.me> (permalink)
References	(8 earlier) <E7KdnWEIOuSUS-LAnZ2dnUU7-UPNnZ2d@giganews.com> <P_QfF.126886$3s7.5819@fx01.fr7> <4qqdnS6fIrqbeuLAnZ2dnUU7-cnNnZ2d@giganews.com> <qlq4f5$55s$1@adenine.netfront.net> <O9udnZZFPoiikxzAnZ2dnUU7-c1QAAAA@giganews.com>

Cross-posted to 4 groups.

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On 17/09/2019 18:48, peteolcott wrote:
> On 9/17/2019 3:11 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> Even if every other aspect of mathematics disagrees, none-the-less the
>>> beginning of this interval: (0, 1] stipulates the point immediately 
>>> after 0.
>>
>> There is no point immediately after 0. You might be incapable of
>> comprehending such a fact of mathematics, but that doesn't mean
>> it's incorrect.
>>
>> An open interval does not mean that there must exist a particular
>> number at the end of that interval. There is no such number. You cannot
>> give its value as it's conceptually and mathematically non-existent.
>> It doesn't exist even in some kind of abstract meta-sense. It doesn't
>> exist even if you introduce the concept of "infinitesimal" into your
>> number system (no matter how you define "infinitesimal").
>>
>> Positing that there exists such a number leads immediately to a
>> contradiction, and it's extraordinarily simple to prove that.
>> Since the assumption was that the number exists, but that assumption
>> leads to a contradiction, that means that the assumption was incorrect.
>> "Proof by contradiction" is not just some wishy-washy nebulous term.
>>
> 
> 
> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>  > Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
>  > something or other.  It includes all the real numbers in (0,1],
>  > and it may also include some other things, each of which is greater
>  > than 0, less than or equal to 1, and somewhere on the "number line"
>  > between 0 and 1.  Some of the non-real elements might be points,
>  > or infinitesimals.  I'll assert that for any two members of *this*
>  > set, exactly one of a<b, a=b, a>b is true.
>  >
>  > I can imagine that *this* set has a least element.  For example
>  > we can define "iota" as an infinitesimal, something that is not a
>  > real number but that has a location on the number line, directly
>  > adjacent to the real number 0.  It is greater than 0 and less than
>  > any positive real number.
> 

You can always define extra elements and add them to sets like this. 
But you can't do so in a way that is consistent in other ways.  So you 
can make the ordered set of "real in (0, 1] plus iota", but you lose 
other properties of the reals.  "reals plus iota" is not a field, nor is 
it complete.

And it does not make sense to talk about "having a location on the 
number line" without defining "the number line" - normally that term is 
used precisely to mean the real numbers.

Thread

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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