Theorem ru 2284

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 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 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 3270 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 3261, Pairing prex 3341, Union uniex 3605, Power Set pwex 3302, and Infinity omex 5542 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 4307 (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 5708 and Cantor's Theorem canth 4923 are provably false! (See ncanth 4924 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 5510 (derived from the Axiom of Regularity), so for us the Russell class equals the universe _V (theorem ruv 5514). See ruALT 5515 for an alternate proof of ru 2284 derived from that fact.

Assertion

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

Proof of Theorem ru

Step	Hyp	Ref	Expression
1		pm5.19 729	. . . . . 6 |- -. (y e. y <-> -. y e. y)
2		eleq1 1794	. . . . . . . 8 |- (x = y -> (x e. y <-> y e. y))
3		id 73	. . . . . . . . . . 11 |- (x = y -> x = y)
4	3, 3	eleq12d 1802	. . . . . . . . . 10 |- (x = y -> (x e. x <-> y e. y))
5	4	notbid 670	. . . . . . . . 9 |- (x = y -> (-. x e. x <-> -. y e. y))
6		df-nel 1857	. . . . . . . . 9 |- (x e/ x <-> -. x e. x)
7	5, 6	syl5bb 588	. . . . . . . 8 |- (x = y -> (x e/ x <-> -. y e. y))
8	2, 7	bibi12d 688	. . . . . . 7 |- (x = y -> ((x e. y <-> x e/ x) <-> (y e. y <-> -. y e. y)))
9	8	a4v 1487	. . . . . 6 |- (A.x(x e. y <-> x e/ x) -> (y e. y <-> -. y e. y))
10	1, 9	mto 120	. . . . 5 |- -. A.x(x e. y <-> x e/ x)
11		abeq2 1836	. . . . 5 |- (y = {x | x e/ x} <-> A.x(x e. y <-> x e/ x))
12	10, 11	mtbir 208	. . . 4 |- -. y = {x | x e/ x}
13	12	nex 1294	. . 3 |- -. E.y y = {x | x e/ x}
14		isset 2129	. . 3 |- ({x | x e/ x} e. _V <-> E.y y = {x | x e/ x})
15	13, 14	mtbir 208	. 2 |- -. {x | x e/ x} e. _V
16		df-nel 1857	. 2 |- ({x | x e/ x} e/ _V <-> -. {x | x e/ x} e. _V)
17	15, 16	mpbir 206	1 |- {x | x e/ x} e/ _V

Syntax hints:  -. wn 2   <-> wb 162  A.wal 1134   = wceq 1136   e. wcel 1138  E.wex 1164  {cab 1708   e/ wnel 1855  _Vcvv 2125

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-10 1146  ax-12 1148  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702

This theorem depends on definitions:  df-bi 163  df-an 241  df-ex 1165  df-sb 1374  df-clab 1709  df-cleq 1714  df-clel 1717  df-nel 1857  df-v 2127

Copyright terms: Public domain