![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > aleph0 | Structured version Visualization version GIF version |
Description: The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers ω (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 ℵ0. 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 |
---|---|
aleph0 | ⊢ (ℵ‘∅) = ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aleph 9935 | . . 3 ⊢ ℵ = rec(har, ω) | |
2 | 1 | fveq1i 6893 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
3 | omex 9638 | . . 3 ⊢ ω ∈ V | |
4 | 3 | rdg0 8421 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
5 | 2, 4 | eqtri 2761 | 1 ⊢ (ℵ‘∅) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∅c0 4323 ‘cfv 6544 ωcom 7855 reccrdg 8409 harchar 9551 ℵcale 9931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 ax-inf2 9636 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-aleph 9935 |
This theorem is referenced by: alephon 10064 alephcard 10065 alephgeom 10077 cardaleph 10084 alephfplem1 10099 pwcfsdom 10578 alephom 10580 winalim2 10691 aleph1re 16188 aleph1min 42308 |
Copyright terms: Public domain | W3C validator |