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Ideas for 'fragments/reports', 'Intro to Gdel's Theorems' and 'Phenomenalism'

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3 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

10599	For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
	Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5)

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

10610	The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
	Full Idea: It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 18.2)

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

10619	The truths of arithmetic are just true equations and their universally quantified versions [Smith,P]
	Full Idea: The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 27.7)
	A reaction: Must each equation be universally quantified? Why can't we just universally quantify over the whole system?