Theorem ru 1437

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} 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 1700 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 1745, Pairing prex 1892, Union uniex 1947, Power Set pwex 1806, and Infinity omex 3475 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 2716 (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 set 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 3571 and Cantor's Theorem canth2 3381 are provably false! 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).

Assertion

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

Proof of Theorem ru

Step	Hyp	Ref	Expression
1		pm4.2 148	. . . . . . 7 |- (y e. y <-> y e. y)
2		pm5.18 497	. . . . . . 7 |- ((y e. y <-> y e. y) <-> -. (y e. y <-> -. y e. y))
3	1, 2	mpbi 164	. . . . . 6 |- -. (y e. y <-> -. y e. y)
4		eleq1 1149	. . . . . . . 8 |- (x = y -> (x e. y <-> y e. y))
5		id 9	. . . . . . . . . . 11 |- (x = y -> x = y)
6	5, 5	eleq12d 1157	. . . . . . . . . 10 |- (x = y -> (x e. x <-> y e. y))
7	6	negbid 463	. . . . . . . . 9 |- (x = y -> (-. x e. x <-> -. y e. y))
8		df-nel 1193	. . . . . . . . 9 |- (x e/ x <-> -. x e. x)
9	7, 8	syl5bb 410	. . . . . . . 8 |- (x = y -> (x e/ x <-> -. y e. y))
10	4, 9	bibi12d 477	. . . . . . 7 |- (x = y -> ((x e. y <-> x e/ x) <-> (y e. y <-> -. y e. y)))
11	10	a4b1 928	. . . . . 6 |- (A.x(x e. y <-> x e/ x) -> (y e. y <-> -. y e. y))
12	3, 11	mto 93	. . . . 5 |- -. A.x(x e. y <-> x e/ x)
13		cleqab 1174	. . . . 5 |- (y = {x | x e/ x} <-> A.x(x e. y <-> x e/ x))
14	12, 13	mtbir 167	. . . 4 |- -. y = {x | x e/ x}
15	14	nex 779	. . 3 |- -. E.y y = {x | x e/ x}
16		isset 1351	. . 3 |- ({x | x e/ x} e. V <-> E.y y = {x | x e/ x})
17	15, 16	mtbir 167	. 2 |- -. {x | x e/ x} e. V
18		df-nel 1193	. 2 |- ({x | x e/ x} e/ V <-> -. {x | x e/ x} e. V)
19	17, 18	mpbir 165	1 |- {x | x e/ x} e/ V

Syntax hints:  -. wn 1   <-> wb 127  A.wal 672  E.wex 678   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092   e/ wnel 1191  Vcvv 1348

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-nel 1193  df-v 1349

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