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Topic: Naive set theory
Replies: 4   Last Post: Apr 9, 2013 3:02 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Naive set theory
Posted: Apr 9, 2013 3:02 PM
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On 4/9/2013 1:04 PM, Zuhair wrote:
> On Apr 9, 5:03 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> Zuhair <zaljo...@gmail.com> writes:
>>> What's the proof of the following in naive set theory?
>>
>>> Not exist x. x is empty
>>
>> By Russell's paradox, there exists a set R such that R in R and R not
>> in R. By ex falso quodlibet, there is no set with no elements.
>>
>> --
>> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>>
>>

> Yes, your response and Smaill's are pretty much the same. Those proofs
> are trivial ones, they depend on Modus Ponens where from P and P->Q we
> infer Q, so if P is false and is a theorem then it qualifies as a step
> in the proof, then since P->Q is trivially true then it is a theorem,
> then obviously Q is a theorem whatever Q is. BUT this proof is TRIVIAL
> and of no importance since we already know that P is **clearly**
> false.


In view of your last remark, the statement you make here is
in error. The proof cannot be trivial if it is making you
question the nature of proof.

The responses you received were given under the presumption of
a classical logic in which material implication is given its
semantics with respect to truth table decision procedures
and classical modus ponens.

There are other logics.

> I want a non trivial proof, i.e. a proof containing no step
> that is TRIVIALLY false. On the other hand the proof of the existence
> of an empty set is SHORTER it is a direct result of naive
> comprehension, and it contains NO trivial step as far as I can see.


Perhaps you could provide that proof without using a
self-contradictory property.

{x|x is red and x is not red}

seems pretty trivial to me.

It is just a matter of where you are applying the
principle of bivalence.

One could, perhaps, speak of an empty set in terms
of intersections

{x|x is red} n {x is not red}

But, once again, the partitions by which the
extensions of concepts may be thought of as
not having overlap are governed by the bivalence
principle.


> Not only that the proof about non existence of the empty set can be
> used to prove the contrary result or any theorem, which is of no
> importance, while the proof of existence of the empty set is not
> shared with any other theorem proved, so it is a genuine proof. One
> can easily see that the proof by principle of explosion is not Content-
> full, i.e. not related specifically to the result it proves.
>


Now you are speaking about relation to the content of
statements. There are views about logic which reject
this notion. I believe that one such logic that attempts
to take this into account may be found here:

http://plato.stanford.edu/entries/logic-relevance/

You will observe that there is a reference to MacColl
as early as 1908. In "Principia Mathematica" there is
a nod to the disagreement over interpretation of the
conditional. This may predate the introduction of
truth tables and certainly predates their wide acceptance
as a decision procedure. So, your concern has been around
for a while.

> This calls for a re-definition of what a *Proof* is. But I don't know
> if this is possible.


There are other logics with other deductive calculi.

The classical first-order logic along with the classical
first-order model theory probably realize the relationship
between a finitary metamathematics and a general mathematics
described by Hilbert as closely as one might hope. Other
logics may not have this kind of strong relationship. Where
they do, it will probably require a bit of searching in the
literature to find the proofs of completeness and soundness.

These considerations may impact what you choose to consider
as proof in your personal investigations.











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