Ahmad Karimi Tel: +98 (0)919 510 2790 Department of Mathematics Fax: +98 (0)21 8288 3493 Tarbiat Modares University E-mail:[email protected] P.O.Box 14115{134 Behbahan KA Univ. of Tech. Tehran, IRAN 61635{151 Behbahan, IRAN Saeed Salehi Tel: +98 (0)411 339 2905 Department of Mathematics Fax: +98 (0)411 334 2102 University of Tabriz E-mail:/[email protected]/ Σα∂ ir Σα`}{ P.O.Box 51666{17766 /[email protected]/ u Tabriz, IRAN Web: http://SaeedSalehi.ir/ Diagonalizing by Fixed{Points Abstract A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema. Here, we fit more theorems in the universal schema of diagonalization, such as Euclid's theorem on the infinitude of the primes and new proofs of G. Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset. Then, in Linear Temporal Logic, we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox. Thus, Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic. Again the diagonal schema of the paper is used in this proof; and also it is shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal/fixed-point schema. We also show the existence of dominating (Ackermann-like) functions arXiv:1303.0730v5 [math.LO] 10 Dec 2016 (which dominate a given countable set of functions|like primitive recursives) using the schema. vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv 2010 Mathematics Subject Classification: 18A10 · 18A15 · 03B44 · 03A05. Keywords: Diagonalization · Self-Reference · Fixed-Points · Cantor's Theorem · Euclid's Theorem · Yablo's Paradox · Ackermann's Function · Dominating Functions · (Linear) Temporal Logic. Date: 30.08.14 (30 August 2014) page 1 (of 12) page 2 (of 12) Diagonalizing by Fixed{Points 1 Introduction Cantor's Diagonal Argument was introduced in his (third proof for the) famous theorem on non{ denumerability of the reals; the argument shows that there can be no surjection from a set A to its powerset P(A): for any function F : A ! P(A) the set DF = fx 2 A j x 62 F (x)g is not in the range of F because for any a 2 A we have a 2 DF ! a 62 F (a), and so a 2 (DF n F (a)) [ (F (a) n DF ), whence DF 6= F (a). This argument shows up also in Russell's Paradox: the collection R = fx j x 62 xg of sets is not a set, since for any set A we have A 2 R ! A 62 A, so A 6= R. One other example is Turing's Halting Problem in Computability Theory: if W0;W1;W2; ··· is the family of all re sets (recursively enu- merable subsets of N), then the set K = fn 2 N j n 62 Wng is not re, because for any re set Wm we have m 2 K ! m 62 Wm, and so m 2 (K n Wm) [ (Wm n K), thus K 6= Wm. It can be seen that the (diagonal) set K = fn 2 N j n 2 Wng is an re but undecidable set. Many other theorems in mathematics (logic, set theory, computability theory, complexity theory, etc.) use diagonal arguments; Tarski's theorem on the undefinability of truth, and Godel¨ 's theorem on the incompleteness of sufficiently strong and (!{)consistent theories are two prominent examples. In 2003, Noson S. Yanofsky published the paper [14] mentioning some earlier descriptions for \many of the classical paradoxes and incompleteness theorems in a categorial fashion", in the sense that by using \the language of category theory (and of cartesian closed categories in particular)" one can demonstrate some paradoxical phenomena and show the above mentioned theorems of Cantor, Tarski and Godel¨ ; the goal of [14] was \to make these amazing results available to a larger audience". In that paper, a universal schema has been considered in the language of sets and functions (not categories) and the paradoxes of the Liar, the strong liar, Russell, Grelling, Richard, Time Travel, and Lob¨ , and the theorems of Cantor (A P(A)), Turing (undecidability of the Halting problem, and existence of a non{ re set), Baker-Gill-Solovay (the existence of an oracle O such that PO 6= NPO), Carnap (the diagonalization lemma), Godel¨ (first incompleteness theorem), Rosser (incompleteness of sufficiently strong and