Self- Referential Paradoxes Noson S. Yanofsky Preliminaries Self-Referential Paradoxes Three Motivating Examples Philosophical Noson S. Yanofsky Interlude Cantor's Inequalities Brooklyn College, CUNY Main Theorem February 17, 2021 Turing's Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading My Agenda Self- Some of the most profound and famous theorems in Referential Paradoxes mathematics and computer science of the past 150 years are Noson S. instances of self-referential paradoxes. Yanofsky Georg Cantor's theorem that shows there are dierent Preliminaries levels of innity; Three Bertrand Russell's paradox which proves that simple set Motivating Examples theory is inconsistent; Philosophical Interlude Kurt Gödel's famous incompleteness theorems that Cantor's demonstrates a limitation of the notion of proof; Inequalities Alan Turing's realization that some problems can never be Main Theorem solved by a computer; Turing's and much more. Halting Problem Amazingly, all these diverse theorems can be seen as instances Fixed Points of a single simple theorem of basic category theory. We describe in Logic this theorem and show some of the instances. No category Two Other Paradoxes theory is needed for this talk. Further Reading My Hidden Agenda Self- Referential Paradoxes Noson S. Yanofsky This talk is a single section (of 58) of an introductory category Preliminaries theory textbook I am writing called Three Motivating MONOIDAL CATEGORIES Examples A Unifying Concept in Mathematics, Physics, and Computing Philosophical Interlude Cantor's The goal is to show the power of category theory (and to get Inequalities you to buy the book!!!) The point is that with a little category Main Theorem theory, one can know a hell-of-a-lot of mathematics, physics, Turing's and computers. Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading Outline of the Talk Self- Referential Paradoxes 1 Some Preliminaries Noson S. Yanofsky 2 Three Motivating Examples Preliminaries 3 Philosophical Interlude Three Motivating 4 Examples Cantor's Inequalities Philosophical Interlude 5 The Main Theorem About Self-Referential Paradoxes Cantor's Inequalities 6 Turing's Halting Problem Main Theorem 7 Fixed Points in Logic Turing's Halting Problem 8 Two Other Paradoxes Fixed Points in Logic 9 Further Reading Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes Before we leap into all the examples, there are some technical Noson S. Yanofsky ideas about sets. Preliminaries Let 2 = f0; 1g be a set with two values which correspond Three to true and false. Motivating Examples Let S be a set. A function g : S −! 2 is a characteristic Philosophical function and describes a subset of . Interlude S Cantor's Consider a set function f : S × S −! 2. The f accepts two Inequalities elements of S and outputs either 0 or 1. Think of Main Theorem Turing's f (a; b) = 1 Halting Problem meaning is a part of or is described by . Fixed Points a b a b in Logic Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes For any element s0 of S, consider the function f where the Noson S. Yanofsky second input is always s0. We say that s0 is hardwired into the function. This gives us a function Preliminaries Three 2 Motivating f ( ; s0): S −! Examples Philosophical with only one input. Since this function goes from S to 2, Interlude it also is a characteristic function and describes a subset of Cantor's Inequalities S. The subset is Main Theorem fs 2 S : f (s; s0) = 1g: Turing's Halting Problem It is all the elements that are part of s0 or all the Fixed Points in Logic elements that are described by s0. Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes Noson S. Yanofsky For dierent f 's and various elements of S, there are Preliminaries dierent characteristic functions which describe various Three subsets. Motivating Examples We now ask a simple question: given g : S −! 2 and Philosophical 2, is there an in such that Interlude f : S × S −! s0 S g Cantor's characterizes the same subset as f ( ; s0)? Inequalities To restate, for a given g and f , does there exist an s0 2 S Main Theorem such that g( ) = f ( ; s0)? If such an s0 exists, then we Turing's say g can be represented by f . Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading Some Preliminaries Self- Referential For every set S there is a set map ∆: S −! S × S called Paradoxes the diagonal map that takes an element t to the ordered Noson S. Yanofsky pair (t; t). This is the core of self reference. Preliminaries If f : S × S −! 