 Russell defined natural numbers as classes of equinumerous classes. Any  pair , a class with two members, can be put into a one to one correspondence with any other, hence all pairs are  equinumerous. The number two is then identified with the class of all pairs. The relation between equinumerous classes when there is such a one to one mapping relating them is called “similarity”. Similarity is defined solely in terms of logical notions of quantifiers and identity. With the natural numbers so defined, Peano axioms can be derived by logical means alone. After natural numbers, Russell adds “series, ordinal numbers and real numbers” (1924, 166) to his list of constructions, and then concludes with the construction of matter.