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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 10099, alephsuc 10101, and alephlim 10100. 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 9969 | . 2 class ℵ | |
2 | char 9589 | . . 3 class har | |
3 | com 7878 | . . 3 class ω | |
4 | 2, 3 | crdg 8438 | . 2 class rec(har, ω) |
5 | 1, 4 | wceq 1533 | 1 wff ℵ = rec(har, ω) |
Colors of variables: wff setvar class |
This definition is referenced by: alephfnon 10098 aleph0 10099 alephlim 10100 alephsuc 10101 |
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