Abstract
In this paper we introduce the notion of elementary numerosity as a special function dened on all subsets of a given set which takes values in a suitable non-Archimedean field, and satises the same formal properties of finite cardinality. We investigate the general compatibility of this notion with the notion of measure. As an application, we present a model for the probability of infinite sequences of coin tosses, directly obtained from a suitable elementary numerosity.
Highlights
In mathematics there are essentially two main ways to estimate the size of a set, depending on whether one is working in a discrete or in a continuous setting
We will introduce the concept of elementary numerosity as a special function defined on all subsets of a given set Ω that takes values in a suitable ordered field F and satisfies the four properties of finite cardinalities itemized above
A comprehensive exposition of nonstandard measure theory and probability theory is given in Ross [14]
Summary
In mathematics there are essentially two main ways to estimate the size of a set, depending on whether one is working in a discrete or in a continuous setting. In the continuous case one uses the notion of (finitely) additive measure, namely a real-valued function (possibly taking the value +∞) which satisfies the following properties:. In the discrete case one uses the notion of cardinality n that strengthens the three properties itemized above as follows:. The goal of this paper is to investigate the relationships between these two notions. To this end, we will introduce the concept of elementary numerosity as a special function defined on all subsets of a given set Ω that takes values in a suitable ordered field F and satisfies the four properties of finite cardinalities itemized above (see Definition 1.1). A comprehensive exposition of nonstandard measure theory and probability theory is given in Ross [14]
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