Combining Philosophers

Ideas for Eubulides, Seymour Lipschutz and Peter Smith

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13 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]
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]
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]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10618 All numbers are related to zero by the ancestral of the successor relation [Smith,P]
10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
10849 Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P]
10850 Baby Arithmetic is complete, but not very expressive [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
10851 Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P]
10852 Robinson Arithmetic (Q) is not negation complete [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
10068 Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
10603 The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10848 Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
10604 Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]