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Ideas for 'Parmenides', 'Logic for Philosophy' and '68: Generation of the soul in 'Timaeus''

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2 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order

13701	A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
	Full Idea: A single second-order sentence has second-order semantic consequences which are all and only the truths of arithmetic, but this is cold comfort because of incompleteness; no axiomatic system draws out the consequences of this axiom.
	From: Theodore Sider (Logic for Philosophy [2010], 5.4.3)

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

16150	One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
	Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
	From: Plato (Parmenides [c.364 BCE], 144a)
	A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.