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Theorem ru 3290
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 {𝑥𝑥𝑥} (the "Russell class") for 𝐴, it asserted {𝑥𝑥𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥𝑥𝑥} ∉ 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 4580 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 4568, Pairing prex 4683, Union uniex 6663, Power Set pwex 4624, and Infinity omex 8233 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 5716 (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 8990 and Cantor's Theorem canth 6322 are provably false! (See ncanth 6323 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 4558 replaces ax-rep 4548) with ax-sep 4558 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 8197 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8200). See ruALT 8201 for an alternate proof of ru 3290 derived from that fact. (Contributed by NM, 7-Aug-1994.)

Assertion
Ref Expression
ru {𝑥𝑥𝑥} ∉ V

Proof of Theorem ru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm5.19 369 . . . . . 6 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 eleq1 2571 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 df-nel 2678 . . . . . . . . 9 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
4 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
54, 4eleq12d 2577 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 303 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
73, 6syl5bb 267 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
82, 7bibi12d 330 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑦𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
98spv 2151 . . . . . 6 (∀𝑥(𝑥𝑦𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
101, 9mto 183 . . . . 5 ¬ ∀𝑥(𝑥𝑦𝑥𝑥)
11 abeq2 2614 . . . . 5 (𝑦 = {𝑥𝑥𝑥} ↔ ∀𝑥(𝑥𝑦𝑥𝑥))
1210, 11mtbir 308 . . . 4 ¬ 𝑦 = {𝑥𝑥𝑥}
1312nex 1707 . . 3 ¬ ∃𝑦 𝑦 = {𝑥𝑥𝑥}
14 isset 3070 . . 3 ({𝑥𝑥𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝑥𝑥})
1513, 14mtbir 308 . 2 ¬ {𝑥𝑥𝑥} ∈ V
1615nelir 2781 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 191  wal 1466   = wceq 1468  wex 1692  wcel 1937  {cab 2491  wnel 2676  Vcvv 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-nel 2678  df-v 3068
This theorem is referenced by: (None)
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