Editing Curry's paradox (section)

== Discussion ==

Curry's paradox can be formulated in any language supporting basic logic operations that also allows a self-recursive function to be constructed as an expression.  Two mechanisms that support the construction of the paradox are [[self-reference]] (the ability to refer to "this sentence" from within a sentence) and [[unrestricted comprehension]] in naive set theory. Natural languages nearly always contain many features that could be used to construct the paradox, as do many other languages. Usually, the addition of metaprogramming capabilities to a language will add the features needed. Mathematical logic generally does not allow explicit reference to its own sentences; however, the heart of [[Gödel's incompleteness theorems]] is the observation that a different form of self-reference can be added—see [[Gödel number]].

The rules used in the construction of the proof are the [[natural deduction|rule of assumption]] for conditional proof, the rule of [[rule of contraction|contraction]], and [[modus ponens]].  These are included in most common logical systems, such as first-order logic.

=== Consequences for some formal logic ===

In the 1930s, Curry's paradox and the related [[Kleene–Rosser paradox]], from which Curry's paradox was developed,<ref>{{cite journal |last=Curry |first=Haskell B. |date=Jun 1942 |title=The Combinatory Foundations of Mathematical Logic |journal=Journal of Symbolic Logic |volume=7 |pages=49&ndash;64 |doi=10.2307/2266302 |jstor=2266302 |s2cid=36344702 |number=2}}</ref><ref name=":0">{{cite journal |last=Curry |first=Haskell B. |date=Sep 1942 |title=The Inconsistency of Certain Formal Logics |journal=The Journal of Symbolic Logic |volume=7 |issue=3 |pages=115–117 |doi=10.2307/2269292 |jstor=2269292 |s2cid=121991184}}</ref> played a major role in showing that various formal logic systems allowing self-recursive expressions are [[Consistency|inconsistent]]. 

The axiom of unrestricted comprehension is not supported by [[Zermelo–Fraenkel set theory|modern set theory]], and Curry's paradox is thus avoided.