Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Word and Object' and 'Theories of Truth: a Critical Introduction'

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2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

17905	Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
	Full Idea: The condition on an explication of number can be put succinctly: any progression will do nicely. Russell once held that one must also be able to measure multiplicity, but this was a mistake; any progression can be fitted to that further condition.
	From: Willard Quine (Word and Object [1960], §54)
	A reaction: [compressed] This is the strongest possible statement that the numbers are the ordinals, and the Peano Axioms will define them. The Fregean view that cardinality comes first is redundant.

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]