consistent theories), Tarski (undefinability of truth in sufficiently strong languages), Parikh (existence of sentences with very long proofs), Kleene (Recursion Theorem), Rice (undecidability of non{ trivial properties of recursive functions), and von Neumann (existence of self{reproducing machines) are shown as instances. In this paper, we fit some other theorems and proofs into the above mentioned universal schema of Yanofsky; these include Euclid's Theorem on the infinitude of the primes, Boolos' proof of the existence of some explicitly definable counterexamples to the non{injectivity of functions F : P(A) ! A for any set A, Yablo's paradox in a form of a mathematical theorem in the framework of linear temporal logic as a non{existence of some certain fixed–points, and the existence of dominating functions for a given countable set of functions like Ackermann's function which dominates all the primitive recursive functions. In the rest of the introduction we fix our notation and introduce the common framework. 1.1 Cantor's Theorem by Fixed{Points Let B, C and D be arbitrary sets. Any function f : B × C ! D corresponds to a function fb : C ! DB where fb(c)(b) = f(b; c) for any b 2 B and c 2 C (the set DB consists of all the functions from B to D). Conversely, for any function F : C ! DB there exists some f : B × C ! D such that fb = F : for any B×C ∼ B C b 2 B and c 2 C let f(b; c) = F (c)(b). In the other words −b : D = (D ) . Let f : B × C ! D be a fixed function. A function g : B ! D is called representable by f at a fixed c0 2 C, when for any x 2 B, B g(x) = f(x; c0) holds. In the other words, g = fb(c0). So, the function fb : C ! D is onto if and only if every function B ! D is representable by f at some c0 2 C. MANUSCRIPT (Submitted) Σα∂ c Ahmad Karimi & Saeed Salehi 2014 ir uΣα`}{ Diagonalizing by Fixed{Points page 3 (of 12) Theorem 1.1 (Cantor) Assume the function α : D −! D, for a set D, does not have any fixed point (i.e., α(d) 6= d for all d 2 D). Then for any set B and any function f : B × B ! D there exists a function g : B ! D that is not representable by f (i.e., for all b 2 B, g(−) 6= f(−; b)). Proof. The desired function g : x 7! α(f(x; x)) can be constructed as follows: f B × B - D 6 4B α ? B - D g where 4B is the diagonal function of B (4B(x) = hx; xi). If g is representable by f at b 2 B, then g(x) = f(x; b) for any x 2 B, and in particular g(b) = f(b; b). On the other hand by the definition of g we have g(x) = α(f(x; x)) and in particular (for x = b) g(b) = α(f(b; b)). It follows that f(b; b) is a fixed–point of α; contradiction. Whence, the function g is not representable by f (at any b 2 B). o For any set A we have P(A) ∼= 2A where 2 = f0; 1g and 2A is the set of all functions from A to 2. So, Cantor's theorem is equivalent to the non{existence of a surjection A ! 2A. Putting it another way, Cantor's theorem says that for any function f : A × A ! 2 there exists a function g : A ! 2 which is not representable by f (at any member of A). In this new setting, Cantor's proof goes as follows: let 4A : A ! A × A be the diagonal function of A (4A(x) = hx; xi) and let α : 2 ! 2 be a fixed function. Define g : A ! 2 by g(x) = α(f(4A(x))). If g is representable by f and fixed a 2 A then f(a; a) = g(a) = α(f(a; a)), which shows that α has a fixed–point (namely, f(a; a)). So, for reaching to a contradiction, we need to take a function α : 2 ! 2 which does not have any fixed–point; and the only such function (without any fixed–point) is the negation function neg : 2 ! 2, neg(i) = 1 − i for i = 0; 1. For a function F : A ! P(A) let f : A × A ! 2 be defined as 1 if a 2 F (a0) f(a; a0) = 0 if a 62 F (a0) The function g constructed by the diagram f A × A - 2 6 4A neg ? A - 2 g is the characteristic function of the set DF = fx 2 A j x 62 F (x)g. That g is not representable by f (at any a 2 A) is equivalent to saying that the set DF is not in the range of F (i.e., DF 6= F (a) for any a 2 A).