Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Content Preservation' and 'Defending the Axioms'

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3 ideas

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

17887	PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
	Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem].
	From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1)
	A reaction: Each expansion brings a limitation, but then you can expand again.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

17891	Arithmetical undecidability is always settled at the next stage up [Koellner]
	Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next.
	From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)

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

17618	Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
	Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness.
	From: Penelope Maddy (Defending the Axioms [2011], 3.4)
	A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics.