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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 4538 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 4526, Pairing prex 4633, Union uniex 6578, Power Set pwex 4577, and Infinity omex 8093 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 5647 (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 8887 and Cantor's Theorem canth 6237 are provably false! (See ncanth 6238 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 4517 replaces ax-rep 4507) with ax-sep 4517 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 8057 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 8060). See ruALT 8061 for an alternate proof of ru 3276 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 358||. . . . . 6|
|2||eleq1 2474||. . . . . . . 8|
|3||df-nel 2601||. . . . . . . . 9|
|4||id 22||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2484||. . . . . . . . . 10|
|6||5||notbid 292||. . . . . . . . 9|
|7||3, 6||syl5bb 257||. . . . . . . 8|
|8||2, 7||bibi12d 319||. . . . . . 7|
|9||8||spv 2038||. . . . . 6|
|10||1, 9||mto 176||. . . . 5|
|11||abeq2 2526||. . . . 5|
|12||10, 11||mtbir 297||. . . 4|
|13||12||nex 1648||. . 3|
|14||isset 3063||. . 3|
|15||13, 14||mtbir 297||. 2|
|Colors of variables: wff setvar class|
|Syntax hints: wn 3 wb 184 wal 1403 wceq 1405 wex 1633 wcel 1842 cab 2387 wnel 2599 cvv 3059|
|This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1639 ax-4 1652 ax-5 1725 ax-6 1771 ax-7 1814 ax-10 1861 ax-11 1866 ax-12 1878 ax-13 2026 ax-ext 2380|
|This theorem depends on definitions: df-bi 185 df-an 369 df-tru 1408 df-ex 1634 df-nf 1638 df-sb 1764 df-clab 2388 df-cleq 2394 df-clel 2397 df-nel 2601 df-v 3061|
|This theorem is referenced by: (None)|
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