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 in nity; 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 = {0, 1} 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 {s ∈ S : f (s, s0) = 1}. 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 ∈ 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 ∈ 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 di cult 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 de ned for 0 Three villagers v and v as Motivating Examples 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 de ned as ∆(v) = (v, v). Preliminaries There is also a negation function NOT : 2 −! 2 de ned 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 de nition 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 {v1, v2, v3,..., v }. We can then Noson S. n Yanofsky describe the function f : Vill × Vill −! 2 as a matrix. Let us say that the barber is v4. Notice that every row has exactly one Preliminaries 1 (every villager gets their haircut in only one place): either Three Motivating along the diagonal (the villager cuts their own hair) or in the 4 Examples v column (the villager goes to the barber.) Since it can only be Philosophical Interlude one or the other, the numbers along the diagonal Cantor's 1 0 1 0 1 are almost the exact opposite of the numbers Inequalities , , , ?, ,...,
Main along the v4 column 0, 1, 0, ?, 1,..., 0. This is a restatement of Theorem the rule of the village. There is only one problem: what is in the Turing's Halting (v4, v4) position. We put a question mark because that entry Problem cannot be the opposite of itself. This way of seeing the problem Fixed Points in Logic will arise over and over again. Here we can see why these
Two Other paradoxes are related to proofs called diagonal arguments. Paradoxes
Further Reading The Barber Paradox Matrix Form
Self- Referential Paradoxes Cutter Noson S. Yanofsky f v1 v2 v3 v4 v5 ··· vn
Preliminaries v1 1 0 0 0 0 ··· 0 Three Motivating Examples v2 0 0 0 1 0 ··· 0 Philosophical Interlude v3 0 0 1 0 0 ··· 0 Cantor's Inequalities 0 0 0 ? 0 0
Cuttee v4 ··· Main Theorem 5 0 0 0 1 0 0 Turing's v ··· Halting . . . . Problem . . . . Fixed Points in Logic vn 0 0 0 0 0 ··· 1 Two Other Paradoxes
Further Reading The Barber Paradox Avoiding Contradictions
Self- Referential Paradoxes What is the resolution to this paradox? There are many Noson S. Yanofsky attempts to solve this paradox, but they are not very successful. For example, the barber resigns as barber before cutting his own Preliminaries hair. (But that means that there is no barber in the town). Or Three Motivating the wife of the barber cuts the barber's hair. (But that means Examples that there are two barbers in the town.) Or the barber is bald. Philosophical Interlude Or the barber is a long-haired hippie. Or the rule is ignored Cantor's while the barber cuts his own hair, etc. All these are saying the Inequalities same thing: the village with this important rule cannot exist. Main Theorem Because if the village with this rule existed, there would be a Turing's contradiction. There are no contradictions in the physical world. Halting Problem The only way the world can be free of contradictions is if this Fixed Points proposed village with this strict rule does not exist. in Logic
Two Other Paradoxes
Further Reading Russell's Paradox
Self- Referential Paradoxes
Noson S. Yanofsky This paradox concerns sets which are considered the foundation of much of mathematics. As is known, sets contain elements. Preliminaries
Three The elements can be anything. In particular an element in a set Motivating can be a set itself. A set containing itself is also not so strange. Examples
Philosophical Here are three examples of sets that contain themselves: Interlude The set of all ideas discussed in this talk Cantor's Inequalities The set that contains all the sets that have more than Main Theorem three objects Turing's The set of abstract ideas. Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Russell's Paradox
Self- Referential Paradoxes
Noson S. If you do not like sets that contain themselves, you might want Yanofsky to consider Russell's set which is the set of all sets that do not
Preliminaries contain themselves. Formally,
Three Motivating Examples R = {set S : S does not contain S} = {S : S ∈/ S}.
Philosophical Interlude Now ask yourself the simple question: does R contain itself? In Cantor's symbols, we ask if ? Let us consider the possible answers. Inequalities R ∈ R
Main If R ∈ R, then since R fails to satisfy the requirements of being Theorem a member of R, we get that R ∈/ R. In contrast, if R ∈/ R, then Turing's Halting since R satis es the requirement of belonging to R, we have Problem that R ∈ R. This is a contradiction. Fixed Points in Logic
Two Other Paradoxes
Further Reading Russell's Paradox
Self- Referential Let us formulate this. There is a collection of all sets called Set. Paradoxes There is also a two-place function f : Set × Set −! 2 that Noson S. Yanofsky describes which sets are elements of which other sets.
Preliminaries 1 : if S ∈ S0 Three 0 Motivating f (S, S ) = Examples 0 : if S 6∈ S0. Philosophical Interlude f Cantor's Set × Set / 2 Inequalities 9 ∆ Main NOT Theorem Turing's Set / 2 Halting g Problem
Fixed Points The value g(S) = NOT (f (S, S)). This means g(S) = 1 if and in Logic only if f (S, S) = 0. g is the characteristic function of those sets Two Other Paradoxes that do not contain themselves.
