Theorem ru 1909

Description: Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as A e. 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 e/ x} (the "Russell class") for A, it asserted {x | x e/ x} e. V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {x | x e/ x} e/ 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 2687 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 2679, Pairing prex 2749, Union uniex 2834, Power Set pwex 2713, and Infinity omex 4551 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 3516 (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 4674 and Cantor's Theorem canth 3846 are provably false! (See ncanth 3847 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 4522 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V.

Assertion

Ref	Expression
ru	|- {x | x e/ x} e/ V

Proof of Theorem ru

Step	Hyp	Ref	Expression
1		pm5.19 666	. . . . . 6 |- -. (y e. y <-> -. y e. y)
2		eleq1 1510	. . . . . . . 8 |- (x = y -> (x e. y <-> y e. y))
3		id 59	. . . . . . . . . . 11 |- (x = y -> x = y)
4	3, 3	eleq12d 1518	. . . . . . . . . 10 |- (x = y -> (x e. x <-> y e. y))
5	4	negbid 609	. . . . . . . . 9 |- (x = y -> (-. x e. x <-> -. y e. y))
6		df-nel 1564	. . . . . . . . 9 |- (x e/ x <-> -. x e. x)
7	5, 6	syl5bb 530	. . . . . . . 8 |- (x = y -> (x e/ x <-> -. y e. y))
8	2, 7	bibi12d 627	. . . . . . 7 |- (x = y -> ((x e. y <-> x e/ x) <-> (y e. y <-> -. y e. y)))
9	8	a4b1 1254	. . . . . 6 |- (A.x(x e. y <-> x e/ x) -> (y e. y <-> -. y e. y))
10	1, 9	mto 106	. . . . 5 |- -. A.x(x e. y <-> x e/ x)
11		abeq2 1544	. . . . 5 |- (y = {x | x e/ x} <-> A.x(x e. y <-> x e/ x))
12	10, 11	mtbir 192	. . . 4 |- -. y = {x | x e/ x}
13	12	nex 1077	. . 3 |- -. E.y y = {x | x e/ x}
14		isset 1789	. . 3 |- ({x | x e/ x} e. V <-> E.y y = {x | x e/ x})
15	13, 14	mtbir 192	. 2 |- -. {x | x e/ x} e. V
16		df-nel 1564	. 2 |- ({x | x e/ x} e/ V <-> -. {x | x e/ x} e. V)
17	15, 16	mpbir 190	1 |- {x | x e/ x} e/ V

Syntax hints:  -. wn 2   <-> wb 146  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440   e/ wnel 1562  Vcvv 1786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-nel 1564  df-v 1787

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