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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 ℵ_0. 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 9099 | . . 3 ⊢ ℵ = rec(har, ω) | |
2 | 1 | fveq1i 6447 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
3 | omex 8837 | . . 3 ⊢ ω ∈ V | |
4 | 3 | rdg0 7800 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
5 | 2, 4 | eqtri 2802 | 1 ⊢ (ℵ‘∅) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∅c0 4141 ‘cfv 6135 ωcom 7343 reccrdg 7788 harchar 8750 ℵcale 9095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-aleph 9099 |
This theorem is referenced by: alephon 9225 alephcard 9226 alephgeom 9238 cardaleph 9245 alephfplem1 9260 pwcfsdom 9740 alephom 9742 winalim2 9853 aleph1re 15378 |
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