Further Reading Russell's Paradox
Self- Now we ask the simple question: does there exist a set R such Referential Paradoxes that g() is represented by f as f ( , R). That is, we want a Noson S. set R such that Yanofsky
Preliminaries g(S) = 1 if and only if f (S, R) = 1
Three Motivating and Examples 0 if and only if 0 Philosophical g(S) = f (S, R) = . Interlude This means that contains only the sets that do not contain Cantor's R Inequalities themselves. The problem is that if such a set R exists, then we Main can ask about , i.e., is . On the one hand is Theorem g(R) R ∈ R g(R)
Turing's de ned as NOT (f (R, R)) and on the other hand, if f Halting represents with , then . That is, Problem g R g(R) = f (R, R)
Fixed Points in Logic f (R, R) = g(R) = NOT (f (R, R)). Two Other Paradoxes This is a contradiction. Further Reading Russell's Paradox Matrix Form
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries Let us look at Russell's paradox from a matrix point of view. Three Consider the in nite collection as . We can Motivating Set {S1, S2, S3,...} Examples then describe the function f : Set × Set −! 2 as a matrix. Philosophical Interlude Notice that the diagonal is dierent than every column of the
Cantor's array. This is a way of saying that that the diagonal (which is Inequalities g) cannot be represented by any column of the array. Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Russell's Paradox Matrix Form
Self- Referential Paradoxes
Noson S. Yanofsky Subset
Preliminaries f S1 S2 S3 S4 S5 ··· Three Motivating 1 0 0 0 0 1 Examples S1 ¬ = ···
Philosophical Interlude S2 0 ¬0 = 1 0 1 0 ··· Cantor's Inequalities S3 0 0 ¬1 = 0 0 0 ··· Main Theorem S4 1 0 0 ¬1 = 0 0 ··· Turing's Element Halting Problem S5 0 1 0 1 ¬0 = 1 ··· Fixed Points . . . .. in Logic . . . .
Two Other Paradoxes
Further Reading Russell's Paradox Avoiding Contradictions
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries The only way to avoid this contradiction is to accept that the Three function g cannot be represented by any element of Set. This Motivating Examples translates into meaning that the collection of all sets that do
Philosophical not contain themselves does not form a set, i.e., this collection Interlude is not an element of Set. While such a collection seems to be a Cantor's Inequalities well-de ned notion, we have shown that if we say that this Main collection is an element of Set, then there is a contradiction. Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Heterological Paradox
Self- Referential Now for a linguistic paradox. The heterological paradox, also Paradoxes called Grelling's paradox after Kurt Grelling, who rst Noson S. Yanofsky formulated it, is about adjectives (words that modify nouns).
Preliminaries Consider several adjectives and ask if they describe themselves. Three English is English. Motivating Examples French is not French (francais is francais.) Philosophical Interlude German is not German (Deutsch is Deutsch.)
Cantor's Inequalities abbreviated is not abbreviated, Main unabbreviated is unabbreviated Theorem
Turing's hyphenated is not hyphenated, etc. Halting Problem Call all adjectives that describe themselves autological. In Fixed Points contrast, call all adjectives that do not describe themselves as in Logic heterological. Two Other Paradoxes
Further Reading Heterological Paradox
Self- Referential Paradoxes autological heterological
Noson S. Yanofsky English non-English Preliminaries French Three Motivating Examples German Philosophical Interlude noun verb Cantor's Inequalities unhyphenated hyphenated Main Theorem unabbreviated abbreviated Turing's Halting Problem polysyllabic monosyllabic Fixed Points . . in Logic . . Two Other Paradoxes
Further Reading Heterological Paradox
Self- Referential Paradoxes
Noson S. Yanofsky Let us ask a simple question? Is heterological heterological? Preliminaries That is, does it belong on the left side or the right side of the Three table? Let us go through the two possibilities. Motivating Examples If heterological is not heterological, then it does not Philosophical Interlude describe itself and therefore it is heterological.
Cantor's Inequalities If heterological is heterological, then it does describe
Main itself and therefor is not heterological. Theorem This is a contradiction. Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Heterological Paradox
Self- Let us formulate this paradox categorically. There is a set Adj Referential Paradoxes of adjectives and a function f : Adj × Adj −! 2 which is Noson S. de ned for adjectives a and a0 as follows: Yanofsky
Preliminaries 1 : if a is described by a0 Three f (a, a0) = Motivating 0 Examples 0 : if a is not described by a . Philosophical Interlude Use f to formulate g as as the composition of the following Cantor's three maps: Inequalities f Main Adj × Adj / 2 Theorem 9 Turing's ∆ NOT Halting Problem 2 Fixed Points Adj g / . in Logic Two Other The function g is the characteristic function of those adjectives Paradoxes that do not describe themselves. Further Reading Heterological Paradox
Self- Can g be represented by some element in Adj? Is there some Referential Paradoxes adjective, say heterological, that we can use in f to represent Noson S. g? That is, is it true that g( ) = f ( , heterological)? Yanofsky heterological Preliminaries g(A) = f (A, ). Three For all including heterological Motivating A A = Examples
Philosophical g(heterological) = f (heterological, heterological). Interlude
Cantor's But that would give a contradiction because by the de nition of Inequalities g we have Main Theorem g(heterological) = NOT (f (heterological, heterological)). Turing's Halting Problem The only conclusion we can come to is that g() cannot be Fixed Points represented by f . That is, the set of all adjectives that do not in Logic describe themselves cannot be represented by heterological. Two Other Paradoxes However, that is exactly the de nition of heterological! Further Reading Heterological Paradox Matrix Form
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries Three The hetrological paradox can be described with a matrix similar Motivating Examples to the earlier matrices. The set Adj = {A1, A2, A3,...}. Again Philosophical we would have a changed diagonal that would be dierent than Interlude every column in the array. Cantor's Inequalities
Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Heterological Paradox Avoiding Contradictions
Self- Referential How do we avoid this little paradox? There are two possible Paradoxes ways of resolving this paradox. Noson S. Yanofsky Many philosophers say that the word heterological cannot
Preliminaries exist. After all, we just showed that it is not always Three well-de ned. We cannot determine if a certain adjective Motivating Examples (hetrological) is heterological or not. Philosophical Interlude Another more obvious solution is to just ignore the
Cantor's problem. Human language is inexact and full of Inequalities contradictions. Every time we use an oxymoron, we are Main Theorem stating a contradiction. Every time we ask for another
Turing's piece of cake while lamenting the fact that we cannot lose Halting Problem weight, we are stating a contradiction. We can safely
Fixed Points ignore the fact that heterological is not well-de ned for in Logic only one adjective. Two Other Paradoxes
Further Reading A philosophical interlude on paradoxes
Self- A paradox is a process where an assumption is made, and Referential Paradoxes through valid reasoning, a contradiction is derived. Noson S. Yanofsky Assumption =⇒ Contradiction. The logician then concludes that since the reasoning was valid Preliminaries
Three and the contradiction cannot happen, it must be that the Motivating Examples assumption was wrong. This is very similar to what
Philosophical mathematicians call proof by contradiction and philosophers Interlude call reductio ad absurdum. A paradox is a method of showing Cantor's Inequalities that the assumption is not part of rational thought.
