Theorem ru 3046

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 in set.mm 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 4111, Pairing prex 4113, Union uniex 4318, Power Set pwex 4330, and Infinity omex in set.mm 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 in set.mm (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 in set.mm and Cantor's Theorem canth in set.mm are provably false! (See ncanth in set.mm 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 in set.mm replaces ax-rep in set.mm) with ax-sep 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 ZF set theory, every set is a member of the Russell class by elirrv in set.mm (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv in set.mm). See ruALT in set.mm for an alternate proof of ru 3046 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		pm5.19 349	. . . . . 6 ⊢ ¬ (y ∈ y ↔ ¬ y ∈ y)
2		eleq1 2413	. . . . . . . 8 ⊢ (x = y → (x ∈ y ↔ y ∈ y))
3		df-nel 2520	. . . . . . . . 9 ⊢ (x ∉ x ↔ ¬ x ∈ x)
4		id 19	. . . . . . . . . . 11 ⊢ (x = y → x = y)
5	4, 4	eleq12d 2421	. . . . . . . . . 10 ⊢ (x = y → (x ∈ x ↔ y ∈ y))
6	5	notbid 285	. . . . . . . . 9 ⊢ (x = y → (¬ x ∈ x ↔ ¬ y ∈ y))
7	3, 6	syl5bb 248	. . . . . . . 8 ⊢ (x = y → (x ∉ x ↔ ¬ y ∈ y))
8	2, 7	bibi12d 312	. . . . . . 7 ⊢ (x = y → ((x ∈ y ↔ x ∉ x) ↔ (y ∈ y ↔ ¬ y ∈ y)))
9	8	spv 1998	. . . . . 6 ⊢ (∀x(x ∈ y ↔ x ∉ x) → (y ∈ y ↔ ¬ y ∈ y))
10	1, 9	mto 167	. . . . 5 ⊢ ¬ ∀x(x ∈ y ↔ x ∉ x)
11		abeq2 2459	. . . . 5 ⊢ (y = {x ∣ x ∉ x} ↔ ∀x(x ∈ y ↔ x ∉ x))
12	10, 11	mtbir 290	. . . 4 ⊢ ¬ y = {x ∣ x ∉ x}
13	12	nex 1555	. . 3 ⊢ ¬ ∃y y = {x ∣ x ∉ x}
14		isset 2864	. . 3 ⊢ ({x ∣ x ∉ x} ∈ V ↔ ∃y y = {x ∣ x ∉ x})
15	13, 14	mtbir 290	. 2 ⊢ ¬ {x ∣ x ∉ x} ∈ V
16		df-nel 2520	. 2 ⊢ ({x ∣ x ∉ x} ∉ V ↔ ¬ {x ∣ x ∉ x} ∈ V)
17	15, 16	mpbir 200	1 ⊢ {x ∣ x ∉ x} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ∉ wnel 2518  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nel 2520  df-v 2862

This theorem is referenced by:  epprc  5828