Theorem ru 3586

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.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 4937 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 4925, Pairing prex 5038, Union uniex 7103, Power Set pwex 4982, and Infinity omex 8707 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 6115 (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, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 9502 and Cantor's Theorem canth 6753 are provably false! (See ncanth 6754 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 4916 replaces ax-rep 4905) with ax-sep 4916 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 8660 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8666). See ruALT 8667 for an alternate proof of ru 3586 derived from that fact. (Contributed by NM, 7-Aug-1994.)

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		pm5.19 374	. . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
2		eleq1w 2833	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
3		df-nel 3047	. . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
4		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5	4, 4	eleq12d 2844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	notbid 307	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
7	3, 6	syl5bb 272	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	2, 7	bibi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)))
9	8	spv 2422	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))
10	1, 9	mto 188	. . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)
11		abeq2 2881	. . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥))
12	10, 11	mtbir 312	. . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
13	12	nex 1879	. . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
14		isset 3359	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥})
15	13, 14	mtbir 312	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
16	15	nelir 3049	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757   ∉ wnel 3046  Vcvv 3351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nel 3047  df-v 3353

This theorem is referenced by: (None)