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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 9357 | . . 3 ⊢ ℵ = rec(har, ω) | |
2 | 1 | fveq1i 6664 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
3 | omex 9094 | . . 3 ⊢ ω ∈ V | |
4 | 3 | rdg0 8046 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
5 | 2, 4 | eqtri 2841 | 1 ⊢ (ℵ‘∅) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∅c0 4288 ‘cfv 6348 ωcom 7569 reccrdg 8034 harchar 9008 ℵcale 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-aleph 9357 |
This theorem is referenced by: alephon 9483 alephcard 9484 alephgeom 9496 cardaleph 9503 alephfplem1 9518 pwcfsdom 9993 alephom 9995 winalim2 10106 aleph1re 15586 aleph1min 39794 |
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