Ideas from 'Set Theory and Its Philosophy' by Michael Potter [2004], by Theme Structure
[found in 'Set Theory and Its Philosophy' by Potter,Michael [OUP 2004,0-19-927041-4]].
green numbers give full details |
back to texts
|
expand these ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
10702
|
Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning
|
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
10713
|
Usually the only reason given for accepting the empty set is convenience
|
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
13044
|
Infinity: There is at least one limit level
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
10708
|
Nowadays we derive our conception of collections from the dependence between them
|
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
13546
|
The 'limitation of size' principles say whether properties collectivise depends on the number of objects
|
4. Formal Logic / G. Formal Mereology / 1. Mereology
10707
|
Mereology elides the distinction between the cards in a pack and the suits
|
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10704
|
We can formalize second-order formation rules, but not inference rules
|
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
10703
|
Supposing axioms (rather than accepting them) give truths, but they are conditional
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
10712
|
If set theory didn't found mathematics, it is still needed to count infinite sets
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17882
|
It is remarkable that all natural number arithmetic derives from just the Peano Axioms
|
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
13043
|
A relation is a set consisting entirely of ordered pairs
|
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
13042
|
If dependence is well-founded, with no infinite backward chains, this implies substances
|
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
13041
|
Collections have fixed members, but fusions can be carved in innumerable ways
|
10. Modality / A. Necessity / 1. Types of Modality
10709
|
Priority is a modality, arising from collections and members
|