Peano's axiomatization treats the induction schema as a primitive axiom from which theorems are derived, not as a grammatical stipulation, and this framework successfully grounds arithmetic.

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Axiomatization(as in the title of Reichenbach's work): The process of taking a theory and organizing it into a set of basic assumptions (called axioms) from which everything else can be logically derived.
Induction schema(An axiom schema of Peano Arithmetic, one instance per expressible formula): For every formula A(x) expressible in the language of Peano Arithmetic: if A(0) holds and for all x, A(x) implies A(x'), then A(x) holds for all x.
Peano(as the mathematician whose system is being discussed): Giuseppe Peano was an Italian mathematician who created a set of basic rules (called axioms) that describe how natural numbers (0, 1, 2, 3...) work and relate to each other.
grammatical stipulation(as a contrast to how Peano's axiom works): A rule about language and how we write symbols, rather than a rule about how the world actually works—like deciding how to spell a word versus explaining what the word means.

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