more from this thinker | more from this text
Full Idea
Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem.
Clarification
How do we know that Julius Caesar is not a number?
Gist of Idea
Property extensions outstrip objects, so shortage of objects caused the Caesar problem
Source
report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.127
A Reaction
So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem?
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman] |
| 8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend] |
| 10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro] |
| 13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C] |
| 18142 | One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock] |
| 11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege] |
| 9046 | Our definition will not tell us whether or not Julius Caesar is a number [Frege] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C] |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| 17997 | Some suggest that the Julius Caesar problem involves category mistakes [Magidor] |