Main Theorem We have so far seen the same pattern of proof in three dierent Turing's areas: (i) villagers, (ii) sets, and (iii) adjectives. The Halting Problem assumption is that the g function can be represented by the f Fixed Points function. A contradiction is then derived and we conclude that in Logic is not represented by . These three examples highlight three Two Other g f Paradoxes dierent areas where the alleged contradictions might be found. Further Reading A philosophical interlude on paradoxes
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries The Physical Universe. A village with a particular rule is part Three of the physical universe. The physical universe does not have Motivating Examples any contradictions. Facts and properties simply are and no Philosophical Interlude object can have two opposing properties. Whenever we come to
Cantor's such contradictions, we have no choice but to conclude that the Inequalities assumption was wrong. Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading A philosophical interlude on paradoxes
Self- Referential Paradoxes Noson S. The Mental and Linguistic Universe. In contrast to the Yanofsky physical universe, the human mind and human language that Preliminaries the mind uses to express itself are full of contradictions. We Three are not perfect machines. We have a lot of dierent Motivating Examples contradictory parts and desires. We all have con icting thoughts Philosophical Interlude in our head and these thoughts are expressed in our speech. So
Cantor's when an assumption brings us to a contradiction in our thought Inequalities or language, we do not need to take it very seriously. If an Main Theorem adjective is in two opposite classi cations, it does not really
Turing's bother us. In such a case, we cannot go back to our assumption Halting Problem and say it is wrong. The entire paradox can be ignored.
Fixed Points in Logic
Two Other Paradoxes
Further Reading A philosophical interlude on paradoxes
Self- Referential Science and Mathematics. There are, however, parts of Paradoxes human thought and language which cannot tolerate Noson S. Yanofsky contradictions: science and mathematics. These areas of exact thought are what we use to discuss the physical world (and Preliminaries more). If science and math are to discuss / describe / model / Three Motivating predict the contradiction-free physical universe, then we better Examples make sure that no contradictions occur there. We rst saw this Philosophical Interlude in the early years of elementary school when our teachers Cantor's proclaimed that we are not permitted to divide by zero. Since Inequalities
Main math and science cannot have contradictions, young edglings Theorem are not permitted to divide by zero. To summarize, science and Turing's Halting mathematics are products of the human mind and language Problem which we do not permit to have contradictions. If an Fixed Points in Logic assumption leads us to a contradiction in science or
Two Other mathematics, then we must abandon the assumption. Paradoxes
Further Reading A philosophical interlude: Where contradictions can occur.
Self- Referential The Mental and Paradoxes The Physical Linguistic Universe Noson S. Universe Yanofsky
Preliminaries Three Science and Motivating Examples Mathematics Philosophical Interlude
Cantor's Inequalities
Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading A philosophical interlude Why categories?
Self- Referential Paradoxes Noson S. Many have felt that these dierent instances of self reference Yanofsky have a similar pattern (witness Bertrand Russell supposedly Preliminaries inventing the barber paradox to illustrate Russell's paradox.) Three The major advance that category theory has to oer the subject Motivating Examples of self-referential paradoxes is to actually show that all these Philosophical Interlude dierent self-referential paradoxes are really instances of the
Cantor's same categorical theorem. F. William Lawvere described a Inequalities simple formalism that showed many of the major self-referential Main Theorem paradoxes and more. This shows that the logic of self-referential
Turing's paradoxes is inherent in many systems. This also shows the Halting Problem unifying power of category theory.
Fixed Points in Logic
Two Other Paradoxes
Further Reading A philosophical interlude Why categories?
Self- Referential There is, however, another positive aspect of our formalism. Paradoxes Lawvere showed us how to have an exact mathematical Noson S. Yanofsky description of the paradoxes while avoiding messy statements about what exists and what does not exist. In the categorical Preliminaries setting, Three Motivating Examples The barber paradox does not say that a village with a rule
Philosophical does not exists. Interlude With Russell's paradox, a category theorist does not say Cantor's Inequalities that a certain collection does not form a set. Main Theorem In the heterological paradox, we avoid the silly analysis as Turing's to whether a word exists or not. Halting Problem In our categorical discussion, we successfully avoid metaphysical Fixed Points in Logic gobbledygook. For this alone, we should be appreciative of the
Two Other categorical formalism. Paradoxes
Further Reading Cantor's Inequalities
Self- Referential At the end of the 19th century Georg Cantor proved some Paradoxes important theorems about the sizes of sets. Noson S. Yanofsky He showed that every set is smaller than its powerset.
Preliminaries Every set S is smaller than the set P(S). Three Motivating A more categorical way of saying this is that for any set S, Examples there cannot exist a surjection h: S −! P(S). Philosophical Interlude Yet another way of saying this, is that for every purported Cantor's surjection , there is some subset of , Inequalities h: S −! P(S) S denoted , that is not in the image of . Main Ch h Theorem This is proven with a proof by contradiction: we are going to Turing's Halting assume (wrongly) that there is such a surjection and derive a Problem contradiction. Since this is formal mathematics, no such Fixed Points in Logic contradiction can exist and hence our assumption that such a Two Other Paradoxes surjection exists must be false.
