Combining Texts

All the ideas for 'works', 'Elements of Set Theory' and 'On Wisdom'

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11 ideas

1. Philosophy / A. Wisdom / 1. Nature of Wisdom

19396

Wisdom is knowing all of the sciences, and their application [Leibniz]

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]

11. Knowledge Aims / A. Knowledge / 1. Knowledge

19397

Perfect knowledge implies complete explanations and perfect prediction [Leibniz]

22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature

22086

The most important aspect of a human being is not reason, but passion [Kierkegaard, by Carlisle]