Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Mathematical logic and theory of types' and 'Understanding the Infinite'

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4 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

15926	Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
	Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed.
	From: Shaughan Lavine (Understanding the Infinite [1994], V.3)
	A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'.

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

15934	Mathematical proof by contradiction needs the law of excluded middle [Lavine]
	Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction.
	From: Shaughan Lavine (Understanding the Infinite [1994], VI.1)
	A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku.

5. Theory of Logic / K. Features of Logics / 5. Incompleteness

17894	We have no argument to show a statement is absolutely undecidable [Koellner]
	Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable.
	From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3)

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox

6407	The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling]
	Full Idea: The class of teaspoons isn't a teaspoon, so isn't a member of itself; but the class of non-teaspoons is a member of itself. The class of all classes which are not members of themselves is a member of itself if it isn't a member of itself! Paradox.
	From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by A.C. Grayling - Russell Ch.2
	A reaction: A very compressed version of Russell's famous paradox, often known as the 'barber' paradox. Russell developed his Theory of Types in an attempt to counter the paradox. Frege's response was to despair of his own theory.