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 2004,0199270414]].
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
10702

Set theory's three roles: taming the infinite, subjectmatter 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. SecondOrder Logic
10704

We can formalize secondorder 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

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
10711

Russell's paradox means we cannot assume that every property is collectivizing

6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
10712

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
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 wellfounded, 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
