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Theorem ru 3428
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 {𝑥𝑥𝑥} (the "Russell class") for 𝐴, it asserted {𝑥𝑥𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥𝑥𝑥} ∉ 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 4793 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 4781, Pairing prex 4900, Union uniex 6938, Power Set pwex 4839, and Infinity omex 8525 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 5964 (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 9282 and Cantor's Theorem canth 6593 are provably false! (See ncanth 6594 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 4772 replaces ax-rep 4762) with ax-sep 4772 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 8489 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8492). See ruALT 8493 for an alternate proof of ru 3428 derived from that fact. (Contributed by NM, 7-Aug-1994.)

Assertion
Ref Expression
ru {𝑥𝑥𝑥} ∉ V

Proof of Theorem ru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm5.19 375 . . . . . 6 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 eleq1 2687 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 df-nel 2895 . . . . . . . . 9 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
4 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
54, 4eleq12d 2693 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 308 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
73, 6syl5bb 272 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
82, 7bibi12d 335 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑦𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
98spv 2258 . . . . . 6 (∀𝑥(𝑥𝑦𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
101, 9mto 188 . . . . 5 ¬ ∀𝑥(𝑥𝑦𝑥𝑥)
11 abeq2 2730 . . . . 5 (𝑦 = {𝑥𝑥𝑥} ↔ ∀𝑥(𝑥𝑦𝑥𝑥))
1210, 11mtbir 313 . . . 4 ¬ 𝑦 = {𝑥𝑥𝑥}
1312nex 1729 . . 3 ¬ ∃𝑦 𝑦 = {𝑥𝑥𝑥}
14 isset 3202 . . 3 ({𝑥𝑥𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝑥𝑥})
1513, 14mtbir 313 . 2 ¬ {𝑥𝑥𝑥} ∈ V
1615nelir 2897 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1479   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wnel 2894  Vcvv 3195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nel 2895  df-v 3197
This theorem is referenced by: (None)
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