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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 (future) 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 4110, Pairing prex 4112, Union uniex 4317, Power Set pwex 4329, and Infinity omex (future) 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 (future) (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 (future) and Cantor's Theorem canth (future) are provably false! (See ncanth (future) 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 (future) replaces ax-rep (future)) with ax-sep (future) 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 (future) (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv (future)). See ruALT (future) for an alternate proof of ru 3045 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 349||. . . . . 6|
|2||eleq1 2413||. . . . . . . 8|
|3||df-nel 2519||. . . . . . . . 9|
|4||id 19||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2421||. . . . . . . . . 10|
|6||5||notbid 285||. . . . . . . . 9|
|7||3, 6||syl5bb 248||. . . . . . . 8|
|8||2, 7||bibi12d 312||. . . . . . 7|
|9||8||spv 1998||. . . . . 6|
|10||1, 9||mto 167||. . . . 5|
|11||abeq2 2458||. . . . 5|
|12||10, 11||mtbir 290||. . . 4|
|13||12||nex 1555||. . 3|
|14||isset 2863||. . 3|
|15||13, 14||mtbir 290||. 2|
|16||df-nel 2519||. 2|
|17||15, 16||mpbir 200||1|
|Colors of variables: wff set class|
|Syntax hints: wn 3 wb 176 wal 1540 wex 1541 wceq 1642 wcel 1710 cab 2339 wnel 2517 cvv 2859|
|This theorem is referenced by: epprc 5855|
|This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334|
|This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nel 2519 df-v 2861|
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