Jan Mycielski

Games with perfect information

Publisher Summary This chapter discusses the games with perfect information (PI). The most seriously played PI-games are Chess and Go. But there are numerous other interesting PI-games: Checkers, Chinese Checkers, Halma, Nim, Hex, their misere variants, etc. Perfect information means that at each time only one of the players moves, that the game depends only on their choices, they remember the past, and in principle they know all possible futures of the game. For example War is not a PI-game since the generals move simultaneously, and Bridge and Backgammon are not PI-games because chance plays a role in them. However, some cases of Pursuit and Evasion can be studied by means of PI-games, in spite of the simultaneity and continuity of the movements of the players. The chapter focuses on infinite PI-games. The theory of infinite PI-games is motivated by its beauty and manifold connections with other parts of mathematics. They give also a natural mathematical theory for some garnes of pursuit and evasion. Finite and infinite PI-games are also used in model theory and in recursion theory.

Hereditarily finite sets and identity trees

Abstract Some asymptotic results about the sizes of certain sets of hereditarily finite sets, identity trees, and finite games are proven.

A model of the neocortex

We try to explain the function of the cerebral cortex by representing it as a large array of small computers, each of which applies a certain algorithm for learning to predict the inputs from the senses or other parts of the brain. This view ignores the role of the brain as a classifying device, but it puts forward its role as a learner of the laws of evolution of sequences of patterns. Memory is not seen here as a list of facts but as an operator predicting certain parameters of the future states of reality from its present and past states. The problem of connecting the classifying or listing capacity of the brain with the structures described here is still open.

Locally Finite Theories

We say that a first order theory T is locally finite if every finite part of T has a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theory T a locally finite theory FIN( T ) which is syntactically (in a sense) isomorphic to T . Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of [5], [6] and [7]. This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.) The first mathematically strong locally finite theory, called FIN, was defined in [1] (see also [2]). Now we get much stronger ones, e.g. FIN(ZF). From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilberts second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects. In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Godels second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions. The results of this paper were announced in [3].

REPRESENTATIONS OF INFINITE PERMUTATIONS BY WORDS (II)

We prove several cases of the following theorem: Every free group word which is not a proper power can represent every permutation of an infinite set. The remaining cases will be proved in a forthcoming paper of R. C. Lyndon. Fx denotes a free group freely generated by the set X. The elements of X are called letters, and the elements of Fx are represented by reduced words in those letters. G denotes an arbitrary group. We say that a word w can represent a in G if a E G and there exists a homomorphism h: Fx G C such that h(w) = a. In other words w can represent a in G iff the equation w = a is solvable in G, where X is the set of unknowns. Notice that, if q is an endomorphism of Fx and +(w) can represent a in G, then w can represent a in G. As was pointed out in [3], the sums of exponents of each letter of w have a greatest common divisior 1 iff w can represent every a in every G. This set of words with g.c.d. 1 is larger than the set of words w for which there exists an automorphism a of Fx such that a(w) E X. E.g., by a theorem of J. H. C. Whitehead (see [5]), the words x2yx-1y-1 and x2y3, for x, y E X, are in the first set but not in the second set. The solvability of equations w = a for various groups G has been considered in several papers, see [2, 3, 4, 6, 7, 10]. In the present paper we will study this question for G = Sy, the group of all permutations of an infinite set Y. In [10] Silberger asks if every w E Fx which is not a power, i.e., is not of the form vk with k > 1, can represent every 7r in Sy, if Y is infinite. He shows that this is true if w = xmyn for x, y E X, x

Can One Solve Equations in Groups

y and m

On boolean functions and connected sets

0

Shadows of the axiom of choice in the universe \(L(\mathbb {R})\)

n. Another proof is given in [2]. (For finite Y sufficient conditions in terms of m and n were given in [4]). Our main result is the following partial solution of Silbergers problem (it was announced in [8]). THEOREM 1. If w E Fx, w is not a power, 7r E Sy, and among the cycles of 7r at least one cycle size appears infinitely many times (fixed points are counted as cycles), then w can represent ir in Sy. REMARK ADDED IN MARCH 1986. R. C. Lyndon has just removed my assumption about the existence of repeating cycles in 7r. Thus Silbergers conjecture is true. My proof uses some rather difficult tools of combinatorial group theory (a theorem of Weinbaum about subwords of a relator and the asphericity theorem for Cayley complexes of groups with one relator) but Lyndons addition is more direct. First he points out that Theorem 1 reduces the problem to the cases when YI = No and either 7r has at least one infinite cycle or 7r has no infinite cycles but Received by the editors June 12, 1985 and, in revised form, October 1985 and March 1986. 1980 Mathematics Subject C(Lsification (1985 Revisnm). Primary 20F05; Secondary 20B07, 20E05. (?)1987 American Mathematical Society 0002-9939/87

New set-theoretic axioms derived from a lean metamathematics

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The meaning of pure mathematics

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