Theorem ru 3710

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 5240 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 5226, Pairing prex 5350, Union uniex 7572, Power Set pwex 5298, and Infinity omex 9331 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 6505 (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 10162 and Cantor's theorem canth 7209 are provably false. (See ncanth 7210 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 5218 replaces ax-rep 5205) with ax-sep 5218 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 9285 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9291). See ruALT 9292 for an alternate proof of ru 3710 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2372. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ru	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Proof of Theorem ru

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.19 387	. . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
2		eleq1w 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
3		df-nel 3049	. . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
4		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5	4, 4	eleq12d 2833	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	notbid 317	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
7	3, 6	syl5bb 282	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	2, 7	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)))
9	8	spvv 2001	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))
10	1, 9	mto 196	. . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)
11		abeq2 2871	. . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥))
12	10, 11	mtbir 322	. . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
13	12	nex 1804	. . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
14		isset 3435	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥})
15	13, 14	mtbir 322	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
16	15	nelir 3051	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ∉ wnel 3048  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nel 3049  df-v 3424

This theorem is referenced by: (None)