On a strict formalist reading (cf. Curry, Detlefsen's 'Hilbert's Program'), there is no well-defined notion of 'natural number' outside a formal system against which model elements can be measured as 'non-standard'.
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An object or thing that fits into a mathematical structure—think of it as a concrete example that satisfies the rules of a formal system.
Natural number(Frege-style logicist definition used in the proof)
A number n such that Precedes⁺(0,n) holds — i.e., 0 bears the strong ancestral of the Precedes relation to n
Non-standard(in logic and mathematics)
Something that doesn't match the usual or expected interpretation—in this context, a number-like object that behaves differently than ordinary natural numbers do.