Sujet : Re: Replacement of Cardinality (real-valued)
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.mathDate : 08. Aug 2024, 20:00:38
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <c9f82f99-6640-40f0-b14b-8e5424056433@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
User-Agent : Mozilla Thunderbird
On 8/7/2024 9:03 PM, Ross Finlayson wrote:
On 08/07/2024 01:06 PM, Jim Burns wrote:
A theory can describe more than one model.
>
Some claims have proofs.
Those claims are true in each model.
>
Some claims are true in each model.
Those claims have proofs.
(That is a very nice result, maybe not super.obvious.)
>
True.and.false in different models
does not make a theory contradictory.
The theory is silent, not wrong.
>
What theory?
| The completeness theorem applies to any first-order theory:
| If T is such a theory, and
| φ is a sentence (in the same language) and
| every model of T is a model of φ,
| then there is a (first-order) proof of φ
| using the statements of T as axioms.
| One sometimes says this as
| "anything true in all models is provable".
| (This does not contradict Gödel's incompleteness theorem,
| which is about a formula φᵤ that is unprovable
| in a certain theory T
| but true in the "standard" model of the natural numbers:
| φᵤ is false in some other, "non-standard" models of T.)
Model theory?
That's exactly what model theory not.is.
| The completeness theorem makes a close link between
| model theory,
| which deals with what is true in different models, and
| proof theory,
| which studies what can be formally proven
| in particular formal systems.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theoremGödel's completeness theorem
Date | Sujet | # | | Auteur |
27 Jul 24 | Re: Replacement of Cardinality | 902 | | Jim Burns |
28 Jul 24 |  Re: Replacement of Cardinality | 892 | | WM |
28 Jul 24 |   Re: Replacement of Cardinality | 891 | | Jim Burns |
29 Jul 24 |    Re: Replacement of Cardinality | 47 | | Ross Finlayson |
29 Jul 24 |     Re: Replacement of Cardinality | 46 | | Ross Finlayson |
29 Jul 24 |      Re: Replacement of Cardinality | 45 | | Ross Finlayson |
29 Jul 24 |       Re: Replacement of Cardinality | 44 | | Jim Burns |
29 Jul 24 |        Re: Replacement of Cardinality | 43 | | Ross Finlayson |
29 Jul 24 |         Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 14 | | Ross Finlayson |
1 Jan 25 |          Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 13 | | Ross Finlayson |
1 Jan 25 |           Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 12 | | Jim Burns |
2 Jan 25 |            Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 11 | | Ross Finlayson |
2 Jan 25 |             Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 4 | | Ross Finlayson |
2 Jan 25 |              Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 3 | | Jim Burns |
2 Jan 25 |               Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 2 | | Ross Finlayson |
2 Jan 25 |                Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 1 | | Jim Burns |
2 Jan 25 |             Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 6 | | Jim Burns |
2 Jan 25 |              Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 5 | | Ross Finlayson |
2 Jan 25 |               Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 2 | | Ross Finlayson |
3 Jan 25 |                Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 1 | | Ross Finlayson |
2 Jan 25 |               Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 2 | | Jim Burns |
3 Jan 25 |                Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum, linear continuum, continuity) | 1 | | Ross Finlayson |
29 Jul 24 |         Re: Replacement of Cardinality | 28 | | Jim Burns |
