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|Description: Russell's Paradox.
Proposition 4.14 of [TakeutiZaring] p.
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 4546 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 set-building axioms of Null Set 0ex 4534, Pairing prex 4641, Union uniex 6584, Power Set pwex 4585, and Infinity omex 8145 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 5659 (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 8902 and Cantor's Theorem canth 6247 are provably false! (See ncanth 6248 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 4524 replaces ax-rep 4514) with ax-sep 4524 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 8109 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 8112). See ruALT 8113 for an alternate proof of ru 3265 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 362||. . . . . 6|
|2||eleq1 2516||. . . . . . . 8|
|3||df-nel 2624||. . . . . . . . 9|
|4||id 22||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2522||. . . . . . . . . 10|
|6||5||notbid 296||. . . . . . . . 9|
|7||3, 6||syl5bb 261||. . . . . . . 8|
|8||2, 7||bibi12d 323||. . . . . . 7|
|9||8||spv 2103||. . . . . 6|
|10||1, 9||mto 180||. . . . 5|
|11||abeq2 2559||. . . . 5|
|12||10, 11||mtbir 301||. . . 4|
|13||12||nex 1677||. . 3|
|14||isset 3048||. . 3|
|15||13, 14||mtbir 301||. 2|
|Colors of variables: wff setvar class|
|Syntax hints: wn 3 wb 188 wal 1441 wceq 1443 wex 1662 wcel 1886 cab 2436 wnel 2622 cvv 3044|
|This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430|
|This theorem depends on definitions: df-bi 189 df-an 373 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-clab 2437 df-cleq 2443 df-clel 2446 df-nel 2624 df-v 3046|
|This theorem is referenced by: (None)|
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