Re: Set and succession

Jan Burse wrote:
John Jones schrieb:
Elements that fall under sets, or objects that fall under concepts, have the same properties, normally represented by a variable. The variable can refer to any one of these elements or objects without explicitly mentioning them, ie. they are hidden, and not created. To create an element or object is to create a whole new type of object, represented by a symbol. Proofs don't bring in new symbols.

How said! Can you proof your claim? I mean
how did you check your claim?

What about Gödels coding. We have infinitely
many symbols there (each natural number is
a symbol), and from these symbols formulas
can be built, and we might reach at a formula
that expresses a totally new definition of
a symbol.

Clearly it is quite unlikely that we might
relate in our real world to this definition.
But we cannot exclude it. Ironically what the
numbers express might speak to us.

Same holds for set theory. By the separation
axiom every formula and set defines a new
subset. And if we can proof the formula non-empty
we will know that there is something in this
set. And this might speak to us.

Like the oracle in greeks time spoke to the
human beings. But we can also use modern words
for it, automated planning (artifical
intelligence), abduction (peirce), logic of
scientific discovery (herbert simon), etc...

Best Regards

http://en.wikipedia.org/wiki/Artificial_Creativity

Surely, aren't the axioms fixed for a particular proof? You can't suddenly introduce a new one, or prove a new one. Godel introduced an axiom of identity half-way through a proof, and tried to get away with it by claiming that it was already part of mathematics and logic.
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Relevant Pages

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