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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 9955 | . . 3 ⊢ ℵ = rec(har, ω) | |
2 | 1 | fveq1i 6892 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
3 | omex 9658 | . . 3 ⊢ ω ∈ V | |
4 | 3 | rdg0 8435 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
5 | 2, 4 | eqtri 2755 | 1 ⊢ (ℵ‘∅) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∅c0 4318 ‘cfv 6542 ωcom 7864 reccrdg 8423 harchar 9571 ℵcale 9951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 ax-inf2 9656 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-aleph 9955 |
This theorem is referenced by: alephon 10084 alephcard 10085 alephgeom 10097 cardaleph 10104 alephfplem1 10119 pwcfsdom 10598 alephom 10600 winalim2 10711 aleph1re 16213 aleph1min 42910 |
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