A Study On Curry’s Paradox

The Curry’s paradox was raised by Haskell Curry in1942. the research was based onthe in-consistency of combinatory logic system in the very beginning and was extended tothe realm of set theory by Prior and Fitch. Geach made the study of Curry’s paradoxextended to the semantic theory. Thanks to the work of Geach, there is a common style in thefield of philosophy of logics “If this statement is true, then God exists.”(This statement refersto “If this statement is true, then God exists.”)There are few papers on the Study of Curry’s paradox at home, these papers are almostbased on some special realm (set theory and theory of truth), and never in the field ofcombinatory logic. The thesis is based on combinatory logic, and Reveal the mechanism ofCurry’s paradox in various theory. Some important results will be given at the end.The first chapter will introduce the thought of combinatory logic, and some basicprinciple we will discuss the original version of Curry’s paradox.The chapter2introduce theminimal illative combinatory logic to discuss the Curry’s paradox based on the extension ofpure combinatory logic. We can see that the combinatorial completeness in combinatorylogic is familiar to the comprehension axiom in na ve set theory, The third chapter describesthe set-theoretic version of the Curry’s paradox, the comprehension axiom will lead to theCurry paradox. Chapter4describes the Curry’s semantic paradox, and tests the Tarski’sun-definability theorem by T-Schema. Chapter5proofs the Curry style L(o|¨)b’s theorem by theDiagonal lemma, and proof Tarski’s un-definability theorem. The part of conclusion willintroduce the significance of Curry’s paradox to philosophical logic and mathematical logic.