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Re: Why is model theory needed?

Newsgroups sci.logic
Date 2023-05-31 11:58 -0700
References (13 earlier) <ab1f8028-11d0-4442-bb83-fe620289a551n@googlegroups.com> <7d2ce2bb-efb1-4bbe-abe0-9e92a0b32dcfn@googlegroups.com> <2f4d8289-0602-4517-a217-210682a0c14cn@googlegroups.com> <df1b3f91-45de-44e6-b7e1-f8d28d15b35fn@googlegroups.com> <92fb36db-7b5b-4267-b182-155ba2a0d03bn@googlegroups.com>
Message-ID <3d8138a9-d276-4171-9e0e-9a6a599b2c16n@googlegroups.com> (permalink)
Subject Re: Why is model theory needed?
From Mild Shock <bursejan@gmail.com>

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The specialization-generalization sign:

> Unicode Character “⊊” (U+228A) 
> https://www.compart.com/en/unicode/U+228A 

Has a technical name, it is called "Subset of with Not
Equal To" in Unicode, it hints that it can be defined as follows:

P ⊊ Q  <=> P ⊆ Q & P =\= Q

But mathematician usually simply say "proper subset"
for ⊊, in contrast to "subset" for ⊆. You can also
write it down with P and Q predicates, first observe

that P = Q <=> P ⊆ Q & Q ⊆ P. Therefore P =\= Q <=>
~(P ⊆ Q) v ~(Q ⊆ P). And therefore P ⊊ Q  <=> P ⊆ Q &
~(Q ⊆ P). So an alternative statement would be:

P ⊊ Q  <=> P ⊆ Q & Q ⊈ P

Which reads with predicates:

P ⊊ Q  <=> ALL(x):[P(x) ⊆ Q(x)] & EXIST(x):[Q(x) & ~P(x)]

If Q=U is the domain of discourse, i.e. ALL(x):U(x), the above 
reduces to, lets also use a set P=s for the other predicate:

s ⊊ U <=> ALL(x):[x e s => U(x)] & EXIST(x):[U(x) & ~(x e s)]

Or simplified, since from ALL(x):U(x) we have ALL(x):[x e s => U(x)] 
anyway true, and EXIST(x):[U(x) & ~(x e s)] doesn't need the U(x). 

So we get:

s ⊊ U <=> EXIST(x):[~(x e s)]

Thats what you proved here, every set s is a specialization
of the domain of discourse U:

http://www.dcproof.com/UniversalSet.htm

Very easy!

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Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-30 15:05 -0700
  Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-30 15:14 -0700
    Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-30 15:57 -0700
    Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-30 18:45 -0700
      Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 01:42 -0700
        Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 01:55 -0700
          Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 02:09 -0700
            Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 08:21 -0700
              Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 08:28 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 08:39 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 11:50 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 11:58 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 13:10 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 21:12 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 21:35 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-01 11:43 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 12:40 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 12:54 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-01 14:24 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 14:55 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 15:18 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-03 08:31 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-03 09:10 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-02 08:19 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-02 11:20 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-03 08:27 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-03 08:31 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-03 13:53 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-03 13:57 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-03 14:04 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-03 14:13 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-04 07:47 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-04 08:26 -0700
                Re: Why is model theory needed? Mild Shock <janburse@fastmail.fm> - 2023-06-04 17:53 +0200
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-04 15:31 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-04 17:21 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-04 17:30 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-04 17:39 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-04 17:49 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-05 08:20 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-05 10:56 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-05 13:16 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-05 13:17 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:10 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 14:07 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-05 15:09 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-05 17:51 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-05 22:54 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-06 18:44 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-06 21:02 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-07 11:42 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-07 13:48 -0700
                Re: Why is model theory needed? Mild Shock <janburse@fastmail.fm> - 2023-06-07 23:43 +0200
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-08 12:14 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-08 12:19 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 10:54 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 10:54 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 11:35 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-09 11:53 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 11:58 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:27 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-09 11:59 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:27 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-06 18:43 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 05:18 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 05:20 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 10:53 -0700
                Re: Why is model theory needed? Mild Shock <janburse@fastmail.fm> - 2023-06-09 21:28 +0200
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:27 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:28 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 15:45 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:51 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-09 15:51 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-09 15:57 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 16:03 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 16:09 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-09 16:19 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-09 16:20 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 16:21 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 16:28 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-09 16:28 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-10 08:25 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-10 09:08 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-10 09:13 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-10 12:52 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-10 13:21 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-11 08:25 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-11 08:44 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-11 09:06 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-11 10:55 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-11 11:54 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-11 13:12 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-11 13:59 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-11 15:26 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-11 15:31 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-11 15:34 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-12 08:31 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-12 08:39 -0700
                Re: Why is model theory needed? Jeffrey Rubard <theleasthappyfella@gmail.com> - 2023-06-12 08:56 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-12 09:00 -0700
                Re: Why is model theory needed? Jeffrey Rubard <theleasthappyfella@gmail.com> - 2023-06-12 09:07 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-12 09:22 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-12 09:37 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-12 08:31 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-12 08:30 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-06-10 12:51 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-10 15:13 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-10 15:19 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-11 08:25 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-10 08:24 -0700
              Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 13:05 -0700
        Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 08:15 -0700
          Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 11:41 -0700
            Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 11:48 -0700
            Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 12:52 -0700
              Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 13:06 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 13:28 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-05-31 18:53 -0700
              Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 13:18 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 13:53 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 13:58 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 14:06 -0700
                Re: Why is model theory needed? Jeffrey Rubard <jeffreydanielrubard@gmail.com> - 2023-05-31 14:08 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 15:57 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-05-31 16:46 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-31 17:31 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 02:48 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 03:02 -0700
                Re: Why is model theory needed? Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-01 08:19 -0700
                Re: Why is model theory needed? Jeffrey Rubard <rehashedrubard@gmail.com> - 2023-06-01 08:42 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-01 07:07 -0700
                Re: Why is model theory needed? Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-01 11:03 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 11:08 -0700
                Re: Why is model theory needed? Mild Shock <bursejan@gmail.com> - 2023-06-01 11:14 -0700

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