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Full Idea
Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects.
Gist of Idea
Arithmetic must allow for the possibility of only a finite total of objects
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148)
Book Ref
-: 'Journal of Philosophy' [-], p.148
A Reaction
This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system.
| 10011 | Identity is a level one relation with a second-order definition [Hodes] |
| 10027 | Mathematics is higher-order modal logic [Hodes] |
| 10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
| 10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
| 10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
| 10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
| 10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
| 10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
| 10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |