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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 4397), 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 4397 | . . 3 | |
2 | ordirr 4452 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | elong 4290 | . . 3 | |
5 | 1, 4 | mpbiri 167 | . 2 |
6 | 3, 5 | mto 651 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wcel 1480 cvv 2681 word 4279 con0 4280 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-sn 3528 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 |
This theorem is referenced by: sucon 4463 |
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