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Mirrors > Home > MPE Home > Th. List > ru | Structured version Visualization version 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 ssex 4629 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 4617, Pairing prex 4735, Union uniex 6732, Power Set pwex 4673, and Infinity omex 8303 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5780 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 9060 and Cantor's Theorem canth 6390 are provably false! (See ncanth 6391 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4607 replaces ax-rep 4597) with ax-sep 4607 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 8267 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8270). See ruALT 8271 for an alternate proof of ru 3305 derived from that fact. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
ru | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 373 | . . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦) | |
2 | eleq1 2580 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦)) | |
3 | df-nel 2687 | . . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥) | |
4 | id 22 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
5 | 4, 4 | eleq12d 2586 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
6 | 5 | notbid 306 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
7 | 3, 6 | syl5bb 270 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
8 | 2, 7 | bibi12d 333 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))) |
9 | 8 | spv 2151 | . . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)) |
10 | 1, 9 | mto 186 | . . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) |
11 | abeq2 2623 | . . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)) | |
12 | 10, 11 | mtbir 311 | . . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
13 | 12 | nex 1709 | . . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} |
14 | isset 3084 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}) | |
15 | 13, 14 | mtbir 311 | . 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V |
16 | 15 | nelir 2790 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 194 ∀wal 1472 = wceq 1474 ∃wex 1694 ∈ wcel 1938 {cab 2500 ∉ wnel 2685 Vcvv 3077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1700 ax-4 1713 ax-5 1793 ax-6 1838 ax-7 1885 ax-10 1966 ax-11 1971 ax-12 1983 ax-13 2137 ax-ext 2494 |
This theorem depends on definitions: df-bi 195 df-an 384 df-tru 1477 df-ex 1695 df-nf 1699 df-sb 1831 df-clab 2501 df-cleq 2507 df-clel 2510 df-nel 2687 df-v 3079 |
This theorem is referenced by: (None) |
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