Further Reading Cantor's Inequalities
Self- Given such an h, let us de ne f : S × S 2 for s, s0 ∈ S as Referential h −! Paradoxes follows Noson S. 1 : s ∈ h(s0) Yanofsky 0 fh(s, s ) = 0 Preliminaries 0 : s ∈/ h(s ) Three Motivating Use fh to construct gh as follows Examples
Philosophical fh Interlude S × S / 2 < Cantor's ∆ NOT Inequalities Main Theorem S / 2 gh Turing's Halting Problem The function gh is the characteristic function of the subset Fixed Points Ch ⊆ S where each element s does not belong to h(s), i.e., in Logic
Two Other Ch = {s ∈ S : s ∈/ h(s)} ⊆ S. Paradoxes
Further Reading Cantor's Inequalities
Self- We claim that the subset C of S is not in the image of h, i.e., Referential h Paradoxes Ch is a witness or a certi cate that h is not surjective. If Ch Noson S. was in the image of h, there would be some s0 ∈ S such that Yanofsky h(s0) = Ch. In that case gh would be represented by fh with s0. Preliminaries That is, for all s ∈ S Three Motivating Examples gh(s) = fh(s, s0) Philosophical Interlude but this would also be true for s0 ∈ S which would mean that Cantor's Inequalities gh(s0) = fh(s0, s0) Main Theorem However, by the de nition of , we have that Turing's gh Halting Problem gh(s0) = NOT (fh(s0, s0)). Fixed Points in Logic Since this cannot be, our assumption that 0 is wrong Two Other h(s ) = Ch Paradoxes and there is a subset of S that is not in the image of h. Further Reading Cantor's Inequalities Avoiding Contradictions
Self- Referential Paradoxes This is part of mathematics and the only resolution is to accept Noson S. Yanofsky the fact no such surjective h exists and that |S| < |P(S)|. Notice that this applies to any set. For nite , this is obvious Preliminaries S n Three since |S| = n implies |P(S)| = 2 . However, this is true for Motivating Examples in nite S also. What this shows is that P(S) is a dierent level
Philosophical of in nity than S. One can iterate this process and get Interlude , Cantor's P(S) Inequalities P(P(S)), Main Theorem P(P(P(S))), Turing's Halting ... Problem This gives many dierent, unequal levels of in nity. Fixed Points in Logic
Two Other Paradoxes
Further Reading Cantor's Inequalities
Self- Referential Related to the above theorem of Cantor, is the theorem that Paradoxes the natural numbers N is smaller than the interval of all real Noson S. Yanofsky numbers between 0 and 1, i.e., (0, 1) ⊆ R. This proof is slightly dierent than the previous examples that Preliminaries we saw. We include it because it has features in it that are Three Motivating closer to the upcoming general theorem. Rather than working Examples with the set 2 = {0, 1}, this proof works with the set Philosophical Interlude 10 = {0, 1, 2, 3,... 9}. Also, rather than working with the Cantor's function NOT : 2 −! 2, we now work with the function Inequalities 10 10 which is de ned as follows: Main α: −! Theorem Turing's α(0) = 1, α(1) = 2, α(2) = 3, . . . , α(8) = 9, α(9) = 0, Halting Problem
Fixed Points i.e., α(n) = n + 1 Mod 10. The most important feature of α in Logic is that, every output is dierent than its input. There are many Two Other Paradoxes such functions from 10 to 10. We choose this one.
Further Reading Cantor's Inequalities Matrix Form
Self- Referential Paradoxes The proof that |N| < |(0, 1)| is, again, a proof by contradiction. Noson S. We assume (wrongly) that there is a surjection h: N (0, 1) Yanofsky −! and come to a contradiction which proves that no such h can Preliminaries possibly exist. With such an h we can de ne a function Three N N 10 which depends on . For N, Motivating fh : × −! h m, n ∈ Examples Philosophical f (m, n) = the mth digit of h(n). Interlude h
Cantor's Inequalities This means that fh gives every digit of the purported function Main h. The next slide will help explain fh. The natural numbers are Theorem on the left tell you the position. The function h assigns to every Turing's Halting natural number on the top, a real number below it. The Problem numbers on the left are the rst inputs to f and the numbers Fixed Points h in Logic on the top are the second inputs to fh. Two Other Paradoxes
Further Reading Cantor's Inequalities Matrix Form
Self- Referential Real Number Paradoxes
Noson S. Yanofsky fh 0 1 2 3 4 5 6 ···
Preliminaries 0 0 0 0 0 0 0 ··· Three Motivating ...... Examples ···
Philosophical Interlude 0 0 0 7 2 7 7 4 ···
Cantor's Inequalities Digit 1 0 1 2 2 7 6 7 ··· Main Theorem 2 5 3 0 3 0 0 0 ··· Turing's Halting 3 0 6 2 0 1 2 0 Problem ···
Fixed Points in Logic 4 0 1 0 2 3 1 3 ··· Two Other Paradoxes 5 0 1 0 3 0 1 5 ··· Further . . . . Reading . . . . ··· Cantor's Inequalities
Self- With such an f , one can go on to describe a function g with Referential h h Paradoxes the by now familiar construction Noson S. Yanofsky fh N × N / 10 < Preliminaries ∆ α Three Motivating Examples N / 10 gh Philosophical Interlude The function also depends on . The next matrix will help Cantor's gh h Inequalities explain the function gh. That is, the nth digit of the nth Main number is changed. The changed numbers are the outputs to Theorem the function . Thinking of the outputs of as the digits of a Turing's gh gh Halting real number, we are describing a real number between 0 and 1. Problem We call this number . In our case, Fixed Points Gh in Logic
