Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Quaestiones de Potentia Dei' and 'Categories'

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3 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

11044	One is prior to two, because its existence is implied by two [Aristotle]
	Full Idea: One is prior to two because if there are two it follows at once that there is one, whereas if there is one there is not necessarily two.
	From: Aristotle (Categories [c.331 BCE], 14a29)
	A reaction: The axiomatic introduction of a 'successor' to a number does not seem to introduce this notion of priority, based on inclusiveness. Introducing order by '>' also does not seem to indicate any logical priority.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

11042	Parts of a line join at a point, so it is continuous [Aristotle]
	Full Idea: A line is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point.
	From: Aristotle (Categories [c.331 BCE], 04b33)
	A reaction: This appears to be the essential concept of a Dedekind cut. It seems to be an open question whether a cut defines a unique number, but a boundary seems to be intrinsically unique. Aristotle wins again.

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]