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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 4174 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 4166, Pairing prex 4233, Union uniex 4532, Power Set pwex 4209, and Infinity omex 7360 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 5346 (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 8118 and Cantor's Theorem canth 6310 are provably false! (See ncanth 6311 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 4157 replaces ax-rep 4147) with ax-sep 4157 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 7327 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7330). See ruALT 7331 for an alternate proof of ru 3003 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 349||. . . . . 6|
|2||eleq1 2356||. . . . . . . 8|
|3||df-nel 2462||. . . . . . . . 9|
|4||id 19||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2364||. . . . . . . . . 10|
|6||5||notbid 285||. . . . . . . . 9|
|7||3, 6||syl5bb 248||. . . . . . . 8|
|8||2, 7||bibi12d 312||. . . . . . 7|
|9||8||spv 1951||. . . . . 6|
|10||1, 9||mto 167||. . . . 5|
|11||abeq2 2401||. . . . 5|
|12||10, 11||mtbir 290||. . . 4|
|13||12||nex 1545||. . 3|
|14||isset 2805||. . 3|
|15||13, 14||mtbir 290||. 2|
|16||df-nel 2462||. 2|
|17||15, 16||mpbir 200||1|
|Colors of variables: wff set class|
|Syntax hints: wn 3 wb 176 wal 1530 wex 1531 wceq 1632 wcel 1696 cab 2282 wnel 2460 cvv 2801|
|This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1536 ax-5 1547 ax-17 1606 ax-9 1644 ax-8 1661 ax-6 1715 ax-7 1720 ax-11 1727 ax-12 1878 ax-ext 2277|
|This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1310 df-ex 1532 df-nf 1535 df-sb 1639 df-clab 2283 df-cleq 2289 df-clel 2292 df-nel 2462 df-v 2803|
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