Editing Cantor's diagonal argument (section)

==Version for Quine's New Foundations==
The above proof fails for [[W. V. Quine]]'s "[[New Foundations]]" set theory (NF). In NF, the [[unrestricted comprehension|naive axiom scheme of comprehension]] is modified to avoid the paradoxes by introducing a kind of "local" [[type theory]].  In this axiom scheme,

:{ ''s'' ∈ ''S'': ''s'' ∉ ''f''(''s'') }

is ''not'' a set — i.e., does not satisfy the axiom scheme.  On the other hand, we might try to create a modified diagonal argument by noticing that

:{ ''s'' ∈ ''S'': ''s'' ∉ ''f''({''s''}) }

''is'' a set in NF.  In which case, if '''''P'''''<sub>1</sub>(''S'') is the set of one-element subsets of ''S'' and ''f'' is a proposed bijection from '''''P'''''<sub>1</sub>(''S'') to '''''P'''''(''S''), one is able to use [[proof by contradiction]] to prove that |'''''P'''''<sub>1</sub>(''S'')| &lt; |'''''P'''''(''S'')|.

The proof follows by the fact that if ''f'' were indeed a map ''onto'' '''''P'''''(''S''), then we could find ''r'' in ''S'', such that  ''f''({''r''}) coincides with the modified diagonal set, above.  We would conclude that if ''r'' is not in ''f''({''r''}), then ''r'' is in ''f''({''r''}) and vice versa.

It is ''not'' possible to put '''''P'''''<sub>1</sub>(''S'') in a one-to-one relation with ''S'', as the two have different types, and so any function so defined would violate the typing rules for the comprehension scheme.