Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Principia Mathematica' and 'The Mysterious Flame'

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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 / 5. Definitions of Number / a. Defining numbers

18152	Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock]
	Full Idea: Although Russell takes numbers to be certain classes, his 'no-class' theory then eliminates all mention of classes in favour of the 'propositional functions' that define them; and in the case of the numbers these just are the numerical quantifiers.
	From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by David Bostock - Philosophy of Mathematics 9.B.4