display all the ideas for this combination of texts
2 ideas
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28) | |
A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is. |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3) |