|Metamath Proof Explorer||
|Mirrors > Home > MPE Home > Th. List > ru||Structured version Unicode version|
|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 4347 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 4339, Pairing prex 4406, Union uniex 4705, Power Set pwex 4382, and Infinity omex 7598 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 5531 (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 8355 and Cantor's Theorem canth 6539 are provably false! (See ncanth 6540 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 4330 replaces ax-rep 4320) with ax-sep 4330 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 7565 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7568). See ruALT 7569 for an alternate proof of ru 3160 derived from that fact. (Contributed by NM, 7-Aug-1994.)
|1||pm5.19 350||. . . . . 6|
|2||eleq1 2496||. . . . . . . 8|
|3||df-nel 2602||. . . . . . . . 9|
|4||id 20||. . . . . . . . . . 11|
|5||4, 4||eleq12d 2504||. . . . . . . . . 10|
|6||5||notbid 286||. . . . . . . . 9|
|7||3, 6||syl5bb 249||. . . . . . . 8|
|8||2, 7||bibi12d 313||. . . . . . 7|
|9||8||spv 1965||. . . . . 6|
|10||1, 9||mto 169||. . . . 5|
|11||abeq2 2541||. . . . 5|
|12||10, 11||mtbir 291||. . . 4|
|13||12||nex 1564||. . 3|
|14||isset 2960||. . 3|
|15||13, 14||mtbir 291||. 2|
|Colors of variables: wff set class|
|Syntax hints: wn 3 wb 177 wal 1549 wex 1550 wceq 1652 wcel 1725 cab 2422 wnel 2600 cvv 2956|
|This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1555 ax-5 1566 ax-17 1626 ax-9 1666 ax-8 1687 ax-6 1744 ax-7 1749 ax-11 1761 ax-12 1950 ax-ext 2417|
|This theorem depends on definitions: df-bi 178 df-an 361 df-tru 1328 df-ex 1551 df-nf 1554 df-sb 1659 df-clab 2423 df-cleq 2429 df-clel 2432 df-nel 2602 df-v 2958|
|Copyright terms: Public domain||W3C validator|