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Re: Expressability in the notation of set theory

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Here are proofs that they are redundant, i.e. that 
some assumptions ∀x∃zR(x,z) or ∀y∃tS(y,t) are 
not needed, already implied for any R or S:

∀x∀y∃z(Rxz ∧ ∃t(Syt ∧ Qxyzt)) → ∀x∃zRxz is valid.
https://www.umsu.de/trees/#~6x~6y~7z%28Rxz~1~7t%28Syt~1Qxyzt%29%29~5~6x~7zRxz

∀x∀y∃z(Rxz ∧ ∃t(Syt ∧ Qxyzt)) → ∀y∃tSyt is valid.
https://www.umsu.de/trees/#~6x~6y~7z%28Rxz~1~7t%28Syt~1Qxyzt%29%29~5~6y~7tSyt

But still nothing of the blog of Terrence Tao was
proved or demonstrated. Since Terrence Tao aims at:

"For every real numbers ...
is true in the non-standard model of the real
numbers, but false in the standard model"
https://terrytao.wordpress.com/2007/08/27/printer-friendly-css-and-nonfirstorderizability/#more-172

An undecidable sentence, because he differs in two
models. Implicitly he thinks this should not happen, so
he thinks it should happen.

And his conclusion is "... this means that the above statement
cannot be expressed in first-order logic.". How do you formalize
such a potato. Thats quite difficult, because Terrence Taos Q

speaks about non-standard analysis. And you would need to
formalize his non-standard analysis that he gave as a link,
plus all the meta mathematics about FOL and SOL:

Ultrafilters, nonstandard analysis, and epsilon management
https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/

Mild Shock schrieb am Samstag, 10. Juni 2023 um 13:30:11 UTC+2:
> What does this have to do with Terrence Taos article? 
> I guess you took the potato, took a potato peeler and 
> ended with a tooth pick of a potato. 
> 
> First order translating "'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'." is extremly trivial. 
> 
> If you allow Skolemization you get:
> /* Terrence Tao Skolemized */ 
> (*') ∃f∃g∀x∀yQ(x,y,f(x),g(y))
> if you use Predicates instead Skolem functions you get: 
> 
> /* Terrence Tao with Predicates */ 
> (*') ∃R∃S∀x∀y(∃z(R(x,z) & ∃t(S(y,t) & Q(x,y,z,t))) 
> 
> I don't think you need to require a priori somewhere 
> ∀x∃zR(x,z) or ∀y∃tS(y,t), because if one of those fails 
> then the whole formula anyway fails, so its somehow 
> 
> already built-in. We have anyway as the first outside 
> quantifiers ∃R and ∃S, so there is automatically some 
> "search" for dependency. Now since it is SOL, you 
> 
> can use this recipe, which I already demonstrated at 
> hand of the Drinker Paradox. Only now R and S are binary, 
> its not anymore monadic second order logic. You can 
> 
> check, do you get the same as Mahesh 
> Viswanathan would get?
> Second Order Logic - Mahesh Viswanathan Fall 2018 
> https://courses.engr.illinois.edu/cs498mv/fa2018/SecondOrderLogic.pdf
> And then? How do you continue the wealth of 
> reasoning in Terrence Taos blog?
> Dan Christensen schrieb am Samstag, 10. Juni 2023 um 03:20:43 UTC+2: 
> > On Friday, June 9, 2023 at 6:42:19 PM UTC-4, Mild Shock wrote: 
> > > Formalization of what? A potato? 
> > > Dan Christensen schrieb am Freitag, 9. Juni 2023 um 22:28:42 UTC+2: 
> > > > On Friday, June 9, 2023 at 3:04:54 PM UTC-4, Mild Shock wrote: 
> > > > > What job? Do you even understand what Terrence Tao 
> > > > > started in his discussion. 
> > > > [snip] 
> > > > 
> > > > It seems you cannot fault my formalization of the problem. It seems your only hope is to change the subject, Mr. Collapse. Oh, well... 
> > > Formalization of what? 
> > 
> > "'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'." 
> > 
> > My suggested formalization using only ordinary rules of logic and axioms of set theory: 
> > ALL(a):[a in u => EXIST(b):[b in u & (a,b) in r1]] 
> > 
> > & ALL(a):[a in u => EXIST(b):[b in u & (a,b) in r21]] 
> > 
> > & ALL(x):ALL(x'):ALL(y):ALL(y'):[x in u & x' in u & y in u & y' in u => [(x,y) in r1 & (x',y') in r2 => Q(x,x',y,y')]] 
> > Still no comments??? 
> > 
> > BTW I have shown that sets of ordered pairs r1 and r2 do exist. I can post it if you like. (117 lines) 
> > 
> > 117. ALL(dom):[Set(dom) 
> > => EXIST(rel):[Set(rel) 
> > & EXIST(a):a in rel 
> > & ALL(a):[a in rel <=> Set'(a) 
> > & ALL(b):ALL(c):[(b,c) in a => b in dom & c in dom] 
> > & ALL(b):[b in dom => EXIST(c):[c in dom & (b,c) in a]]]]] 
> > Conclusion, 1 
> > 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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