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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 , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . 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 4347 asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, leading him to introduce the setbuilding axioms of Null Set 0ex 4339, Pairing prex 4406, Union uniex 4705, Power Set pwex 4382, and Infinity omex 7598 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 5531 (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 set 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 MorseKelley (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 8355 and Cantor's Theorem canth 6539 are provably false! (See ncanth 6540 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which axsep 4330 replaces axrep 4320) with axsep 4330 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 9:119 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 7565 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7568). See ruALT 7569 for an alternate proof of ru 3160 derived from that fact. (Contributed by NM, 7Aug1994.) 
Ref  Expression 

ru 
Step  Hyp  Ref  Expression 

1  pm5.19 350  . . . . . 6  
2  eleq1 2496  . . . . . . . 8  
3  dfnel 2602  . . . . . . . . 9  
4  id 20  . . . . . . . . . . 11  
5  4, 4  eleq12d 2504  . . . . . . . . . 10 
6  5  notbid 286  . . . . . . . . 9 
7  3, 6  syl5bb 249  . . . . . . . 8 
8  2, 7  bibi12d 313  . . . . . . 7 
9  8  spv 1965  . . . . . 6 
10  1, 9  mto 169  . . . . 5 
11  abeq2 2541  . . . . 5  
12  10, 11  mtbir 291  . . . 4 
13  12  nex 1564  . . 3 
14  isset 2960  . . 3  
15  13, 14  mtbir 291  . 2 
16  15  nelir 2698  1 
Colors of variables: wff set class 
Syntax hints: wn 3 wb 177 wal 1549 wex 1550 wceq 1652 wcel 1725 cab 2422 wnel 2600 cvv 2956 
This theorem was proved from axioms: ax1 5 ax2 6 ax3 7 axmp 8 axgen 1555 ax5 1566 ax17 1626 ax9 1666 ax8 1687 ax6 1744 ax7 1749 ax11 1761 ax12 1950 axext 2417 
This theorem depends on definitions: dfbi 178 dfan 361 dftru 1328 dfex 1551 dfnf 1554 dfsb 1659 dfclab 2423 dfcleq 2429 dfclel 2432 dfnel 2602 dfv 2958 
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