Re: Moore on Skolem's Paradox

William of Ockham wrote: 
Aatu;

So the terms "the set of all quadruplets of naturals x,y,z,n with n>2
such that x^n + y^n = z^n" and "the empty set" have the same meaning?


Since I don't know mathematics, I don't know which set is meant by
"the set of all quadruplets of naturals x,y,z,n with n>2 such that x^n
+ y^n = z^n".  Presumably you know the answer.  Which set do you mean
by "the set of all quadruplets of naturals x,y,z,n with n>2 such that
x^n + y^n = z^n"?

A quadruplet is a mathematical object consisting of four other mathematical objects in some order. It's usually written as <x,y,z,u> where x,y,z and u are the mathematical objects in the quadruplet, in that order. So the term "the set of all quadruplets of naturals x,y,z,n with n>2 such that x^n + y^n = z^n" refers to the set

 {<x,y,z,n> | x^n + y^n = z^n and n > 2}

(Here ^ denotes exponentiation).

Andrew Wiles proved in the middle 1990's that there are no such quadruplets, i.e.

 {<x,y,z,n> | x^n + y^n = z^n and n > 2} = the empty set

So it's not possible for the extensions of "the set of all quadruplets of naturals x,y,z,n with n>2 such that x^n + y^n = z^n" and "the empty set" to be different. By your account we should conclude that the meaning of "the set of all quadruplets of naturals x,y,z,n with n>2 such that x^n + y^n = z^n" is the same as that of "the empty set". This conclusion strikes me as a bit odd.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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