Combining Texts

Ideas for 'Unconscious Cerebral Initiative', 'Philosophies of Mathematics' and 'Reasoning and the Logic of Things'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

5 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
     Full Idea: The first-order version of the induction axiom is weaker than the second-order, because the latter applies to all concepts, but the first-order applies only to concepts definable by a formula in the first-order language of number theory.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7 n7)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
     Full Idea: The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)
     A reaction: The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic.
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17902 A successor is the union of a set with its singleton [George/Velleman]
     Full Idea: For any set x, we define the 'successor' of x to be the set S(x) = x U {x}.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
     A reaction: This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76.
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
     Full Idea: The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs).
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1)
     A reaction: Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10130 Set theory can prove the Peano Postulates [George/Velleman]
     Full Idea: The Peano Postulates can be proven in ZFC.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)