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]].

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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
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
Infinity: There is at least one limit level
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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
The 'limitation of size' principles say whether properties collectivise depends on the number of objects
4. Formal Logic / G. Formal Mereology / 1. Mereology
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
We can formalize second-order formation rules, but not inference rules
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Supposing axioms (rather than accepting them) give truths, but they are conditional
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
Russell's paradox means we cannot assume that every property is collectivizing
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
If set theory didn't found mathematics, it is still needed to count infinite sets
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
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
A relation is a set consisting entirely of ordered pairs
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
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
Collections have fixed members, but fusions can be carved in innumerable ways
10. Modality / A. Necessity / 1. Types of Modality
Priority is a modality, arising from collections and members