Two Other Gh = 0.121126 ... Paradoxes
Further Reading Cantor's Inequalities Matrix Form
Self- Referential Paradoxes Real Number Noson S. Yanofsky fh 0 1 2 3 4 5 6 ··· Preliminaries Three 0 0 0 0 0 0 0 ··· Motivating Examples ...... Philosophical ··· Interlude
Cantor's 0 α(0) = 1 0 7 2 7 7 4 ··· Inequalities
Main Digit 1 0 α(1) = 2 2 2 7 6 7 ··· Theorem Turing's 2 5 3 0 1 3 0 0 0 Halting α( ) = ··· Problem
Fixed Points 3 0 6 2 α(0) = 1 1 2 0 ··· in Logic Two Other 4 0 1 0 2 3 α(1) = 2 3 ··· Paradoxes Further 5 0 1 0 3 0 1 5 6 Reading α( ) = ··· ...... ··· Cantor's Inequalities
Self- The claim is that g is not represented by f . This means that Referential h h Paradoxes the number represented by gh will not be the number Noson S. represented by f ( , n0) for any n0. Another way to say this is Yanofsky h that the number Gh will not be any column in the scheme Preliminaries described by the matrix. This is obviously true because Gh was Three formed to be dierent than the rst column because the rst Motivating Examples digit is dierent. It is dierent than the second column because Philosophical it was formed to be dierent at the second digit. It is dierent Interlude
Cantor's than the third column because it was formed to be dierent at Inequalities the third digit, etc. Main Theorem Gh is saying Turing's I am not on column because my th digit is dierent from Halting n n Problem the nth column's nth digit. Fixed Points in Logic or
Two Other Paradoxes I am not in the image of h. Further Conclusion: is not on our list and hence is not surjective. Reading Gh h Cantor's Inequalities
Self- Let us show the end of the proof formally. If there was some n0 Referential Paradoxes that represented gh, then for all m Noson S. Yanofsky gh(m) = fh(m, n0)
Preliminaries (i.e., G is the same as column n .) But if this was true for all Three h o Motivating m, then it is true for n0 also (that is, it is true by every digit Examples including the one on the diagonal.) But that says that Philosophical Interlude Cantor's gh(n0) = fh(n0, n0). Inequalities Main However g was de ned for n0 as Theorem h
Turing's Halting gh(n0) = α(fh(n0, n0)). Problem Fixed Points We conclude that no such n0 exists and g describes a number in Logic h in (0, 1) but is not in the image of h. That is, h cannot be Two Other Paradoxes surjective and |N| < |(0, 1)|. Further Reading Main Theorem Some Preliminaries
Self- De nition. First a simple de nition in Set. Consider a set Referential Y Paradoxes and a set function α: Y −! Y . We call s0 ∈ Y a xed point Noson S. of if . That is, the output is the same (or xed) as Yanofsky α α(s0) = s0 the input. We can write the element s0 by talking about a Preliminaries function p : {∗} −! Y such that p(∗) = s0. Saying that s0 is a Three Motivating xed point of α amounts to saying that α ◦ p = p, i.e., the Examples following diagram commutes: Philosophical Interlude p Cantor's {∗} / Y Inequalities
Main Theorem p α ! ~ Turing's Y . Halting Problem
Fixed Points Let us generalize this to any category A with a terminal object in Logic 1. Let y be an object in A and α: y −! y be a morphism in Two Other Paradoxes A. Then we say p : 1 −! y is a xed point of α if α ◦ p = p. Further Reading Lawvere's Theorem
Self- Referential Paradoxes
Noson S. Yanofsky Now for the main theorem as given by Lawvere in 1969. Preliminaries
Three Motivating Examples Lawvere's Theorem. Let A be a category with a terminal
Philosophical object and binary products. Let y be an object in the category Interlude and α: y −! y be a morphism in the category. If α does not Cantor's Inequalities have a xed point, then for all objects a and for all Main f : a × a −! y there exists a g : a −! y such that g is not Theorem representable by f . Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Lawvere's Theorem
Self- Referential Paradoxes
Noson S. Yanofsky Proof. Let α: y −! y not have a xed point, then for any a and for any we can compose with and to Preliminaries f : a × a −! y f ∆ α form as below. Three g Motivating Examples f Philosophical a × a / y Interlude < ∆ α Cantor's Inequalities a y. Main g / Theorem
Turing's Halting g is not representable by f . Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Turing's Halting Problem
Self- Referential Paradoxes In the early 1930's, long before the engineers actually created
Noson S. computers, Alan Turing, the father of computer science, Yanofsky showed what computers cannot do. Loosely speaking, he proved
Preliminaries that no program can decide whether or not any program will go Three into an in nite loop or not. Already from this inexact statement Motivating Examples one can see the self reference: programs deciding properties of Philosophical programs. Interlude Let us state a more exact version of Turing's theorem. First Cantor's Inequalities some preliminaries. Programs come in many dierent forms. Main Here we are concerned with programs that only accept a single Theorem
Turing's natural number. To every such program, there is a unique Halting natural number that describes that program. This fact that Problem
Fixed Points programs that act on numbers can be represented by numbers in Logic shows that programs can be self referential. Two Other Paradoxes
