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Mirrors > Home > ILE Home > Th. List > ru | GIF version |
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 3866. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
ru | ⊢ {x ∣ x ∉ x} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 621 | . . . . . 6 ⊢ ¬ (y ∈ y ↔ ¬ y ∈ y) | |
2 | eleq1 2097 | . . . . . . . 8 ⊢ (x = y → (x ∈ y ↔ y ∈ y)) | |
3 | df-nel 2204 | . . . . . . . . 9 ⊢ (x ∉ x ↔ ¬ x ∈ x) | |
4 | id 19 | . . . . . . . . . . 11 ⊢ (x = y → x = y) | |
5 | 4, 4 | eleq12d 2105 | . . . . . . . . . 10 ⊢ (x = y → (x ∈ x ↔ y ∈ y)) |
6 | 5 | notbid 591 | . . . . . . . . 9 ⊢ (x = y → (¬ x ∈ x ↔ ¬ y ∈ y)) |
7 | 3, 6 | syl5bb 181 | . . . . . . . 8 ⊢ (x = y → (x ∉ x ↔ ¬ y ∈ y)) |
8 | 2, 7 | bibi12d 224 | . . . . . . 7 ⊢ (x = y → ((x ∈ y ↔ x ∉ x) ↔ (y ∈ y ↔ ¬ y ∈ y))) |
9 | 8 | spv 1737 | . . . . . 6 ⊢ (∀x(x ∈ y ↔ x ∉ x) → (y ∈ y ↔ ¬ y ∈ y)) |
10 | 1, 9 | mto 587 | . . . . 5 ⊢ ¬ ∀x(x ∈ y ↔ x ∉ x) |
11 | abeq2 2143 | . . . . 5 ⊢ (y = {x ∣ x ∉ x} ↔ ∀x(x ∈ y ↔ x ∉ x)) | |
12 | 10, 11 | mtbir 595 | . . . 4 ⊢ ¬ y = {x ∣ x ∉ x} |
13 | 12 | nex 1386 | . . 3 ⊢ ¬ ∃y y = {x ∣ x ∉ x} |
14 | isset 2555 | . . 3 ⊢ ({x ∣ x ∉ x} ∈ V ↔ ∃y y = {x ∣ x ∉ x}) | |
15 | 13, 14 | mtbir 595 | . 2 ⊢ ¬ {x ∣ x ∉ x} ∈ V |
16 | df-nel 2204 | . 2 ⊢ ({x ∣ x ∉ x} ∉ V ↔ ¬ {x ∣ x ∉ x} ∈ V) | |
17 | 15, 16 | mpbir 134 | 1 ⊢ {x ∣ x ∉ x} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 98 ∀wal 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 ∉ wnel 2202 Vcvv 2551 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nel 2204 df-v 2553 |
This theorem is referenced by: (None) |
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