Combining Texts

All the ideas for 'The Problem of Empty Names', 'Philosophical Grammar' and 'Elements of Set Theory'

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11 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]

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]

5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names

18946

Unreflectively, we all assume there are nonexistents, and we can refer to them [Reimer]

6. Mathematics / A. Nature of Mathematics / 1. Mathematics

18281

In mathematics everything is algorithm and nothing is meaning [Wittgenstein]

21. Aesthetics / A. Aesthetic Experience / 2. Aesthetic Attitude

6606	Consider: "Imagine this butterfly exactly as it is, but ugly instead of beautiful" [Wittgenstein]