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The numeral axioms

What is pedagogically prior is not necessarily epistemologically prior, but certainly one is taught the numerals before one's "intuition" of the succession of numbers is "awakened." Regardless, it is possible to introduce the numerals without appeal to some intuitively given concept of the natural number sequence. The following axioms and axiom schemata may, for convenience, be given the title of 'the theory of numeral succession,' (NS). The following axiom and definition schemata provide for the usual correlation of "numbers" with numerals and a characterization of the successor function. Definition of 'numeral': Ό . . . '9' are simple numerals. If fti and Uz are simple numerals and nλ Φ '0' then Γnx n2 Ί is a compound numeral. If ΐii and n2 are compound numerals, rn1 n2 Λ is a compound numeral. (In the following Ύiχ and n3 Φ '0'.)