2 is a function that evaluates the Three relationship of S elements to S elements, then Motivating Examples f Philosophical S × S / 2 Interlude < ∆ Cantor's Inequalities Main S Theorem Turing's takes every element as follows Halting t 2 S Problem Fixed Points t 7−! (t; t) 7−! f (t; t): in Logic Two Other Paradoxes This evaluates t with itself. Further Reading The Barber Paradox Self- Bertrand Russell was a great expositor. The barber paradox is Referential Paradoxes attributed to Russell and is used to explain some of the central Noson S. ideas of self-referential systems. Imagine an isolated village in Yanofsky the Austrian alps where it is dicult for villagers to leave and Preliminaries for itinerant barbers to come to the village. This village has Three exactly one barber and there is a strict rule that is enforced: Motivating Examples Philosophical A villager cuts his own hair i he does not go to the barber. Interlude Cantor's If the villager will cut his own hair, why should he go to the Inequalities barber? On the other hand, if the villager goes to the barber, Main Theorem he will not need to cut his own hair. This works out very well Turing's for the villagers except for one: the barber. Who cuts the Halting Problem barber's hair? If the barber cuts his own hair, then he is Fixed Points violating the village ordinance by cutting his own hair and in Logic having his hair cut by the barber. If he goes to the barber, then Two Other Paradoxes he is also cutting his own hair. This is illegal! Further Reading The Barber Paradox Self- Let us formalize the problem. Let the set Vill consist of all the Referential Paradoxes villagers in the village. The function Noson S. Yanofsky f : Vill × Vill −! 2 Preliminaries describes who cuts whose hair in the village. It is dened for 0 Three villagers v and v as Motivating Examples 8 1 : if the hair of v is cut by v 0 Philosophical 0 < Interlude f (v; v ) = 0 Cantor's : 0 : if the hair of v is not cut by v : Inequalities Main We can now express the village ordinance as saying that for all v Theorem Turing's f (v; v) = 1 if and only if f (v; barber) = 0: Halting Problem This is true for all v including v = barber. In this case we get: Fixed Points in Logic f (barber; barber) = 1 if and only if f (barber; barber) = 0: Two Other Paradoxes This is a contradiction! Further Reading The Barber Paradox Self- Referential Let us be more categorical. There is Paradoxes The diagonal set function that is Noson S. ∆: Vill −! Vill × Vill Yanofsky dened as ∆(v) = (v; v). Preliminaries There is also a negation function NOT : 2 −! 2 dened as Three NOT (0) = 1 and NOT (1) = 0. Motivating Examples Composing f with ∆ and NOT gives us g : Vill −! 2 as Philosophical Interlude in the following commutative diagram: Cantor's Inequalities f Vill × Vill / 2 Main 9 Theorem ∆ NOT Turing's Halting 2 Problem Vill g / Fixed Points in Logic g = NOT ◦ f ◦ ∆: Two Other Paradoxes Further Reading The Barber Paradox Self- For a villager v, Referential Paradoxes Noson S. g(v) == NOT (f (∆(v))) = NOT (f (v; v)): Yanofsky So Preliminaries g(v) = 1 Three Motivating if and only if Examples Philosophical NOT (f (v; v)) = 1 Interlude if and only if Cantor's Inequalities f (v; v) = 0 Main Theorem if and only if Turing's the hair of is not cut by . Halting v v Problem In other words g(v) = 1 if and only if v does not cut his own Fixed Points hair. in Logic Two Other g is the characteristic function of the subset of villagers who do Paradoxes not cut their own hair. Further Reading The Barber Paradox Self- We now ask the simple question: can g be represented by f ? In Referential Paradoxes other words, is there a villager v0 such that g( ) = f ( ; v0)? It Noson S. stands to reason that the barber is the villager who can Yanofsky represent g. After all, f ( ; barber) describes all the villagers Preliminaries who get their hair cut by the barber. Three Motivating Examples g(v) = f (v; barber) Philosophical Interlude What about v = barber. Cantor's Inequalities barber barber barber Main g( ) = f ( ; ): Theorem Turing's But the denition of g is given as Halting Problem g(barber) = NOT (f (barber; barber)). We conclude that g is Fixed Points not represented by f ( ; barber) That is, the set of villagers in Logic who do not cut their own hair can not be the same as the set of Two Other Paradoxes villagers who get their hair cut by anyone. Further Reading The Barber Paradox Matrix Form Self- Referential It is helpful to describe this problem in matrix form. Let us Paradoxes consider the set Vill as fv1; v2; v3;:::; v g.