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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 4470), 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 4470 | . . 3 | |
2 | ordirr 4526 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | elong 4358 | . . 3 | |
5 | 1, 4 | mpbiri 167 | . 2 |
6 | 3, 5 | mto 657 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wcel 2141 cvv 2730 word 4347 con0 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-sn 3589 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 |
This theorem is referenced by: sucon 4537 |
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