Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'A Subject with No Object' and 'The Periodic Table'

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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 / 7. Mathematical Structuralism / c. Nominalist structuralism

9925	Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen]
	Full Idea: Usually structuralism and nominalism are considered rivals. But structuralism can also be the first step in a strategy of nominalist reconstrual or paraphrase.
	From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.0)
	A reaction: Hellman and later Chihara seem to be the main proponents of nominalist structuralism. My sympathies lie with this strategy. Are there objects at the nodes of the structure, or is the structure itself platonic? Mill offers a route.