FOM: Re: Arbitrary Objects

[Franklin Vera Pacheco]

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>I proposed a plausible semantical solution of the question,
>a few days ago, in terms of a discussion between two
>opponents (after all, any science develops as a discussion
>between opponents), which, to repeat, is the following:
>
>A: I claim that any x\in X satisfies F
>B: Prove!
>A: give me any x\in X
>
>Now, if B quits then A wins by default.
>If B gives an element, then if A demonstrates F then A wins
>and if A fails then B wins.
>
>Note: semantically, there is ONLY ONE choice of an arbitrary
>x here.
>
>V.Kanovei
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This is indeed my point of view of the arbitrary objects. 
The Kanovei's "discussion" is in my terms the rule R which selects from X 
all it's elements
one by each time.
There is a cognitive relation between proves and music.
To give a prove (without "arbitrary objects) is as a musical composition, 
to prove something
with a prove (without arbitrary objects) is to play the music. Now, a 
prove with arbitrary 
objects is to play a music several times with the variation of a note, 
then the
arbitrary-objects-prove is the aggregation of all that playings.
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[My posting]
> I think that you can think in arbitrary objects as a "Rule":
>
>   Let's translate some mathematical sentences...
>
>" Let F:N--->R be an arbitrary biyection from N (natural numbers) to R
>(real numbers),then you can make a list with all the real numbers...."
>                    Cantor.
>
>Translation:
>
> " Are given the set  F(N,R)  of all biyections from N to R , an element
>of F(N,R)^F(N,R) i.e. an enumeration of all the elements of
>F(N,R), and a "Rule" that gives in each step the elements of F(N,R) in 
>the
>order of the enumeration." Then the rest of the proof follows with that
>element given by the "Rule".
>
> Then any proof using arbitrary objects can be taken as a pair of
>proof-Rule or a secuence of proof. Each time you give an element of the
>set from where you take the "arbitrary objects" there will be a proof
>without arbitrary objects with the element that you have gived.



-- 
Franklin Vera Pacheco
45 #10029 e/100 y 104
Marianao, C Habana,
Cuba.
e-mail:franklin at ghost.matcom.uh.cu
tel:2606043