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Description: Russell's Paradox.
Proposition 4.14 of [TakeutiZaring] p.
14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that is a set only when it is smaller than some other set . The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 3896. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
ru |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 654 | . . . . . 6 | |
2 | eleq1 2141 | . . . . . . . 8 | |
3 | df-nel 2340 | . . . . . . . . 9 | |
4 | id 19 | . . . . . . . . . . 11 | |
5 | 4, 4 | eleq12d 2149 | . . . . . . . . . 10 |
6 | 5 | notbid 624 | . . . . . . . . 9 |
7 | 3, 6 | syl5bb 190 | . . . . . . . 8 |
8 | 2, 7 | bibi12d 233 | . . . . . . 7 |
9 | 8 | spv 1781 | . . . . . 6 |
10 | 1, 9 | mto 620 | . . . . 5 |
11 | abeq2 2187 | . . . . 5 | |
12 | 10, 11 | mtbir 628 | . . . 4 |
13 | 12 | nex 1429 | . . 3 |
14 | isset 2605 | . . 3 | |
15 | 13, 14 | mtbir 628 | . 2 |
16 | df-nel 2340 | . 2 | |
17 | 15, 16 | mpbir 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 103 wal 1282 wceq 1284 wex 1421 wcel 1433 cab 2067 wnel 2339 cvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nel 2340 df-v 2603 |
This theorem is referenced by: (None) |
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