display all the ideas for this combination of texts
5 ideas
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) |
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. |
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. |
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. |
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) |