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Ideas for 'fragments/reports', 'De arcanus motus' and 'Introduction to the Philosophy of Mathematics'

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3 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

17924	Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
	Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false.
	From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3)
	A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa.

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17929	Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]
	Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences.
	From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

17930	Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
	Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
	From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
	A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.