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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

10067	Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave]
	Full Idea: Gentzen proved the consistency of arithmetic from assumptions which transcend arithmetic.
	From: report of Gerhard Gentzen (works [1938]) by Alan Musgrave - Logicism Revisited §5
	A reaction: This does not contradict Gödel's famous result, but reinforces it. The interesting question is what assumptions Gentzen felt he had to make.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic

11041	Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle]
	Full Idea: Of quantities, some are discrete, others continuous. ...Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place.
	From: Aristotle (Categories [c.331 BCE], 04b20)
	A reaction: This distinction seems to me to be extremely illuminating, when comparing natural numbers with real numbers, and it is the foundation of the Greek view of mathematics.