Ideas from 'What are Sets and What are they For?' by Oliver,A/Smiley,T [2006], by Theme Structure

[found in 'Metaphysics (Philosophical Perspectives 20)' (ed/tr Hawthorne,John) [Blackwell 2006,978-1-4051-6792-5]].

Click on the Idea Number for the full details    |     back to texts     |     expand these ideas

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The empty set is something, not nothing!
The empty set is usually derived from Separation, but it also seems to need Infinity
We don't need the empty set to express non-existence, as there are other ways to do that
Maybe we can treat the empty set symbol as just meaning an empty term
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Set
The unit set may be needed to express intersections that leave a single member
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
Cantor's sets were just collections, but Dedekind's were containers
5. Theory of Logic / G. Quantification / 6. Plural Quantification
If you only refer to objects one at a time, you need sets in order to refer to a plurality
We can use plural language to refer to the set theory domain, to avoid calling it a 'set'
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths are true no matter what exists - but predicate calculus insists that something exists
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
If mathematics purely concerned mathematical objects, there would be no applied mathematics
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Sets might either represent the numbers, or be the numbers, or replace the numbers