Groups | Search | Server Info | Login | Register

Groups > sci.logic > #252702

Re: Smullyan's Proof of the Drinkers Principle V3

Show all headers | View raw

A little History
The year was 1901. Just when the world of mathematics
had dealt with one set of paradoxes and foundational
crises regarding limits in calculus, a new batch had
emerged from set theory. Like many of his peers at
the time, young philosopher and mathematician Bertrand
Russell was hard at work trying to construct a firm
logical foundation for all of mathematics.

While working on the first of his books in the area,
the Principles of Mathematics, Russell came upon the
idea of a set of all sets which do not include
themselves. He soon realized that when you consider
whether this, in his words, "very peculiar class"
includes itself, "each alternative leads to its
opposite and there is a contradiction".
https://mathimages.swarthmore.edu/index.php/Russell's_Antinomy

In Naive Comprehension you could
deduce an Universal Set:

 > 2    EXIST(a):[Set(a) & ALL(b):[b in a <=> P(b)]]
 >      Subset, 1

Just use for P(b) for example b=b. Thats the Zermelo
Fix, that Comprehension doesn't give the Universal Set
by introducing the side conditon b in x, and thus

that there is no contradiction with the
Russell Paradox. if you had Naive Comprehension,
you could deduce the Universal Set as follows:

2    EXIST(u):[Set(u) & ALL(b):[b in u <=> b=b]]
      Subset, 1

Which is logically equivalent to:

2    EXIST(u):[Set(u) & ALL(b):[b in u]]
      Subset, 1

And with the Russell Paradox this would
render your system inconsistent. In the old
times this inconsistency was called an Antinomy.

Mostowski Collapse schrieb:
> Dang you are dumb, Dan-O-Matik, you even don't
> know what Naive and Non-Naive Comprehension is?
> And how Russell made Frege stumble?
> 
> Did you even go to school?
> 
> It seems you don't understand what is Naive and
> what not. Here again in example of the subset axiom.
> Its also naive if you use Set(_):
> 
> Naive [the b in x is missing]
> 
> 2    EXIST(a):[Set(a) & ALL(b):[b in a <=> P(b)]]
>      Subset, 1
> 
> 
> Non-Naive [the b in x is there]
> 
> 
> 1    Set(x)
>      Axiom
> 
> 2    EXIST(a):[Set(a) & ALL(b):[b in a <=> b in x & P(b)]]
>      Subset, 1
> 
> 
> I am not objecting that you use Set(_). Why do
> you think the usage of Set(_) is problematic?
> Where did I say that. The Naive thing is using Set(_)
> 
> only, and not an upper bound as well [the b in x is missing].
> The Non-Naive thing is using Set(_) and an upper bound.
> [the b in x is there].
> 
> Now analyze this sentence by Terrence Tao:
> 
> Exercise 10.3.5 from Analysis Vol.1 by Terence Tao.
> Give an example of a subset X⊂R and a function f:X→R
> which is differentiable on X, is such that f′(x)>0
> for all x∈X, but f is not strictly monotone increasing.
> 
> Does he say onle Set(X) or does he say more?
> 
> Dan Christensen schrieb:> Wrong again, Mr. Collapse.
>  >
>  >> Do you think Terrence Tao would say, eh lets have a set Set(s)?
>  >>
>  > [snip]
>  >
>  > He has established that there are objects that are sets (e.g. the set 
> of real numbers)) and some that are not (e.g. 4). If he was writing a 
> formal proof using set theory, he would then have to formally indicate 
> which objects are subject to the axioms of set theory (the sets). A 
> predicate is a good way to do that.
> 
> Mostowski Collapse schrieb:
>> With SET SPACES you cannot prove your "junk theorem".
>>
>> /* Junk Theorem Currently Provable */
>> 28 ALL(s):[Set(s) => EXIST(a):~a e s]
>> Rem DNeg, 27
>> http://www.dcproof.com/UniversalSet.htm
>>
>> if you for example replace Set(s) by s ∈ 𝒫(U). Then what
>> would go through in Coq with properly typing would amount to.
>> You need to type the existential quantifier:
>>
>> /* Junk Theorem Not Provable Anymore */
>> ALL(s):[s ∈ 𝒫(U) => EXIST(a):[U(a) & ~a e s]]
>>
>> The above is not provable anymore. Sets s from 𝒫(U)
>> do not always have elements inside U that are outside
>> these sets. This is because we have that U ∈ 𝒫(U),
>>
>> which then violates the above theorem. So the
>> Junk Theorem is not anymore provable if you
>> use properly SET SPACES as mathematicians do
>>
>> one way or the other in most math books. Just
>> check out Terrence Tao what he writes. Or things
>> like sigma spaces in probability theory etc.. etc..
>>
> 

Back to sci.logic | Previous | Next — Previous in thread | Next in thread | Find similar

Thread

Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-03 16:16 -0700
  Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-04 20:19 +0200
    Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-04 20:27 +0200
      Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-04 20:32 +0200
        Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-04 12:15 -0700
          Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 16:33 -0700
            Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 16:50 -0700
            Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-04 16:54 -0700
              Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 17:09 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 17:12 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-04 22:34 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-05 03:35 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-05 14:37 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-06 03:09 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-06 03:43 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-06 10:29 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-07 02:41 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-07 06:59 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-07 08:31 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-07 11:19 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-07 20:34 +0200
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-07 20:41 +0200
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-07 20:47 +0200
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-07 20:48 +0200
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-07 11:51 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-08 05:36 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-08 07:22 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-11 14:34 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-11 16:10 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-11 17:00 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <janburse@fastmail.fm> - 2023-05-12 12:04 +0200
                Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-12 03:13 -0700
                Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-12 07:12 -0700
    Re: Smullyan's Proof of the Drinkers Principle V3 Dan Christensen <Dan_Christensen@sympatico.ca> - 2023-05-04 11:40 -0700
      Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 11:43 -0700
        Re: Smullyan's Proof of the Drinkers Principle V3 Mostowski Collapse <bursejan@gmail.com> - 2023-05-04 11:49 -0700

csiph-web