30 Jul 24 |          Re: Replacement of Cardinality (ubiquitous ordinals) | 27 | | Ross Finlayson |
30 Jul 24 |           Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Ross Finlayson |
30 Jul 24 |           Re: Replacement of Cardinality (ubiquitous ordinals) | 25 | | Jim Burns |
30 Jul 24 |            Re: Replacement of Cardinality (ubiquitous ordinals) | 24 | | Ross Finlayson |
30 Jul 24 |             Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Ross Finlayson |
31 Jul 24 |             Re: Replacement of Cardinality (ubiquitous ordinals) | 22 | | Jim Burns |
1 Aug 24 |              Re: Replacement of Cardinality (ubiquitous ordinals) | 21 | | Ross Finlayson |
1 Aug 24 |               Re: Replacement of Cardinality (ubiquitous ordinals) | 19 | | Jim Burns |
1 Aug 24 |                Re: Replacement of Cardinality (ubiquitous ordinals) | 18 | | Ross Finlayson |
2 Aug 24 |                 Re: Replacement of Cardinality (ubiquitous ordinals) | 17 | | Jim Burns |
2 Aug 24 |                  Re: Replacement of Cardinality (ubiquitous ordinals) | 16 | | Ross Finlayson |
2 Aug 24 |                   Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Jim Burns |
2 Aug 24 |                   Re: Replacement of Cardinality (ubiquitous ordinals) | 13 | | FromTheRafters |
2 Aug 24 |                    Re: Replacement of Cardinality (ubiquitous ordinals) | 12 | | Ross Finlayson |
3 Aug 24 |                     Re: Replacement of Cardinality (ubiquitous ordinals) | 11 | | Jim Burns |
4 Aug 24 |                      Re: Replacement of Cardinality (ubiquitous ordinals) | 10 | | Ross Finlayson |
4 Aug 24 |                       Re: Replacement of Cardinality (ubiquitous ordinals) | 9 | | Jim Burns |
4 Aug 24 |                        Re: Replacement of Cardinality (ubiquitous ordinals) | 8 | | Ross Finlayson |
4 Aug 24 |                         Re: Replacement of Cardinality (ubiquitous ordinals) | 7 | | Jim Burns |
4 Aug 24 |                          Re: Replacement of Cardinality (ubiquitous ordinals) | 6 | | Ross Finlayson |
4 Aug 24 |                           Re: Replacement of Cardinality (ubiquitous ordinals) | 5 | | Jim Burns |
4 Aug 24 |                            Re: Replacement of Cardinality (ubiquitous ordinals) | 4 | | Ross Finlayson |
4 Aug 24 |                             Re: Replacement of Cardinality (ubiquitous ordinals) | 3 | | Jim Burns |
4 Aug 24 |                              Re: Replacement of Cardinality (ubiquitous ordinals) | 2 | | Ross Finlayson |
4 Aug 24 |                               Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Jim Burns |
2 Aug 24 |                   Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Jim Burns |
1 Jan 25 |               Re: Replacement of Cardinality (ubiquitous ordinals) | 1 | | Ross Finlayson |
29 Jul 24 |    Re: Replacement of Cardinality | 842 | | WM |
29 Jul 24 |     Re: Replacement of Cardinality | 841 | | Jim Burns |
30 Jul 24 |      Re: Replacement of Cardinality | 840 | | WM |
30 Jul 24 |       Re: Replacement of Cardinality | 839 | | Jim Burns |
30 Jul 24 |        Re: Replacement of Cardinality | 13 | | Moebius |
31 Jul 24 |         Re: Replacement of Cardinality --- infinitesimal number system | 11 | | olcott |
31 Jul 24 |          Re: Replacement of Cardinality --- infinitesimal number system | 6 | | Moebius |
31 Jul 24 |           Re: Replacement of Cardinality --- infinitesimal number system | 5 | | olcott |
31 Jul 24 |            Re: Replacement of Cardinality --- infinitesimal number system | 1 | | Chris M. Thomasson |
31 Jul 24 |            Re: Replacement of Cardinality --- infinitesimal number system | 3 | | Moebius |
31 Jul 24 |             Re: Replacement of Cardinality --- infinitesimal number system | 2 | | olcott |
