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Description: Russell's Paradox.
Proposition 4.14 of [TakeutiZaring] p.
14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as A ∈ V, asserted that any collection of sets A is a set i.e. belongs to the universe V of all sets. In particular, by substituting {x ∣ x ∉ x} (the "Russell class") for A, it asserted {x ∣ x ∉ x} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {x ∣ x ∉ x} ∉ V. 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 A is a set only when it is smaller than some other set B. However, Zermelo was then faced with a "chicken and egg" problem of how to show B is a set, leading him to introduce the setbuilding 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 MorseKelley (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 axsep (future) replaces axrep (future)) with axsep (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 9:119 (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 V (theorem ruv (future)). See ruALT (future) for an alternate proof of ru 3045 derived from that fact. (Contributed by NM, 7Aug1994.) 
Ref  Expression 

ru  ⊢ {x ∣ x ∉ x} ∉ V 
Step  Hyp  Ref  Expression 

1  pm5.19 349  . . . . . 6 ⊢ ¬ (y ∈ y ↔ ¬ y ∈ y)  
2  eleq1 2413  . . . . . . . 8 ⊢ (x = y → (x ∈ y ↔ y ∈ y))  
3  dfnel 2519  . . . . . . . . 9 ⊢ (x ∉ x ↔ ¬ x ∈ x)  
4  id 19  . . . . . . . . . . 11 ⊢ (x = y → x = y)  
5  4, 4  eleq12d 2421  . . . . . . . . . 10 ⊢ (x = y → (x ∈ x ↔ y ∈ y)) 
6  5  notbid 285  . . . . . . . . 9 ⊢ (x = y → (¬ x ∈ x ↔ ¬ y ∈ y)) 
7  3, 6  syl5bb 248  . . . . . . . 8 ⊢ (x = y → (x ∉ x ↔ ¬ y ∈ y)) 
8  2, 7  bibi12d 312  . . . . . . 7 ⊢ (x = y → ((x ∈ y ↔ x ∉ x) ↔ (y ∈ y ↔ ¬ y ∈ y))) 
9  8  spv 1998  . . . . . 6 ⊢ (∀x(x ∈ y ↔ x ∉ x) → (y ∈ y ↔ ¬ y ∈ y)) 
10  1, 9  mto 167  . . . . 5 ⊢ ¬ ∀x(x ∈ y ↔ x ∉ x) 
11  abeq2 2458  . . . . 5 ⊢ (y = {x ∣ x ∉ x} ↔ ∀x(x ∈ y ↔ x ∉ x))  
12  10, 11  mtbir 290  . . . 4 ⊢ ¬ y = {x ∣ x ∉ x} 
13  12  nex 1555  . . 3 ⊢ ¬ ∃y y = {x ∣ x ∉ x} 
14  isset 2863  . . 3 ⊢ ({x ∣ x ∉ x} ∈ V ↔ ∃y y = {x ∣ x ∉ x})  
15  13, 14  mtbir 290  . 2 ⊢ ¬ {x ∣ x ∉ x} ∈ V 
16  dfnel 2519  . 2 ⊢ ({x ∣ x ∉ x} ∉ V ↔ ¬ {x ∣ x ∉ x} ∈ V)  
17  15, 16  mpbir 200  1 ⊢ {x ∣ x ∉ x} ∉ V 
Colors of variables: wff set class 
Syntax hints: ¬ wn 3 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {cab 2339 ∉ wnel 2517 Vcvv 2859 
This theorem is referenced by: epprc 5855 
This theorem was proved from axioms: ax1 5 ax2 6 ax3 7 axmp 8 axgen 1546 ax5 1557 ax17 1616 ax9 1654 ax8 1675 ax6 1729 ax7 1734 ax11 1746 ax12 1925 axext 2334 
This theorem depends on definitions: dfbi 177 dfan 360 dftru 1319 dfex 1542 dfnf 1545 dfsb 1649 dfclab 2340 dfcleq 2346 dfclel 2349 dfnel 2519 dfv 2861 
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