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Ideas for 'On the Question of Absolute Undecidability', 'The Advancement of Learning' and 'works'

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3 ideas

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity

17890	There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
	Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable).
	From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
	A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway]

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)