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Ideas for 'fragments/reports', 'Reals by Abstraction' and 'Model Theory'

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2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10632 The real numbers may be introduced by abstraction as ratios of quantities [Hale, by Hale/Wright]
     Full Idea: The real numbers may be introduced by abstraction as ratios of quantities. ..They are not defined by Dedekind cuts; rather, the cuts constitute a domain with the properties that are a necessary precondition.
     From: report of Bob Hale (Reals by Abstraction [1998]) by B Hale / C Wright - Intro to 'The Reason's Proper Study' 3.3
     A reaction: This is Hale's neo-logicist attempt to derive the real numbers from Hume's Principle.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
10480 First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
     Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another.
     From: Wilfrid Hodges (Model Theory [2005], 4)
     A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them.