Full Idea
Mill says arithmetic has two axioms, that 'things which are equal to the same thing are equal to each other', and 'equals added to equals make equal sums', plus a definition for each numeral as 'formed by the addition of a unit to the previous number'.
Gist of Idea
The only axioms needed are for equality, addition, and successive numbers
Source
report of John Stuart Mill (System of Logic [1843], p.610?) by Stewart Shapiro - Thinking About Mathematics 4.3
Book Reference
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.95
A Reaction
The difficulty here seems to be the definition of 1, and (even worse for an empiricist), of 0. Then he may have a little trouble when he reaches infinity.