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Mirrors > Home > ILE Home > Th. List > onprc | GIF version |
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 4481), 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 | ⊢ ¬ On ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 4481 | . . 3 ⊢ Ord On | |
2 | ordirr 4537 | . . 3 ⊢ (Ord On → ¬ On ∈ On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ¬ On ∈ On |
4 | elong 4369 | . . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On)) | |
5 | 1, 4 | mpbiri 168 | . 2 ⊢ (On ∈ V → On ∈ On) |
6 | 3, 5 | mto 662 | 1 ⊢ ¬ On ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2148 Vcvv 2737 Ord word 4358 Oncon0 4359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-sn 3597 df-uni 3808 df-tr 4099 df-iord 4362 df-on 4364 |
This theorem is referenced by: sucon 4548 |
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