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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 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. 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 𝐴 is a set only when it is smaller than some other set 𝐵. 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 4105. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
ru | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 701 | . . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦) | |
2 | eleq1 2233 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦)) | |
3 | df-nel 2436 | . . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥) | |
4 | id 19 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
5 | 4, 4 | eleq12d 2241 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
6 | 5 | notbid 662 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
7 | 3, 6 | syl5bb 191 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
8 | 2, 7 | bibi12d 234 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))) |
9 | 8 | spv 1853 | . . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)) |
10 | 1, 9 | mto 657 | . . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) |
11 | abeq2 2279 | . . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)) | |
12 | 10, 11 | mtbir 666 | . . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
13 | 12 | nex 1493 | . . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
14 | isset 2736 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}) | |
15 | 13, 14 | mtbir 666 | . 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V |
16 | df-nel 2436 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V) | |
17 | 15, 16 | mpbir 145 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∀wal 1346 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 ∉ wnel 2435 Vcvv 2730 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nel 2436 df-v 2732 |
This theorem is referenced by: (None) |
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