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Description: No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4445), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
Ref | Expression |
---|---|
onprc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 4445 | . . 3 | |
2 | ordirr 4501 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | elong 4333 | . . 3 | |
5 | 1, 4 | mpbiri 167 | . 2 |
6 | 3, 5 | mto 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wcel 2128 cvv 2712 word 4322 con0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 |
This theorem is referenced by: sucon 4512 |
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