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## What is an integer ? A crafty representation

Integers are an extension of the natural numbers where the existence of subtraction provides a more appropriate framework for certain questions of arithmetic. They can be described axiomatically, but can also be constructed from the set of natural numbers and some...

## What is a natural number? Defining or axiomatising

Mathematical science does not seek to define the notion of a natural number, but to understand the set of natural numbers. “Natural numbers have been made by God, everything else is the work of men”. Leopold Kronecker 1.We don’t define the natural...

## An infinity of prime numbers : Euclid’s theorem

The prime natural numbers are those which have no divisors other than 1 and themselves. They exist in infinite number by Euclid’s theorem, which is not difficult to prove. 1.Prime numbers 1.1.Divisors and primes A prime number is a non-zero natural number (see...

## Finiteness and Mathematical Infinity : Comparing and Enumerating

A finite set is a set that can be counted using the natural numbers $$1,\ldots,n$$ for a certain natural number $$n$$. But what is counting ? And then, what is an infinite set? 1.Comparing sets : the notion of bijection The notions of finite set and infinite set, and...

## What is a set? Founding mathematics in intuition

Naive set theory or “potato science” is the natural (and understandable!) foundation of mathematical science. “I know what time is. If you ask me, I don’t know anymore.” Augustine. This insightful quote from Augustine emphasises that...