### 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
 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
###### 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 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