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Re: Expressability in the notation of set theory

Newsgroups sci.logic
Date 2023-06-10 04:30 -0700
References (3 earlier) <fa2bf214-77c4-497c-b8d4-fb03f6b16842n@googlegroups.com> <u5vt4i$149dp$1@solani.org> <756a6232-e267-46f5-a0fd-86f5f1377f39n@googlegroups.com> <406ef698-8190-41e7-9dc7-15416defd9c8n@googlegroups.com> <33ed59fc-415f-49cb-94c9-19f53adbec46n@googlegroups.com>
Message-ID <38f8ea3f-6438-4295-98f5-370aefb00e5dn@googlegroups.com> (permalink)
Subject Re: Expressability in the notation of set theory
From Mild Shock <bursejan@gmail.com>

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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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