The uniform distributions puzzle.

This note deepens a problem proposed and discussed by Kadane and O'Hagan (JASA, 1995). Kadane and O'Hagan discuss the existence of a uniform probability on the set of natural numbers (they provide a su_cient and necessary condition for the existence of such a uniform probability). I question the practical relevance of their solution. I show that a purely _nitely additive measure on the set of natural numbers cannot be constructed (its existence needs some form of the Axiom of Choice, the prototype of a nonconstructive axiom).

Purely finitely additive measures are non-constructible objects

The existence of a purely finitely additive measure cannot be proved in Zermelo-Frankel set theory if the use of the Axiom of Choice is disallowed.Finitely additive probabilities; Charges; Axiom of choice; Constructivism.

On additive properties of general sequences

AbstractThe authors give a survey of their papers on additive properties of general sequences and they prove several further results on the range of additive representation functions and on difference sets. Many related unsolved problems are discussed

Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn2/log2n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate

Towards an epistemic theory of probability.

The main concern of this thesis is to develop an epistemic conception of probability. In chapter one we look at Ramsey's work. In addition to his claim that the axioms of probability ace laws of consistency for partial beliefs, we focus attention on his view that the reasonableness of our probability statements does not consist merely in such coherence, but is to be assessed through the vindication of the habits which give rise to them. In chapter two we examine de Finetti's account, and compare it with Ramsey's. One significant point of divergence is de Finetti's claim that coherence is the only valid form of appraisal for probability statements. His arguments for this position depend heavily on the implementation of a Bayesian model for belief change; we argue that such an approach fails to give a satisfactory account of the relation between probabilities and objective facts. In chapter three we stake out the ground for oar own positive proposals - for an account which is non-objective in so far as it does not require the postulation of probabilistic facts, but non-subjective in the sense that probability statements are open to objective forms of appraisal. we suggest that a certain class of probability statements are best interpreted as recommendations of partial belief; these being measurable by the betting quotients that one judges to be fair. Moreover, we argue that these probability statements are open to three main forms of appraisal (each quantifiable through the use of proper scoring rules), namely: (i) Coherence (ii) Calibration (iii) Refinement. The latter two forms of appraisal are applicable both in an ex ante sense (relative to the information known by the forecaster) and an ex post one (relative to the results of the events forecast). In chapters four and five we consider certain problems which confront theories of partial belief; in particular, (1) difficulties surrounding the justification of the rule to maximise one's information, and (2) problems with the ascription of probabilities to mathematical propositions. Both of these issues seem resolvable; the first through the principle of maximising expected utility (SEU), and the second either by amending the axioms of probability, or by making use of the notion that probabilities are appraisable via scoring rules. There do remain, however, various difficulties with SEU, in particular with respect to its application in real-life situations. These are discussed, but no final conclusion reached, except that an epistemic theory such as ours is not undermined by the inapplicability of SEU in certain situations

Turán-Ramsey theorems and simple asymptotically extremal structures

This paper is a continuation of [10], where P. Erdos, A. Hajnal, V. T. Sos. and E. Szemeredi investigated the following problem: Assume that a so called forbidden graph L and a function f(n) = o(n) are fixed. What is the maximum number of edges a graph G(n) on n vertices can have without containing L as a subgraph, and also without having more than f(n) independent vertices? This problem is motivated by the classical Turan and Ramsey theorems, and also by some applications of the Turin theorem to geometry, analysis (in particular, potential theory) [27 29], [11-13]. In this paper we are primarily interested in the following problem. Let (G(n)) be a graph sequence where G(n) has n vertices and the edges of G(n) are coloured by the colours chi1,...,chi(r), so that the subgraph of colour chi(nu) contains no complete subgraph K(pnu), (nu = 1,...,r). Further, assume that the size of any independent set in G(n) is o(n) (as n --> infinity). What is the maximum number of edges in G(n) under these conditions? One of the main results of this paper is the description of a procedure yielding relatively simple sequences of asymptotically extremal graphs for the problem. In a continuation of this paper we shall investigate the problem where instead of alpha(G(n)) = o(n) we assume the stronger condition that the maximum size of a K(p)-free induced subgraph of G(n) is o(n)

"The Mathematics of Economic Growth"

Traditionally, economists have considered that mathematics acts as a universal language that lends clarity to theoretical statements. This paper proposes that mathematics does not function as a mere language. Rather, the advocacy of particular theoretical views and the choice of mathematical formalisms go hand-in-hand. The paper explores this issue by investigating the role of mathematics in developments of the theory of economic growth.