Combining Texts

Ideas for 'fragments/reports', 'Constructibility and Mathematical Existence' and 'The Principles of Mathematics'

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6 ideas

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

14113	The null class is a fiction [Russell]
	Full Idea: The null class is a fiction.
	From: Bertrand Russell (The Principles of Mathematics [1903], §079)
	A reaction: This does not commit him to regarding all classes as fictions - though he seems to have eventually come to believe that. The null class seems to have a role something like 'Once upon a time...' in story-telling. You can then tell truth or fiction.

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets

15894	Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
	Full Idea: Russell was the inventor of the naïve set theory so often attributed to Cantor.
	From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Shaughan Lavine - Understanding the Infinite I

4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets

14126	Order rests on 'between' and 'separation' [Russell]
	Full Idea: The two sources of order are 'between' and 'separation'.
	From: Bertrand Russell (The Principles of Mathematics [1903], §204)

14127	Order depends on transitive asymmetrical relations [Russell]
	Full Idea: All order depends upon transitive asymmetrical relations.
	From: Bertrand Russell (The Principles of Mathematics [1903], §208)

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory

8758	We could talk of open sentences, instead of sets [Chihara, by Shapiro]
	Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'.
	From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2
	A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets.

4. Formal Logic / G. Formal Mereology / 1. Mereology

14121	The part-whole relation is ultimate and indefinable [Russell]
	Full Idea: The relation of whole and part is, it would seem, an indefinable and ultimate relation, or rather several relations, often confounded, of which one at least is indefinable.
	From: Bertrand Russell (The Principles of Mathematics [1903], §135)
	A reaction: This is before anyone had produced a mathematical account of mereology (qv).