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Ideas for 'Dthat', 'Introduction to the Theory of Logic' and 'Cardinality, Counting and Equinumerosity'

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

17459	Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
	Full Idea: The interest of Frege's Theorem is that it offers us an explanation of the fact that the numbers satisfy the Dedekind-Peano axioms.
	From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6)
	A reaction: He says 'explaining' does not make it more fundamental, since all proofs explain why their conclusions hold.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction

10891	If a set is defined by induction, then proof by induction can be applied to it [Zalabardo]
	Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property.
	From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3)