Further Reading Turing's Halting Problem
Self- Referential Paradoxes
Noson S. Yanofsky Programs that accept a single number can take an input and Preliminaries halt or they can go into an in nite loop. The halting problem Three Motivating asks for (i) a number of a program that accepts a single number Examples and (ii) an input to that program. It returns 1 or 0 depending Philosophical Interlude on if it halts or goes into an in nite loop. Turing's Halting Cantor's theorem says that no such program can possibly exist. This is Inequalities not a limitation of modern technology or of our current ability. Main Theorem Rather, this is an inherent limitation of computation. Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Turing's Halting Problem
Self- Referential Paradoxes
Noson S. The proof is, once again, a proof by contradiction. Assume Yanofsky (wrongly) that there does exist a program that accepts a
Preliminaries program number and an input, and can determine if that
Three program will halt or go into an in nite loop when that number Motivating Examples is entered into that program. Formally, such a program
Philosophical describes a total computable function. The function is named Interlude Halt : N × N −! Bool de ned on natural numbers m, n ∈ N is Cantor's Inequalities Main 1 : if input m into program n halts Theorem Halt(m, n) = Turing's Halting 0 : if input m into program n goes into a loop Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Turing's Halting Problem Matrix Form
Self- Referential Paradoxes Program Noson S. Yanofsky Halt 0 1 2 3 4 5 ···
Preliminaries 0 0 1 0 0 0 1 ··· Three Motivating Examples 1 1 1 1 1 1 1 ··· Philosophical Interlude 2 0 1 0 0 0 0 ··· Cantor's Inequalities Input 3 0 1 0 0 1 0 ··· Main Theorem 4 0 0 1 1 1 1 Turing's ··· Halting Problem 5 1 0 1 0 1 1 ··· Fixed Points . . . in Logic . . .. Two Other Paradoxes
Further Reading Turing's Halting Problem
Self- It is not hard to see that the function ∆: N N × N de ned Referential −! Paradoxes as ∆(n) = (n, n) is a computable function. Consider the partial Noson S. NOT function ParNOT : Bool Bool de ned as follows: Yanofsky −! 1 if 0 Preliminaries : n = ParNOT (n) = Three Motivating " : if n = 1 Examples Philosophical where " means it goes into an in nite loop. It is not hard to see Interlude that ParNOT is a computable function. Since Halt is assumed Cantor's Inequalities computable, and the function ∆ and ParNOT are computable, Main then their composition as follows is also computable function. Theorem
Turing's Halt Halting N × N / Bool Problem < ∆ ParNOT Fixed Points in Logic N # Two Other 0 / Bool Paradoxes Halt
Further Reading Turing's Halting Problem Matrix Form
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries The new computable function, Halt0, accepts a number n as Three input and does the opposite of what program on input Motivating n n Examples does. That is, if program n on input n halts, then Halt0(n) will Philosophical Interlude go into an in nite loop. Otherwise, if program n on input n 0 Cantor's goes into an in nite loop, then Halt (n) will halt. We can see Inequalities the way Halt0 is de ned in the next slide. Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Turing's Halting Problem Matrix Form
Self- Referential Paradoxes Program Noson S. Yanofsky Halt 0 1 2 3 4 5 ··· Preliminaries Three 0 α(0) = 1 1 0 0 0 1 ··· Motivating Examples 1 1 1 1 1 1 1 Philosophical α( ) =" ··· Interlude
Cantor's 2 0 1 α(0) = 1 0 0 0 ··· Inequalities
Main Input 3 0 1 0 α(0) = 1 1 0 ··· Theorem Turing's 4 0 0 1 1 1 1 Halting α( ) =" ··· Problem Fixed Points 5 1 0 1 0 1 α(1) =" ··· in Logic . . .. Two Other . . . Paradoxes
Further Reading Turing's Halting Problem
Self- Since Halt0 is a computable function, the program for this Referential Paradoxes computable function must have a number and be somewhere on Noson S. our list of computable functions. However, it is not. Halt0 was Yanofsky formed to be dierent than every column in the chart. What is Preliminaries wrong? We know that ∆ and ParNOT are computable. We Three assumed that was computable. It must be that our Motivating Halt Examples assumption about Halt was wrong. Halt is not computable. Philosophical Let us formally show that 0 is dierent than every column Interlude Halt 0 Cantor's in the chart. Imagine that Halt is computable and the number Inequalities 0 0 of Halt is n0. This means that Halt is the n0 column of our Main 0 Theorem chart. Another way to say this is that Halt is representable by Turing's Halt( , n0), i.e., for all n, Halting Problem 0 Halt (n) = Halt(n, n0). Fixed Points in Logic 0 Now let us ask about Halt (n0)? We get Two Other 0 Paradoxes Halt (n0) = Halt(n0, n0). Further 0 Reading But we de ned Halt (n0) to be 0 Halt (n0) = ParNOT (Halt(n0, n0)). This is a contradiction. Turing's Halting Problem
Self- Referential Paradoxes
Noson S. Yanofsky In a sense, we can say that the computational task that 0 Preliminaries Halt 0 Three (and in particular Halt (n0)) performs is: Motivating Examples If you ask me whether I will halt or go into an in nite loop, Philosophical Interlude then I will give the wrong answer.
