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Theorem ru 3276
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  e.  _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  e/  x } (the "Russell class") for  A, it asserted  { x  |  x  e/  x }  e.  _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove  { x  |  x  e/  x }  e/  _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 4538 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 4526, Pairing prex 4633, Union uniex 6578, Power Set pwex 4577, and Infinity omex 8093 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 5647 (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 8887 and Cantor's Theorem canth 6237 are provably false! (See ncanth 6238 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 4517 replaces ax-rep 4507) with ax-sep 4517 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 8057 (derived from the Axiom of Regularity), so for us the Russell class equals the universe 
_V (theorem ruv 8060). See ruALT 8061 for an alternate proof of ru 3276 derived from that fact. (Contributed by NM, 7-Aug-1994.)

Assertion
Ref Expression
ru  |-  { x  |  x  e/  x }  e/  _V

Proof of Theorem ru
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pm5.19 358 . . . . . 6  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 eleq1 2474 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 df-nel 2601 . . . . . . . . 9  |-  ( x  e/  x  <->  -.  x  e.  x )
4 id 22 . . . . . . . . . . 11  |-  ( x  =  y  ->  x  =  y )
54, 4eleq12d 2484 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
65notbid 292 . . . . . . . . 9  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
73, 6syl5bb 257 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
82, 7bibi12d 319 . . . . . . 7  |-  ( x  =  y  ->  (
( x  e.  y  <-> 
x  e/  x )  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
98spv 2038 . . . . . 6  |-  ( A. x ( x  e.  y  <->  x  e/  x
)  ->  ( y  e.  y  <->  -.  y  e.  y ) )
101, 9mto 176 . . . . 5  |-  -.  A. x ( x  e.  y  <->  x  e/  x
)
11 abeq2 2526 . . . . 5  |-  ( y  =  { x  |  x  e/  x }  <->  A. x ( x  e.  y  <->  x  e/  x
) )
1210, 11mtbir 297 . . . 4  |-  -.  y  =  { x  |  x  e/  x }
1312nex 1648 . . 3  |-  -.  E. y  y  =  {
x  |  x  e/  x }
14 isset 3063 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  <->  E. y  y  =  {
x  |  x  e/  x } )
1513, 14mtbir 297 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
1615nelir 2740 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387    e/ wnel 2599   _Vcvv 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nel 2601  df-v 3061
This theorem is referenced by: (None)
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