Tutorial 10: Set Theory V: Russell's Paradox, Axiomatic Set Theory, and Criticisms

Logical System

Russell's Paradox.

Unfortunately (naive) set theory is inconsistent, as was discovered by Bertrand Russell. The problem lies with the Schema of Abstraction, then all that is needed is to consider the set of those objects which are not members of themselves. This seems to be a perfectly good defining property for a set (most sets seem to be not members of themselves; for example, the null set ∅ is not a member of ∅ ie ∅∉ ∅). But if we give this set a name, say b, and ask if b is a member of itself: it is iff it isn't (which is a contradiction).

(∀y)(yε{x:Φ[x]}≡Φ[y]) Axiom Schema of Abstraction




b={x:x∉y} ∴ bεb≡ b∉b

And the proof is simple and straightforward (just an identity substitution into an instance of Abstraction).

This inconsistency or paradox ruins (naive) set theory.

If you decide to use the web application for the exercises you can launch it from here Deriver [Gentzen] — username 'logic' password 'logic'. Then either copy and paste the above formulas into the Journal or use the Deriver File Menu to Open Web Page with this address https://softoption.us/test/Deriver/CombinedTutorialsGentzen.html . 


You will need to set some Preferences for this.

  • set identity to true (and that will give you the identity rules)
  • set firstOrder to true (to get first order theories)
  • set setTheory to true (to get set theory)
  • and  you can check that the parser is set to gentzen.