Asia Pacific Mathematics Newsletter 1 ViciousVicious Queues Queues and and ViciousVicious Circles Circles N Raja N Raja 1. Introduction 2. Paradox and Circularity How wonderful that we have met with a We have to be willing to wrestle with para- paradox. Now we have some hope of making dox in pursuing understanding. progress. — Harold Evans — Niels Bohr This article presents an introduction to some of Paradoxes may be classified into two basic types the fascinating paradoxes that arise in logic and — logical paradoxes and semantic paradoxes. the foundations of mathematics. The study of Logical paradoxes involve notions only from the paradoxes spans a recorded history of nearly two theory of sets. The most well known among the thousand six hundred years of intellectual stud- logical paradoxes is Russell’s paradox [3] from Set ies in philosophy, mathematics, logic, and other theory: disciplines. It is not surprising that the number of The set of all sets that are not members of them- known paradoxes is quite large, and we can only selves is both a member of itself, and also not a member touch upon a tiny fraction of them in this article. of itself. However, we try to discern a pattern even among Other examples of logical paradoxes are Can- the few paradoxes that we do present, and also tor’s paradox and Burali-Forti’s paradox. Unlike point out several connections between paradoxes logical paradoxes, the semantic paradoxes make and the proofs of a number of important theorems use of notions which do not occur in the standard in mathematics. language of set theory. In this article, we will focus A bird’s eye view of the paradoxes shows on some semantic paradoxes. recurrent use of certain features such as self- reference, indexicals, negation, and circularity 2.1. Grelling’s paradox among others, and also the prevalence of certain patterns of reasoning. A closer look at the for- There are some adjectives that describe them- mulations of the paradoxes shows that some of selves and some that do not. “English” is an En- them consciously restrict themselves, as it were, glish word; while “French” is not a French word. to the use of a smaller palette of such features “Polysyllabic” is polysyllabic; but “monosyllabic” and yet succeed in the formulation of the paradox. is not monosyllabic. Call all adjectives that de- Such formulations have a greater impact than scribe themselves “autological”. Call all adjectives those formulations which are prodigal with basic that do not describe themselves “heterological”. features, since they are a clear demonstration of Grelling’s paradox [3] is brought about by the the fact that the absence of the certain features question: Is “heterological” heterological? And the in the underlying theory is no guarantee of their answer is: It is if only if it is not. eventual freedom from paradox. Grelling’s paradox shows that the words “au- The connections between paradoxes and tological” and “heterological” do not form two proofs of mathematical theorems arises due to well-defined categories into which all adjectives the fact that there are several similarities and fall. It demonstrates that natural language does common elements between the formulation and not necessarily partition all objects of thought analysis of paradoxes on the one hand and re- into well-defined categories that will stand up to ductio ad absurdum proofs of various theorems arbitrary logical scrutiny. Grelling’s paradox can in mathematics on the other hand. We indicate, be translated into Russell’s paradox. Identify each very briefly, some of the interesting connections adjective with the set of objects to which that that have been intensely explored by many other adjective applies. Thus an autological word is a mathematicians. set, one of whose elements is the set itself. The 12 January 2013, Volume 3 No 1 Asia Pacific Mathematics Newsletter2 question of whether the word “heterological” is integer which to be specified requires more characters heterological becomes the question of whether the than there are in this sentence. set of all sets not containing themselves contains Does this sentence specify a positive integer? itself as an element. The sentence has 114 characters (counting spaces between words and the period but not the quota- 2.2. Epimenides’s paradox tion marks). Yet it supposedly specifies an integer that, by definition, requires more than 114 char- The Epimenides’s paradox is also known as the acters to be specified. This is clearly paradoxical. Liar’s paradox. The very first person known to It can also rewritten as: The smallest positive contemplate the Liar’s paradox [4] was the Greek integer that cannot be defined in less than fourteen philosopher Eubulides