Re: Expressability in the notation of set theory

Newsgroups	sci.logic
Date	2023-06-10 10:52 -0700
References	(4 earlier) <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> <38f8ea3f-6438-4295-98f5-370aefb00e5dn@googlegroups.com>
Message-ID	<af60bd69-dd89-4d55-97fd-2f90b4a2ccf2n@googlegroups.com> (permalink)
Subject	Re: Expressability in the notation of set theory
From	Dan Christensen <Dan_Christensen@sympatico.ca>

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On Saturday, June 10, 2023 at 7:30:11 AM UTC-4, Mild Shock wrote:

> 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 

> What does this have to do with Terrence Taos article? 

Pay attention. Tao wrote:

"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

I am suggesting a formalization using the ordinary rules of logic and axioms of set theory, which you have yet to comment on.


> 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:

 See my suggestion above. No Skolemization required. Just the ordinary rules of logic and set theory.

> /* Terrence Tao Skolemized */ 
> (*') ∃f∃g∀x∀yQ(x,y,f(x),g(y))
> if you use Predicates instead Skolem functions you get: 
> 

I think you want relations, not functions here. 

> /* Terrence Tao with Predicates */ 
> (*') ∃R∃S∀x∀y(∃z(R(x,z) & ∃t(S(y,t) & Q(x,y,z,t))) 
> 

See my suggestion above. It does not require quantification of propositions or predicates, just the ordinary rules of logic and set theory.

> 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 
> 

[snip]

You can prove the existence of any the required relations using ordinary set theory, i.e. the axioms for Cartesian products, powersets and subsets. (See my previous posting.)

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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