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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 | |- ( aleph ` (/) ) = _om |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-aleph | |- aleph = rec ( har , _om ) |
|
| 2 | 1 | fveq1i | |- ( aleph ` (/) ) = ( rec ( har , _om ) ` (/) ) |
| 3 | omex | |- _om e. _V |
|
| 4 | 3 | rdg0 | |- ( rec ( har , _om ) ` (/) ) = _om |
| 5 | 2 4 | eqtri | |- ( aleph ` (/) ) = _om |