Theorem ru 3743

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 5291 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 5270, Pairing prex 5409, Union uniex 7739, Power Set pwex 5351, and Infinity omex 9608 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 6623 (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 10463 and Cantor's theorem canth 7364 are provably false. (See ncanth 7365 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 5257 replaces ax-rep 5238) with ax-sep 5257 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 9555 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9566). See ruALT 9567 for an alternate proof of ru 3743 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2404. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2176, ax-11 2192, and ax-12 2213. (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 2162	. . . . 5 ⊢ ¬ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦)
2		id 23	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	2, 2	neleq12d 3069	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ 𝑦 ∉ 𝑦))
4		df-nel 3065	. . . . . . 7 ⊢ (𝑦 ∉ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
5	3, 4	bitrdi 290	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
6	5	eqabbw 2836	. . . . 5 ⊢ (𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦))
7	1, 6	mtbir 326	. . . 4 ⊢ ¬ 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
8	7	nex 1830	. . 3 ⊢ ¬ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥}
9		isset 3469	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑧 𝑧 = {𝑥 ∣ 𝑥 ∉ 𝑥})
10	8, 9	mtbir 326	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
11	10	nelir 3067	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ∉ wnel 3064  Vcvv 3455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nel 3065  df-v 3457

This theorem is used by: (None)