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

[mailbombbin]

On 9/13/23, mailbombbin <mailbombbin at gmail.com> wrote:
>> 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]].)

S(m) is true only when m == phi, where phi(m) is false
S(m) uses True(n)
simplify
S(x) == (x == phi)
phi(phi) == false
True("S(S)") ?
S(S) == S == phi
True("S == phi") ?
S() uses True(): S(x) ==

i think i need to write S(x) in terms of phi and True