Cantor's Inequalities Since computers cannot give the wrong answer, Halt0 cannot Main exist and hence cannot exist. Theorem Halt
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Fixed Points in Logic
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries The Contrapositive of Lawvere's Theorem. Let A be a Three category with a terminal object and binary products. Let be Motivating y Examples an object in the category and α: y −! y be a morphism in the Philosophical Interlude category. If there is an object a and a morphism Cantor's f : a × a −! y, such that g = α ◦ f ◦ ∆ is representable by f , Inequalities then α has a xed point. Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading Fixed Points in Logic Matrix Form
Self- Referential Second Input Paradoxes
Noson S. Yanofsky f p ···
Preliminaries α f ( , p) ··· Three Motivating Examples α f ( , p) ···
Philosophical Interlude α f ( , p) ···
Cantor's Inequalities α f ( , p) ···
Main First Input Theorem p αf (p, p) = f (p, p) ··· Turing's Halting Problem f ( , p) α ··· Fixed Points . . . .. in Logic . . . f ( , p) . Two Other Paradoxes The crossing point is a xed point. Further Reading Fixed Points in Logic
Self- Now apply this theorem to logic. We use the contrapositive of Referential Paradoxes Lawvere's Theorem to nd xed points in logic. First, some Noson S. elementary logic. We are working in a system that can handle Yanofsky basic arithmetic. We will deal with logical formulas that accept Preliminaries at most one value which is a number. Three Motivating Examples A(x), B(x), C(x),... Philosophical Interlude A logical formula that accepts no value, sentences, Cantor's Inequalities
Main A, B, C,... Theorem Turing's We are interested in equivalence classes of these sets: two Halting Problem formulas are equivalent if they are provably logically equivalent. Fixed Points We will call the equivalence classes of predicates Lind1 for the in Logic Lindenbaum classes of predicates. The equivalence classes of Two Other Paradoxes sentences will be denoted Lind0. Further Reading Fixed Points in Logic
Self- All logical formulas can be encoded as a natural number. We Referential Paradoxes will write the natural number of a logical formula A(x) as Noson S. A(x) and the number of a sentence A is A . Logical Yanofsky p q p q formulas about numbers will be able to evaluate logical Preliminaries formulas about numbers. It is these numbers that will help Three logical formulas be self-referential. Motivating Examples We are going to get xed points of logical predicates. For every Philosophical predicate, , there is a way of constructing a xed point Interlude E(x)
Cantor's which is a logical sentence C such that Inequalities Main E(pCq) ≡ C Theorem The process that goes from a to will be called a xed Turing's E(x) C Halting point machine. is a logical sentence that says Problem C Fixed Points I have property E. in Logic
Two Other With this xed point machine we will nd some of the most Paradoxes fascinating aspects of logic. Further Reading Fixed Points in Logic
Self- Back to the contrapositive of the Lawvere Theorem. Referential Paradoxes The category is Set. Noson S. Yanofsky The a of the theorem is the set of equivalence classes Lind1. Preliminaries The y of the theorem is the set of equivalence classes Three Motivating Lind0. Examples There is a function de ned as Philosophical f : Lind1 × Lind1 −! Lind0 Interlude follows: Cantor's Inequalities f (A(x), B(x)) = B(pA(x)q). Main The of the theorem depends on some predicate so we Theorem α E
Turing's write it as αE . The function αE applies the predicate to Halting the number of a sentence . It is a function Problem A Fixed Points αE : Lind0 −! Lind0 which is de ned for the sentence A as in Logic Two Other αE (A) = E( A ). Paradoxes p q
Further Reading Fixed Points in Logic Matrix Form
Self- Referential Paradoxes
Noson S. Yanofsky Second Input
Preliminaries f A0(x) A1(x) A2(x) ··· A5(x) ··· Three Motivating Examples A0(x) E(A0(pA0(x)q) A1(pA0(x)q) A2(pA0(x)q) ··· Ap(pA0(x)q) ··· Philosophical Interlude A1(x) A0(pA0(x)q) E(A1(pA1(x)q) A2(pA1(x)q) ··· Ap(pA1(x)q) ··· Cantor's Inequalities A2(x) A0(pA2(x)q) A1(pA2(x)q) E(A2(pA2(x)q) ··· Ap(pA2(x)q) ··· Main . . . . . Theorem ...... Ap(pA3(x)q) ··· Turing's Halting First Input Problem Ap(x) A0(pAp(x)q) A1(pAp(x)q) A2(pAp(x)q) ··· E(Ap(pAp(x)q) ··· Fixed Points ...... in Logic . . . . .
Two Other Paradoxes
Further Reading Application: Gödel's Incompleteness Theorem
Self- We use the xed point machine to make interesting Referential Paradoxes self-referential statements. Let Prov(x, y) be the two place Noson S. predicate that is true when y is the Gödel number of a proof of Yanofsky a statement whose Gödel number is x. Now we form the Preliminaries statement Three Prov Motivating E(x) = (∀y)¬ (y, x). Examples G ≡ E( G ) = (∀y)¬Prov(y, G ) Philosophical p q p q Interlude G is a logical statement that essentially says Cantor's Inequalities I am a statement for which any y is not a proof of me Main Theorem I am unprovable. Turing's Halting Problem If G was false then there would be a proof of G and hence there Fixed Points would be a proof of a false statement. In that case the system in Logic is not consistent. On the other hand, if is true, then it Two Other G Paradoxes essentially says that G is true but unprovable. Further Reading Application: Gödel's Incompleteness Theorem
Self- Referential Paradoxes True = Provable True Noson S. Yanofsky
Preliminaries Provable Three Motivating Examples
Philosophical Interlude
Cantor's Inequalities
Main Theorem Gödel sentence Turing's Halting Problem Fixed Points What was believed before and after Gödel's Theorem. in Logic
Two Other Paradoxes
Further Reading Application: Tarski's Theorem
Self- Alfred Tarski's theorem shows that a logical system cannot tell Referential Paradoxes which of its predicates are true. Assume (wrongly) that there is Noson S. some logical formula T (x) that accepts a number and tells if Yanofsky the statement is true. This formula will be true when x is the Preliminaries Gödel number of a true statement in the theory. We can then Three use to form the statement Motivating T (x) Examples
Philosophical E(x) = ¬T (x) Interlude
Cantor's This says that E(x) is true when T (x) is false. Now place E(x) Inequalities into the xed point machine. We will get a statement C such Main Theorem that Turing's C ≡ E( C ) = ¬T ( C ). Halting p q p q Problem The logical sentence C essentially says Fixed Points in Logic I am false. Two Other Paradoxes It is a logical version of the liar paradox. Further Reading Application: Parikh Sentences
