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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)