Theorem ru 1934

Description: Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. 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 Comprehension 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 2715 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 2707, Pairing prex 2777, Union uniex 2869, Power Set pwex 2741, and Infinity omex 4619 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 3579 (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 set 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 the very strong 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 4742 and Cantor's Theorem canth 3909 are provably false! (See ncanth 3910 for some intuition behind the latter.) Nonetheless, NF has not been shown to be inconsistent and has its advocates - who's to say which set theory is "right"? 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 4590 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V.

Assertion

Ref	Expression
ru	⊢ {x∣x ∉ x} ∉ V

Proof of Theorem ru

Step	Hyp	Ref	Expression
1		pm5.19 668	. . . . . 6 ⊢ ¬ (y ∈ y ↔ ¬ y ∈ y)
2		eleq1 1531	. . . . . . . 8 ⊢ (x = y → (x ∈ y ↔ y ∈ y))
3		id 59	. . . . . . . . . . 11 ⊢ (x = y → x = y)
4	3, 3	eleq12d 1539	. . . . . . . . . 10 ⊢ (x = y → (x ∈ x ↔ y ∈ y))
5	4	negbid 610	. . . . . . . . 9 ⊢ (x = y → (¬ x ∈ x ↔ ¬ y ∈ y))
6		df-nel 1585	. . . . . . . . 9 ⊢ (x ∉ x ↔ ¬ x ∈ x)
7	5, 6	syl5bb 531	. . . . . . . 8 ⊢ (x = y → (x ∉ x ↔ ¬ y ∈ y))
8	2, 7	bibi12d 628	. . . . . . 7 ⊢ (x = y → ((x ∈ y ↔ x ∉ x) ↔ (y ∈ y ↔ ¬ y ∈ y)))
9	8	a4v 1270	. . . . . 6 ⊢ (∀x(x ∈ y ↔ x ∉ x) → (y ∈ y ↔ ¬ y ∈ y))
10	1, 9	mto 106	. . . . 5 ⊢ ¬ ∀x(x ∈ y ↔ x ∉ x)
11		abeq2 1565	. . . . 5 ⊢ (y = {x∣x ∉ x} ↔ ∀x(x ∈ y ↔ x ∉ x))
12	10, 11	mtbir 192	. . . 4 ⊢ ¬ y = {x∣x ∉ x}
13	12	nex 1099	. . 3 ⊢ ¬ ∃y y = {x∣x ∉ x}
14		isset 1810	. . 3 ⊢ ({x∣x ∉ x} ∈ V ↔ ∃y y = {x∣x ∉ x})
15	13, 14	mtbir 192	. 2 ⊢ ¬ {x∣x ∉ x} ∈ V
16		df-nel 1585	. 2 ⊢ ({x∣x ∉ x} ∉ V ↔ ¬ {x∣x ∉ x} ∈ V)
17	15, 16	mpbir 190	1 ⊢ {x∣x ∉ x} ∉ V

Syntax hints:  ¬ wn 2   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  {cab 1461   ∉ wnel 1583  Vcvv 1807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-nel 1585  df-v 1808

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