Understanding Liar Paradox and Set Theory
Delve into the intriguing paradox of the liar, explored through historical and mathematical perspectives. Uncover solutions through Cantor's theorem, Russell's principles, and the axiom of regularity in set theory. Discover how these concepts offer insights into resolving logical contradictions.
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The Paradox of the Liar This sentence is false. Today is the 14th July.
A well-known variant of the liar One of Crete s own prophets has said it: Cretans are always liars, evil brutes, lazy gluttons. This saying is true. Apostle Paul (Titus I. 12-13) (1) If (1) is true, then (1) is false. as told by Epimenides If (1) as told by Epimenides is false, then there is some Cretan who made some true statement. Buridan This is O.K.?
Back to the Liar Bolzano: this can only refer to some other proposition (Satz an sich). But what does that other proposition say? (2) This sentence is false. This refers now to some (3). Etc. (3) (4) (1) (2)
If (1) is true, then (2) is false. Hence (3) is true. Hence (4) is false. Hence (5) is true. Hurrah! Every even-numbered sentence is false, every odd- numbered is true. No contradiction. But why not conversely? s is a liar sentence (under conditions F), if (s is true iff s is false (under conditions F)).
Cantors theorem: There is no mapping from H onto pot(H). Let f be a mapping from H into pot(H). Let be Hf = {x H: x f(x)} Let us suppose that there is an h H such that f(h) = Hf. Under this condition h Hf is a liar sentence. The solution is simple: the condition is false. Russell: Let us forget about powersets, let H be the set of all sets, f = Id and let us define Hf on the same way. HId={x H : x x In this case, we certainly have an h: HId itself. h HId (i. e. ' HId HId ) is a liar sentence again, and there is no (explicit) condition that we could reject...
Solutions: 1. H is not a set standard set theory. 2. Russell s vicious-circle principle: "no totality can contain members defined in terms of itself . In other words: the domain of a variable must not contain members that can be defined by referring to the whole domain. The Liar (and some other) paradoxes can be eliminated by this principle. See Russell, Mathematical logic as based on the theory of types
A similar principle in set theory: axiom of regularity (well- foundedness). (1) A nonempty set should have a member with which it has no common members. (A definition: a set H is wellfounded iff it has a member which doesn t have a common member with H.) In other words: there is no infinite chain. (2) There is no such series of sets: h0 h1 h2 [(1) and (2) are equivalent.]
Some instructive examples: 1. Representation of the notion of ordered pair in set theory: (3) Let <a, b> = {{a}, {a, b}} Easy to show: if <a, b> = <c, d> , then a=c and b=d. That s why (3) is an acceptable representation of the notion of ordered pair. Another usual representation: <a, b> = {a, {a, b}} Can you show the same property in this case, too? (Yes, but you must use the axiom of regularity.) Moral: You need regularity sometimes in cases you don t expect.
2. Let f(0) = 0, f(n) = (n, f(n-1)) = (n, (n-1, f(n-2)) . = (n, (n-1, (n-2, (n-3, 0) ))) 3. Another series: f(n) = (n, f(n+1)) f(0) = (0, (1, (2, ))) It seems that this series is a well-defined object, too. Moral: Sometimes we can allow infinite regress.
4. Let us have the following series of sentences: S1: All members of the series from S2 on are false. S2: All members from S3 are false. Etc. (Stephen Yablo 1998) Moral: Infinite regress can lead to paradoxical scenarios even without any sort of circularity.
The Liar in the 20th century: The G del-sentence of a consistent first-order theory is true iff it is not provable. (Of course, only theories in that G del numbering is possible can have a G del sentence.) If mathematical truth be the same as provability, then the G del-sentence is a liar sentence. The concept of provability in a G del-numbered theory is necessarily different from the concept of truth in the same theory. Of course, the concept of truth for a G del-numbered theory can be provability in another theory. This is what we do when we construct models for theories within set theory.
Tarskis theorem: We cannot introduce into the language of a formalized theory a truth predicate without making the theory inconsistent (under some reasonable conditions for the truth predicate, including that every sentence be either true or false). Cause: in such theories, the Liar paradox can be formulated. Solution until the seventies: prohibition of semantical predicates in the object language. This is a workable solution, but we must pay a too high price for it. Kripke (1975) : Outline of a theory of truth introduces partial truth predicate(s) into the object language.
