Theorem aleph0

Description: The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers _om (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written aleph_0. Exercise 3 of TakeutiZaring p. 91. Also Definition 12(i) of Suppes p. 228. From Moshé Machover,Set Theory, Logic, and Their Limitations, p. 95: "Aleph ... the first letter in the Hebrew alphabet ... is also the first letter of the Hebrew word ... (_einsoph_, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003) (Revised by Mario Carneiro, 13-Sep-2013)

	Ref	Expression
Assertion	aleph0	⊢ ( ℵ ‘ ∅ ) = ω

Proof

Step	Hyp	Ref	Expression
1		df-aleph	⊢ ℵ = rec ( har , ω )
2	1	fveq1i	⊢ ( ℵ ‘ ∅ ) = ( rec ( har , ω ) ‘ ∅ )
3		omex	⊢ ω ∈ V
4	3	rdg0	⊢ ( rec ( har , ω ) ‘ ∅ ) = ω
5	2 4	eqtri	⊢ ( ℵ ‘ ∅ ) = ω