Theorem ru 3802

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 5339 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 5325, Pairing prex 5452, Union uniex 7770, Power Set pwex 5398, and Infinity omex 9706 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 6661 (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 10538 and Cantor's theorem canth 7396 are provably false. (See ncanth 7397 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 5317 replaces ax-rep 5303) with ax-sep 5317 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 9659 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9665). See ruALT 9666 for an alternate proof of ru 3802 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2380. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2141, ax-11 2158, and ax-12 2178. (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 2127	. . . . 5 ⊢ ¬ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦)
2		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	2, 2	neleq12d 3057	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ 𝑦 ∉ 𝑦))
4		df-nel 3053	. . . . . . 7 ⊢ (𝑦 ∉ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
5	3, 4	bitrdi 287	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
6	5	eqabbw 2818	. . . . 5 ⊢ (𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦))
7	1, 6	mtbir 323	. . . 4 ⊢ ¬ 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
8	7	nex 1798	. . 3 ⊢ ¬ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
9		isset 3502	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥})
10	8, 9	mtbir 323	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11	10	nelir 3055	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ∉ wnel 3052  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nel 3053  df-v 3490

This theorem is referenced by: (None)