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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 4158 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 4150, Pairing prex 4217, Union uniex 4516, Power Set pwex 4193, and Infinity omex 7344 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 5330 (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 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 8102 and Cantor's Theorem canth 6294 are provably false! (See ncanth 6295 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 4141 replaces ax-rep 4131) with ax-sep 4141 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 7311 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7314). See ruALT 7315 for an alternate proof of ru 2990 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 349||. . . . . 6|
|2||eleq1 2343||. . . . . . . 8|
|3||df-nel 2449||. . . . . . . . 9|
|4||id 19||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2351||. . . . . . . . . 10|
|6||5||notbid 285||. . . . . . . . 9|
|7||3, 6||syl5bb 248||. . . . . . . 8|
|8||2, 7||bibi12d 312||. . . . . . 7|
|9||8||spv 1938||. . . . . 6|
|10||1, 9||mto 167||. . . . 5|
|11||abeq2 2388||. . . . 5|
|12||10, 11||mtbir 290||. . . 4|
|13||12||nex 1542||. . 3|
|14||isset 2792||. . 3|
|15||13, 14||mtbir 290||. 2|
|16||df-nel 2449||. 2|
|17||15, 16||mpbir 200||1|
|Colors of variables: wff set class|
|Syntax hints: wn 3 wb 176 wal 1527 wex 1528 wceq 1623 wcel 1684 cab 2269 wnel 2447 cvv 2788|
|This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1533 ax-5 1544 ax-17 1603 ax-9 1635 ax-8 1643 ax-6 1703 ax-7 1708 ax-11 1715 ax-12 1866 ax-ext 2264|
|This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1310 df-ex 1529 df-nf 1532 df-sb 1630 df-clab 2270 df-cleq 2276 df-clel 2279 df-nel 2449 df-v 2790|
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