[FOM] Logic and metalanguage - a question (fwd)

[Hao Zhaokuan]

We never define sets and numbers by logic, just describe them by some axioms. Actually, since our axioms are not strong enough, we just grasp a part of the meaning of the concept of set! When Frege said that he define the concept of number with pure logic, he assumed that it is known what the extension of a concept is(Grundlagen,68), hence he used concept of set already. 

Zhaokuan


----- Original Message ----- 
From: "Thomas Forster" <T.Forster at dpmms.cam.ac.uk>
To: <fom at cs.nyu.edu>
Sent: Friday, July 21, 2006 6:41 AM
Subject: [FOM] Logic and metalanguage - a question (fwd)


> 
> I got this rather nice email from someone who appears to be in Bohemia or 
> Moravia.  Is there anyone on this list based in those places who could 
> take him in hand..?
> 
>     Thomas
> 
> -- 
> www.dpmms.cam.ac.uk/~tf;  Home phone +64-3-348-6609 (that's a fax too!);
> mobile: +64-21-0580093. work +64-3-3642385
> 
> 
> 
> 
> 
> 
> 
> ---------- Forwarded message ----------
> Date: Thu, 20 Jul 2006 20:10:13 +0200
> From: "Adam  [windows-1250]  Li?ka" <adam.liska at centrum.cz>
> To: tf at dpmms.cam.ac.uk
> Subject: Logic and metalanguage - a question
> 
> Dear Mr. Forster,
> 
> I'm a high school student with an interest in Logic. I've been studying it lately intensively and encountered some problems to which I couldn't find solutions in available literature or on the Internet. In my search for information, I came across your personal webpage and subsequently, your e-mail address. I'd like to ask you if you could tell me whether the following reasoning is correct?
> 
> When talking about logic, when studying logic, we use numerous concepts whose precise definitions make use of logic. For example sets (axiomatic set theory) or numbers (Frege's concept of numbers).
> 
> We use these concepts on an intuitive base while describing logic - i.e. we use these concepts without defining them rigorously. But this is of no concern since we're using only methods and concepts that should be "convincing to everyone qualified to engage in such activities." (Robert R Stoll defines in his book "Set Theory and Logic" metamathematics as "the study of formal theories by methods which should be convincing to everyone qualified to engage in such activities".) Thus the way how to prevent loops* from occurring. The meanings of these concepts are considered to be (intuitively) clear to everyone.
> 
> *a) metalanguage level - using sets and numbers while desribing logic
> b) formal language level - using logic to define sets and consequently numbers => a loop occurs.
> 
> This is the way I see and understand it. Kindly correct me if I'm (completely) wrong, any help will be greatly appreciated. Is there any literature that deals with this problem?
> 
> Sincerely,
> 
> Adam Liska
> 
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