Self- Rohit Parikh used the xed point machine to formulate some Referential Paradoxes fascinating sentences that express properties about the length Noson S. of its own proof. Consider the two-place predicate Pr en(m, x) Yanofsky which is true if there exists a proof of length m (in symbols) of Preliminaries a statement whose Gödel number is x. Three Motivating Pr en Examples En(x) = ¬(∃m < n (m, x)). Philosophical Pr en Interlude Cn ≡ En(pCnq) = ¬(∃m < n (m, pCnq)). Cantor's The logical sentence essentially says Inequalities Cn Main I do not have a proof of length less than n. Theorem
Turing's As long as the logical system is consistent, Cn will be true and Halting Problem will not have a proof of length less than n. Parikh showed that Fixed Points although Cn does not have a short proof (you can make n as in Logic large as you want), there does exist a short proof of the fact Two Other Paradoxes that Cn is provable. Further Reading Epimenides and the Liar
Self- Referential Before we close this talk it pays to look at two famous Paradoxes paradoxes that are not exactly instances of Cantor's theorem Noson S. Yanofsky but are close enough that they are easy to describe. The two examples are (i) the Epimenides paradox (the liar's paradox) Preliminaries and (ii) time travel paradoxes. Three Motivating Examples
Philosophical Chronologically, the granddaddy of all the self-referential Interlude paradoxes is the Epimenides paradox. Epimenides (6th or 7th Cantor's Inequalities century BC), a philosopher from Crete was a curmudgeon who
Main did not like his neighbors in Crete. He is quoted as saying that Theorem All Cretans are liars. The problem is that he is a Cretan. He is Turing's Halting talking about himself and his statement. If his statement is Problem true, then this very utterance is also a lie and hence is not true. Fixed Points in Logic On the other hand, if what he is saying is false, then he is not a Two Other liar and what he said is true. This seems to be a contradiction. Paradoxes
Further Reading Epimenides and the Liar
Self- There is a set of English sentences which we call Sent. Referential Paradoxes f : Sent × Sent −! 2. The function f is de ned for sentences s Noson S. and s0 as Yanofsky 1 : s is negated by s0 Preliminaries f (s, s0) = Three 0 Motivating 0 : s is not negated s . Examples Philosophical f Interlude Sent × Sent / 2 8 Cantor's ∆ NOT Inequalities Main 2 Theorem Sent g / Turing's Halting g is the characteristic function of the subset of sentences that Problem negate themselves. Till here, we have been mimicking the Fixed Points in Logic set-up of Lawvere's theorem. It is not clear what we would Two Other mean by talking about g being representable by f . What would Paradoxes it mean for a sentence S to represent a subset of sentences? Further Reading Nevertheless, the sentences that negate themselves are declarative sentences that are contradictions. Time Travel Paradoxes
Self- Referential If time travel was possible, a time traveler might go back in Paradoxes time and shoot his bachelor grandfather, guaranteeing that the Noson S. Yanofsky time traveler was never born. Homicidal behavior is not necessary to achieve such paradoxical results. The time traveler Preliminaries might just make sure that his parents never meet, or he might Three Motivating simply go back in time and make sure that he does not enter Examples the time machine. These actions would imply a contradiction Philosophical Interlude and hence cannot happen. The time traveler should not shoot Cantor's his own grandfather (moral reasons notwithstanding) because if Inequalities
Main he shoots his own grandfather, he will not exist and will not be Theorem able to travel back in time to shoot his own grandfather. So by Turing's Halting performing an action he is guaranteeing that the action cannot Problem be performed. Here an event negatively aects itself. Since the Fixed Points in Logic physical universe does not permit contradictions, we must deny
Two Other the assumption that time travel exists. Paradoxes
Further Reading Time Travel Paradoxes
Self- There is a collection of all physical events, Events. Referential Paradoxes f : Events × Events −! 2 is de ned for two events e, and e0 as Noson S. Yanofsky 1 : if e is negated by e0 0 Preliminaries f (e, e ) = 0 if is not negated 0 Three : e e . Motivating Examples f Philosophical Events × Events / 2 Interlude 6 ∆ NOT Cantor's Inequalities 2 Main Events g / Theorem Turing's g is the characteristic function of those events that negate Halting Problem themselves. Such events cannot exist. Till here the pattern has Fixed Points been the same with Lawvere's theorem. However, we do not go in Logic on to talk about representing g by f . What would it mean for Two Other Paradoxes f ( , e) to represent a subset of events? Further Reading Further Reading
Self- Referential F. William Lawvere. Diagonal arguments and cartesian Paradoxes closed categories. In Category Theory, Homology Theory Noson S. Yanofsky and their Applications, II pages 134-145. Springer, Berlin, 1969. Preliminaries F. William Lawvere and Stephen H. Schanuel. Conceptual Three Motivating mathematics, Second edition. Cambridge University Press, Examples 2009. (Session 29). Philosophical Interlude F. William Lawvere and Robert Rosebrugh. Sets for Cantor's Inequalities mathematics. Cambridge University Press, 2003. (Section
Main 7.3). Theorem Rohit Parikh. Existence and feasibility in arithmetic. J. Turing's Halting Symbolic Logic, 36:494-508, 1971. Problem Noson S. Yanofsky. A universal approach to self-referential Fixed Points in Logic paradoxes, incompleteness and xed points. Bull. Symbolic Two Other Logic, 9(3):362-386, 2003. Paradoxes
Further Reading Further Reading
Self- Referential Paradoxes
Noson S. Yanofsky Noson S. Yanofsky. The Outer Limits of Reason: What Preliminaries Science, Mathematics, and Logic Cannot Tell Us. MIT Three Motivating Press, Cambridge, MA, 2013. Examples Noson S. Yanofsky. Resolving paradoxes. Philosophy Now, Philosophical Interlude 016:10-12, 2015. Cantor's Inequalities Noson S. Yanofsky. Paradoxes, contradictions, and the
Main limits of science. American Scientist, pages 166-173, Theorem May-June 2016. Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading The End
Self- Referential Paradoxes
Noson S. Yanofsky
Preliminaries
Three Motivating Examples Thank You Philosophical Interlude
Cantor's Inequalities
Main Theorem
Turing's Halting Problem
Fixed Points in Logic
Two Other Paradoxes
Further Reading