Combining Texts

Ideas for 'Parmenides', 'Metaphysical Dependence' and 'Foundations without Foundationalism'

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2 ideas

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

13657	First-order arithmetic can't even represent basic number theory [Shapiro]
	Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28)
	A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is.

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

13656	Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
	Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic.
	From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3)