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Mirrors > Home > MPE Home > Th. List > df-aleph | Structured version Visualization version GIF version |
Description: Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as Theorems aleph0 10007, alephsuc 10009, and alephlim 10008. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.) |
Ref | Expression |
---|---|
df-aleph | ⊢ ℵ = rec(har, ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cale 9877 | . 2 class ℵ | |
2 | char 9497 | . . 3 class har | |
3 | com 7803 | . . 3 class ω | |
4 | 2, 3 | crdg 8356 | . 2 class rec(har, ω) |
5 | 1, 4 | wceq 1542 | 1 wff ℵ = rec(har, ω) |
Colors of variables: wff setvar class |
This definition is referenced by: alephfnon 10006 aleph0 10007 alephlim 10008 alephsuc 10009 |
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