31 Jul 24 |              Re: Replacement of Cardinality --- infinitesimal number system | 1 | | Moebius |
31 Jul 24 |          Re: Replacement of Cardinality --- infinitesimal number system | 3 | | FromTheRafters |
31 Jul 24 |           Re: Replacement of Cardinality --- infinitesimal number system | 1 | | olcott |
31 Jul 24 |           Re: Replacement of Cardinality --- infinitesimal number system | 1 | | Ross Finlayson |
31 Jul 24 |          Re: Replacement of Cardinality --- infinitesimal number system | 1 | | Jim Burns |
31 Jul 24 |         Re: Replacement of Cardinality | 1 | | WM |
31 Jul 24 |        Re: Replacement of Cardinality | 825 | | WM |
31 Jul 24 |         Re: Replacement of Cardinality | 824 | | Jim Burns |
1 Aug 24 |          Re: Replacement of Cardinality | 823 | | WM |
1 Aug 24 |           Re: Replacement of Cardinality | 822 | | Jim Burns |
1 Aug 24 |            Re: Replacement of Cardinality | 1 | | Moebius |
2 Aug 24 |            Re: Replacement of Cardinality | 820 | | WM |
2 Aug 24 |             Re: Replacement of Cardinality | 819 | | Jim Burns |
2 Aug 24 |              Re: Replacement of Cardinality | 24 | | Moebius |
3 Aug 24 |               Re: Replacement of Cardinality | 23 | | WM |
3 Aug 24 |                Re: Replacement of Cardinality | 4 | | joes |
4 Aug 24 |                 Re: Replacement of Cardinality | 3 | | WM |
4 Aug 24 |                  Re: Replacement of Cardinality | 2 | | Jim Burns |
4 Aug 24 |                   Re: Replacement of Cardinality | 1 | | Moebius |
3 Aug 24 |                Re: Replacement of Cardinality | 18 | | Chris M. Thomasson |
3 Aug 24 |                 Re: Replacement of Cardinality | 15 | | FromTheRafters |
4 Aug 24 |                  Re: Replacement of Cardinality (real-valued) | 14 | | Ross Finlayson |
4 Aug 24 |                   Re: Replacement of Cardinality (real-valued) | 12 | | FromTheRafters |
4 Aug 24 |                    Re: Replacement of Cardinality (real-valued) | 11 | | Ross Finlayson |
4 Aug 24 |                     Re: Replacement of Cardinality (real-valued) | 10 | | Ross Finlayson |
4 Aug 24 |                      Re: Replacement of Cardinality (real-valued) | 9 | | Ross Finlayson |
4 Aug 24 |                       Re: Replacement of Cardinality (real-valued) | 3 | | Ross Finlayson |
4 Aug 24 |                        Re: Replacement of Cardinality (real-valued) | 1 | | Ross Finlayson |
4 Aug 24 |                        Re: Replacement of Cardinality (real-valued) | 1 | | Ross Finlayson |
7 Aug 24 |                       Re: Replacement of Cardinality (real-valued) | 5 | | Jim Burns |
8 Aug 24 |                        Re: Replacement of Cardinality (real-valued) | 4 | | Ross Finlayson |
8 Aug 24 |                         Re: Replacement of Cardinality (real-valued) | 3 | | Jim Burns |
8 Aug 24 |                          Re: Replacement of Cardinality (real-valued) | 2 | | Ross Finlayson |
13 Aug 24 |                           Re: Replacement of Cardinality (real-valued) | 1 | | Ross Finlayson |
30 Dec 24 |                   Re: Replacement of Cardinality (real-valued) | 1 | | Ross Finlayson |
4 Aug 24 |                 Re: Replacement of Cardinality | 2 | | Moebius |
4 Aug 24 |                  Re: Replacement of Cardinality | 1 | | WM |
3 Aug 24 |              Re: Replacement of Cardinality | 794 | | WM |
3 Aug 24 |               Re: Replacement of Cardinality | 793 | | Jim Burns |
4 Aug 24 |                Re: Replacement of Cardinality | 44 | | Moebius |
4 Aug 24 |                Re: Replacement of Cardinality | 748 | | WM |
29 Jul 24 |    Re: Replacement of Cardinality | 1 | | Moebius |
29 Jul 24 |  Re: Replacement of Cardinality | 9 | | Moebius |