Theorem ru 3726

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 ∈ 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 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ 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 his Grundgesetze der Arithmetik.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5262 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 5242, Pairing prex 5380, Union uniex 7695, Power Set pwex 5322, and Infinity omex 9564 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 6586 (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 10397 and Cantor's theorem canth 7321 are provably false. (See ncanth 7322 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 5231 replaces ax-rep 5212) with ax-sep 5231 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 9512 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9522). See ruALT 9523 for an alternate proof of ru 3726 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2376. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2147, ax-11 2163, and ax-12 2185. (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ru	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Proof of Theorem ru

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ru0 2133	. . . . 5 ⊢ ¬ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦)
2		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	2, 2	neleq12d 3041	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ 𝑦 ∉ 𝑦))
4		df-nel 3037	. . . . . . 7 ⊢ (𝑦 ∉ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
5	3, 4	bitrdi 287	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
6	5	eqabbw 2809	. . . . 5 ⊢ (𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦))
7	1, 6	mtbir 323	. . . 4 ⊢ ¬ 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
8	7	nex 1802	. . 3 ⊢ ¬ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
9		isset 3443	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥})
10	8, 9	mtbir 323	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11	10	nelir 3039	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714   ∉ wnel 3036  Vcvv 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nel 3037  df-v 3431

This theorem is referenced by: (None)