Theorem ru

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, which Frege acknowledged in the second edition of hisGrundgesetze der Arithmetik.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 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 , Pairing prex , Union uniex , Power Set pwex , and Infinity omex 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 (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 setvar 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 his New Foundations set theory (axiom system NF of Quine p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 and Cantor's theorem canth are provably false. (See ncanth for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep replaces ax-rep ) with ax-sep restricted to only bounded quantifiers. 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 (derived from the Axiom of Regularity), so for us the Russell class equals the universe _V (Theorem ruv ). See ruALT for an alternate proof of ru derived from that fact. (Contributed by NM, 7-Aug-1994) Remove use of ax-13 . (Revised by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	ru	$$⊢\left\{x|x\notin x\right\}\notin \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		pm5.19	$$⊢\neg \left(y\in y\leftrightarrow \neg y\in y\right)$$
2		eleq1w	$$⊢x=y\to \left(x\in y\leftrightarrow y\in y\right)$$
3		df-nel	$$⊢x\notin x\leftrightarrow \neg x\in x$$
4		id	$$⊢x=y\to x=y$$
5	4 4	eleq12d	$$⊢x=y\to \left(x\in x\leftrightarrow y\in y\right)$$
6	5	notbid	$$⊢x=y\to \left(\neg x\in x\leftrightarrow \neg y\in y\right)$$
7	3 6	bitrid	$$⊢x=y\to \left(x\notin x\leftrightarrow \neg y\in y\right)$$
8	2 7	bibi12d	$$⊢x=y\to \left(\left(x\in y\leftrightarrow x\notin x\right)\leftrightarrow \left(y\in y\leftrightarrow \neg y\in y\right)\right)$$
9	8	spvv	$$⊢\forall x\left(x\in y\leftrightarrow x\notin x\right)\to \left(y\in y\leftrightarrow \neg y\in y\right)$$
10	1 9	mto	$$⊢\neg \forall x\left(x\in y\leftrightarrow x\notin x\right)$$
11		abeq2	$$⊢y=\left\{x|x\notin x\right\}\leftrightarrow \forall x\left(x\in y\leftrightarrow x\notin x\right)$$
12	10 11	mtbir	$$⊢\neg y=\left\{x|x\notin x\right\}$$
13	12	nex	$$⊢\neg \exists yy=\left\{x|x\notin x\right\}$$
14		isset	$$⊢\left\{x|x\notin x\right\}\in \mathrm{V}\leftrightarrow \exists yy=\left\{x|x\notin x\right\}$$
15	13 14	mtbir	$$⊢\neg \left\{x|x\notin x\right\}\in \mathrm{V}$$
16	15	nelir	$$⊢\left\{x|x\notin x\right\}\notin \mathrm{V}$$