Proof
Let f: N→N be the function defined by:
- f(#(θ)) = #(θ(#(θ))
for each T-formula θ in one free variable, and f(n) = 0 otherwise. The function f is computable, so there is a formula δ representing f in T. Thus for each formula θ, T proves
- (∀y) ,
which is to say
- (∀y) .
Now define the formula β(z) as:
- β(z) = (∀y) ,
then
- β(#(θ)) ⇔ (∀y) ,
which is to say
- β(#(θ)) ⇔ ψ(#(θ(#(θ))))
Let φ be the sentence β(#(β)). Then we can prove in T that:
- (*) φ ⇔ (∀y) ⇔ (∀y) .
Working in T, analyze two cases:
1. Assuming φ holds, substitute #(β(#(β)) for y in the rightmost formula in (*), and obtain:
- (#(β(#(β)) = #(β(#(β))) → ψ(#(β(#(β))),
Since φ = β(#(β)), it follows that ψ(#(φ)) holds.
2. Conversely, assume that ψ(#(β(#(β)))) holds. Then the final formula in (*) must be true, and φ is also true.
Thus φ ↔ ψ(#(φ)) is provable in T, as desired.
Read more about this topic: Diagonal Lemma
Famous quotes containing the word proof:
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)
“From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
—Johan Huizinga (1872–1945)
“Right and proof are two crutches for everything bent and crooked that limps along.”
—Franz Grillparzer (1791–1872)