of Miletus who said “A words. It is reasonable to assume that this is a spec- man says that he is lying. Is what he says true or ification for a number. There are a finite number false?.” One of the simplest ways of formulating of sentences of fewer than fourteen words, and the liar’s paradox, is by the statement: This sen- some finite subset of them specify unique positive tence is false. integers. So there is clearly some positive number It might seem at first that a sentence must that is the smallest integer not in that finite set. directly refer to its own truth value, in order to But the Berry sentence itself is a specification for construct a paradox. But this is not necessary. It that number in only thirteen words. is possible to avoid having a sentence directly Berry’s paradox is the starting point for the refer to its own truth value, and still construct proof of a result in the area of algorithmic infor- the paradox using a pair of sentences: The following sentence is true. mation theory, called Chaitin’s theorem [1], which The preceding sentence is false. uses programs or proofs of bounded lengths to Neither of the above sentences refer to their own construct a rigorous version of this paradox. It is truth values, but together they construct the Liar’s also closely related to the notions of Kolmogorov paradox. complexity and Martin-L¨of randomness. The Liar’s paradox, when it is removed from the domain of truth to that of provability, is closely 2.5. L¨ob’s paradox associated to G¨odel’s Incompleteness Theorem [5]. L¨ob’s paradox [6] shows that it is possible to derive any arbitrary sentence from a sentence 2.3. Quine’s paradox with self-reference and some apparently innocu- ous logical deduction rules. Willard Quine [3] showed that it is possible to Let A be any sentence. Let B be the sentence: avoid direct self-reference, as embodied e.g. in the “If this sentence is true then A”. So, B asserts: “If B word “this”, and yet construct the liar’s paradox is true, then A”. in a single sentence, as follows: Now consider the following argument: As- “yields falsehood when appended to its own quotation” sume B is true (hypothesis); then by B, since B yields falsehood when appended to its own quotation. is true, A holds. This argument shows that, if B Quine’s paradox is an algorithm for construct- is true, then A. But this is exactly what B asserts. ≡ ing yet another sentence. Say X yields false- Hence B is true. Therefore, by B, since B is true, hood when appended to its own quotation. Then X’s A is true. Thus, every sentence is true. ≡ quotation “yields falsehood when appended to its L¨ob’s paradox has close connections to L¨ob’s own quotation”. X appended to X’s quotation gives theorem [6] in provability logic. L¨ob’s theorem “yields falsehood when appended to its own quotation” states that: yields falsehood when appended to its own quotation. In other words, the sentence says that it is false. (PA ⊢ Bew(#P) → P) → (PA ⊢ P) The liar’s paradox once again! where PA denotes Peano Arithmetic, Bew(#P) means that the formula with G¨odel number P is 2.4. Berry’s paradox provable, and the symbol ⊢ stands for provability. Berry’s paradox was published in 1908 by L¨ob’s theorem in provability logic states that, in a Bertrand Russell [3]. It reads: The smallest positive theory with Peano arithmetic, for any formula P, January 2013, Volume 3 No 1 13 Asia Pacific Mathematics Newsletter 3 if it is provable that “if P is provable then P”, then logics which place restrictions on the use of the “P is provable”. Provability logic abstracts away negation operator, thereby managing to avoid from the details of encodings used in G¨odel’s Russell’s paradox easily. However, these systems incompleteness theorems. This is achieved by ex- still need to take care to avoid falling into Curry’s pressing the provability of φ in the given system paradox. The resolution of Curry’s paradox is in the language of modal logic, by means of the often a contentious issue because nontrivial res- modality φ. L¨ob’s theorem is formalised by the olutions are difficult and unintuitive. axiom: ( P → P) → P 2.7. Zwicker’s paradox known as the G¨odel–L¨ob axiom. It is sometimes William Zwicker formulated the Hypergame also formalised by means of an inference rule that paradox [11] in the setting of game theory. As a infers P from (P → P). preliminary to formulating the paradox, we need the notion of a finite game. 2.6. Curry’s paradox A two-person game may be defined to be finite if it satisfies the following conditions: Haskell Curry [2], in 1942, was the first to show that the negation operator is not a necessary (i) Two players, A and B, move alternately, A element in formulating Russell-type paradoxes.