Theorem ru 1982

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 2789 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 2781, Pairing prex 2853, Union uniex 3090, Power Set pwex 2818, and Infinity omex 4763 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 3679 (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 4887 and Cantor's Theorem canth 4200 are provably false! (See ncanth 4201 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 4732 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 4735). See ruALT 4736 for an alternate proof of ru 1982 derived from that fact.

Assertion

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

Proof of Theorem ru

Step	Hyp	Ref	Expression
1		pm5.19 671	. . . . . 6 |- -. (y e. y <-> -. y e. y)
2		eleq1 1575	. . . . . . . 8 |- (x = y -> (x e. y <-> y e. y))
3		id 59	. . . . . . . . . . 11 |- (x = y -> x = y)
4	3, 3	eleq12d 1583	. . . . . . . . . 10 |- (x = y -> (x e. x <-> y e. y))
5	4	notbid 613	. . . . . . . . 9 |- (x = y -> (-. x e. x <-> -. y e. y))
6		df-nel 1629	. . . . . . . . 9 |- (x e/ x <-> -. x e. x)
7	5, 6	syl5bb 534	. . . . . . . 8 |- (x = y -> (x e/ x <-> -. y e. y))
8	2, 7	bibi12d 631	. . . . . . 7 |- (x = y -> ((x e. y <-> x e/ x) <-> (y e. y <-> -. y e. y)))
9	8	a4v 1308	. . . . . 6 |- (A.x(x e. y <-> x e/ x) -> (y e. y <-> -. y e. y))
10	1, 9	mto 105	. . . . 5 |- -. A.x(x e. y <-> x e/ x)
11		abeq2 1609	. . . . 5 |- (y = {x | x e/ x} <-> A.x(x e. y <-> x e/ x))
12	10, 11	mtbir 190	. . . 4 |- -. y = {x | x e/ x}
13	12	nex 1134	. . 3 |- -. E.y y = {x | x e/ x}
14		isset 1858	. . 3 |- ({x | x e/ x} e. V <-> E.y y = {x | x e/ x})
15	13, 14	mtbir 190	. 2 |- -. {x | x e/ x} e. V
16		df-nel 1629	. 2 |- ({x | x e/ x} e/ V <-> -. {x | x e/ x} e. V)
17	15, 16	mpbir 188	1 |- {x | x e/ x} e/ V

Syntax hints:  -. wn 2   <-> wb 144  A.wal 987   = wceq 989   e. wcel 991  E.wex 1013  {cab 1503   e/ wnel 1627  Vcvv 1855

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 995  ax-gen 996  ax-8 997  ax-10 999  ax-12 1001  ax-17 1004  ax-4 1006  ax-5o 1008  ax-6o 1011  ax-9o 1156  ax-10o 1174  ax-16 1244  ax-11o 1252  ax-ext 1498

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1014  df-sb 1206  df-clab 1504  df-cleq 1509  df-clel 1512  df-nel 1629  df-v 1856

Copyright terms: Public domain