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Single Idea 10702

[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory ]

Full Idea

Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning.

Gist of Idea

Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning

Source

Michael Potter (Set Theory and Its Philosophy [2004], Intro 1)

Book Ref

Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.4

A Reaction

These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers.

The 14 ideas from 'Set Theory and Its Philosophy'

10702 Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
10703 Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter]
10704 We can formalize second-order formation rules, but not inference rules [Potter]
13041 Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
10707 Mereology elides the distinction between the cards in a pack and the suits [Potter]
10708 Nowadays we derive our conception of collections from the dependence between them [Potter]
13042 If dependence is well-founded, with no infinite backward chains, this implies substances [Potter]
10709 Priority is a modality, arising from collections and members [Potter]
10712 If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
10713 Usually the only reason given for accepting the empty set is convenience [Potter]
13043 A relation is a set consisting entirely of ordered pairs [Potter]
13044 Infinity: There is at least one limit level [Potter]
17882 It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
13546 The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter]