[spam][crazy][fiction][random] Non-Canon MCBoss Spinoffs

[mailbombbin]

> Let L be the language of [[first-order arithmetic]].
> Let N be the standard structure for L, i.e. N consists of the ordinary
> set of natural numbers and their addition and multiplication. Each
> sentence in L can be interpreted in N and then becomes either true
> or false.
> Each formula phi in L has a [[Gödel number]] g(phi). This is a natural number that "encodes" <math>\varphi.</math> In that way, the language <math>L</math> can talk about formulas in <math>L,</math> not just about numbers. Let <math>T</math> denote the set of  <math>L</math>-sentences true in <math>N,</math> and <math>T^*</math> the set of Gödel numbers of the sentences in <math>T.</math> The following theorem answers the question: Can <math>T^*</math> be defined by a formula of first-order arithmetic?


> To prove the theorem, we proceed by contradiction and assume that an
> L-formula True(n) exists which is true for the natural number n in N if
> and only if n is the Gödel number of a sentence in L that is true in N.
> We could then use True(n) to define a new L-formula S(m) which is true
> for the natural number m if and only if m is the Gödel number of a
> formula phi(x) (with a free variable x) such that phi(m) is false when
> interpreted in N (i.e. the formula phi(x), when applied to its own Gödel
> number, yields a false statement). If we now consider the Gödel number
> g of the formula S(m), and ask whether the sentence S(g) is true in N,
> we obtain a contradiction.
> (This is known as a [[Diagonal lemma|diagonal argument]].)