Combining Texts

All the ideas for 'The Logic of Boundaryless Concepts', 'Elements of Set Theory' and 'Popular Scientific Lectures'

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11 ideas

1. Philosophy / G. Scientific Philosophy / 2. Positivism

15873

Laws of nature are just records of regularities and correlations, with concepts to make recording them easier [Mach, by Harré]

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]

5. Theory of Logic / A. Overview of Logic / 3. Value of Logic

9390	Logic guides thinking, but it isn't a substitute for it [Rumfitt]

9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects

9389	Vague membership of sets is possible if the set is defined by its concept, not its members [Rumfitt]