Gödel’s Legacy

Abstract

In his truly remarkable paper of 1931, Kurt Gödel casts grave doubts on the viability of Hilbert’s program for providing a consistent foundation for mathematics. In addition, it compelled logicians to view their subject through entirely new lenses. Two years later, Gödel was invited by the Mathematical Association of America to address one of its meetings. In his address, he spoke as follows:

Appendix: The Zermelo-Fraenkel Axioms

For brevity, I have omitted universal quantifiers at the left of these statements. The axioms of separation and collection are each really an infinite sequence of axioms in which the letter ϕ can be replaced by any legitimate formula written in terms of logical operations and the symbols occurring in the other axioms.

Empty set	x ∉ ∅
Extensionality	$\forall t\,(t \in x \leftrightarrow t \in y) \rightarrow x = y$
Pairs	t ∈ {x, y} ↔ t = x ∨ t = y
Definition	{x} = {x, x} ;   < x, y > = {{ x}, {x, y}}
Definition	$x \subseteq y\Longleftrightarrow\forall t\! \in \! x\,(t \in y)$
Power	$t \in \mathcal{P}(x) \leftrightarrow t \subseteq x$
Union	$t \in \bigcup x \leftrightarrow \exists z\! \in \! x\,(t \in z)$
Definition	$x \cup y =\bigcup \{ x,y\}$
Infinity	∅ ∈ ω ;  x ∈ ω → x ∪{ x} ∈ ω
Separation	$\forall u\exists x\forall t\,[t \in x \leftrightarrow t \in u \wedge \phi (t,\vec{p}\,)]$
Definition	Fn(f) ⇔ < x, y 1 >, < x, y 2 > ∈ f → y 1 = y 2
	f(x) = y⇔ < x, y > ∈ f
Choice	$\forall x\exists f\,(\mbox{ Fn}(f) \wedge \forall z\! \in \! x\,[z\neq \emptyset \rightarrow f(z) \in z)])$
Foundation	$x\neq \emptyset \rightarrow \exists y\! \in \! x\forall z\! \in \! y\,(z\not\in x)$
Collection	$\forall x\! \in \! u\exists y\phi (x,y,\vec{p}) \rightarrow \exists z\forall x\! \in \! u\exists y\! \in \! z\phi (x,y,\vec{p})$