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