Combining Texts

All the ideas for 'fragments/reports', 'Elements of Set Theory' and 'Identity'

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10 ideas

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST

13201

∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]

13206

A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]

13204

The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set

13200

Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]

13199

The empty set may look pointless, but many sets can be constructed from it [Enderton]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets

13203

The singleton is defined using the pairing axiom (as {x,x}) [Enderton]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII

13202

Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

13205

We can only define functions if Choice tells us which items are involved [Enderton]

9. Objects / F. Identity among Objects / 1. Concept of Identity

8970	Our notion of identical sets involves identical members, which needs absolute identity [Hawthorne]

21. Aesthetics / C. Artistic Issues / 7. Art and Morality

468	Musical performance can reveal a range of virtues [Damon of Ath.]