Combining Texts

All the ideas for 'poems', 'Of Organum or Ars Magna of Thinking' and 'Elements of Set Theory'

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12 ideas

1. Philosophy / F. Analytic Philosophy / 1. Nature of Analysis

5021	An idea is analysed perfectly when it is shown a priori that it is possible [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]

13204

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

13206

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

13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / d. Rational foundations

5020	Our thoughts are either dependent, or self-evident. All thoughts seem to end in the self-evident [Leibniz]

22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention

6017	Nomos is king [Pindar]

22. Metaethics / C. The Good / 2. Happiness / d. Routes to happiness

5019	Supreme human happiness is the greatest possible increase of his perfection [Leibniz]