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Expressability in the notation of set theory -- Follow-up

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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(x):ALL(x'):[x e u & x' e u  => EXIST(y):EXIST(y'):[y e u & (x,y) e r1 & y' e u & (x',y') e r2 => Q(x,x',y,y')]]

Where 

Set'(r1) & ALL(b):ALL(c):[(b,c) in r1 => b e u & c in u] & ALL(b):[b in u => EXIST(c):[c in u & (b,c) in r1]]

Set'(r2) & ALL(b):ALL(c):[(b,c) in r2 => b e u & c in u] & ALL(b):[b in u => EXIST(c):[c in u & (b,c) in r2]]

https://www.dcproof.com/Tao2ndOrderLogic.htm (25 lines + lemma)

Comments?

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 -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-11 18:25 -0700
  Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 00:28 -0700
    Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 03:36 -0700
      Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 07:12 -0700
    Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 07:04 -0700
      Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 08:16 -0700
        Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 08:48 -0700
          Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 08:56 -0700
            Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 08:59 -0700
              Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 10:16 -0700
        Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 10:05 -0700
  Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 09:58 -0700
  Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-12 15:04 -0700
    Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 16:03 -0700
      Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 16:27 -0700
        Re: Expressability in the notation of set theory -- Follow-up Mild Shock <bursejan@gmail.com> - 2023-06-12 16:37 -0700
    Re: Expressability in the notation of set theory -- Follow-up Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-06-13 08:09 -0700

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