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Re: Expressability in the notation of set theory

Newsgroups sci.logic
Date 2023-06-07 22:11 -0700
References <1941110f-b45f-4e42-8290-9f7949082816n@googlegroups.com>
Message-ID <599bf252-8d5e-4995-9d64-1bd69751d9b2n@googlegroups.com> (permalink)
Subject Re: Expressability in the notation of set theory
From Dan Christensen <Dan_Christensen@sympatico.ca>

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On Wednesday, June 7, 2023 at 11:25:40 PM UTC-4, Dan Christensen wrote:
> The following from Terence Tao's blog dated 2007-08-27: 
> 
> "It seems that one cannot express 
> 
> "'For every x and x’, there exists a y depending only on x and a y’ depending only on x’ such that Q(x,x’,y,y’) is true' 
> 
> "in first order logic!" 
> 
> https://terrytao.wordpress.com/2007/08/27/printer-friendly-css-and-nonfirstorderizability/#more-172 
> 
> Following is my attempt to formalize this statement using the notation of set theory in DC Proof: 
> 
> ALL(a):[a in u => EXIST(b):[b in u & (a,b) in r]] 
> 
> & ALL(x):ALL(y):[x in u & y in u => ALL(a):ALL(b):ALL(c):ALL(d):[(a,b,c,d) in q 
> <=> (a,b,c,d) in u4 & [a=x & b=y & (x,c) in r & (y,d) in r]]] 
> 
> where u4 = u*u*u*u (Cartesian product) 
> 

For a simpler example of why predicate logic (FOL) is insufficient, and why we need something like set theory, see my blog posting "The Barber Paradox Revisited" at 
https://dcproof.wordpress.com/2017/01/11/the-barber-paradox-revisited-why-we-need-sets/

There, I present 3 variations of the Barber Paradox. In the final variation based on set theory, I prove that, in an village with a resident barber, that barber can shave those and only those men in the village who do not shave themselves if and only if that barber is not a man. This would be seem to be unprovable in FOL. It requires something like set theory.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

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Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-07 20:25 -0700
  Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-07 22:11 -0700
    Re: Expressability in the notation of set theory Julio Di Egidio <julio@diegidio.name> - 2023-06-07 23:34 -0700
      Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 07:31 -0700
  Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 00:51 -0700
    Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 01:03 -0700
      Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 01:19 -0700
      Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 08:04 -0700
        Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 11:10 -0700
          Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 11:40 -0700
          Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 11:51 -0700
            Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 12:16 -0700
              Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 12:34 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 13:47 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 13:57 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 14:11 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 14:42 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 14:45 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 14:51 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 14:57 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 15:04 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 15:09 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 15:28 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 15:48 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-08 15:54 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 19:43 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 19:32 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 14:38 -0700
  Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-08 21:19 -0700
    Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 00:21 -0700
      Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 00:28 -0700
        Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 00:34 -0700
          Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 00:48 -0700
      Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-09 06:57 -0700
        Re: Expressability in the notation of set theory Mild Shock <janburse@fastmail.fm> - 2023-06-09 21:04 +0200
          Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-09 13:28 -0700
            Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 15:42 -0700
              Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 15:58 -0700
                Re: Expressability in the notation of set theory Ross Finlayson <ross.a.finlayson@gmail.com> - 2023-06-10 13:36 -0700
              Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-09 18:17 -0700
              Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-09 18:20 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 04:30 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 04:46 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-10 10:52 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 11:06 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 11:11 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 11:17 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-12 00:37 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-12 00:38 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 07:22 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-12 08:17 -0700
                Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-10 11:21 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 11:27 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-10 15:10 -0700
                Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-11 02:02 -0700
    Re: Expressability in the notation of set theory Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-09 12:03 -0700
      Re: Expressability in the notation of set theory Mild Shock <janburse@fastmail.fm> - 2023-06-09 21:12 +0200
        Re: Expressability in the notation of set theory Mild Shock <janburse@fastmail.fm> - 2023-06-09 21:22 +0200
          Re: Expressability in the notation of set theory Mild Shock <bursejan@gmail.com> - 2023-06-09 12:33 -0700

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