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3 ideas
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) |
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) |
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? |