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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 31 ideas from Michael Potter

22273 Traditionally there are twelve categories of judgement, in groups of three [Potter]
22279 Frege's sign |--- meant judgements, but the modern |- turnstile means inference, with intecedents [Potter]
22281 A material conditional cannot capture counterfactual reasoning [Potter]
22283 Compositionality should rely on the parsing tree, which may contain more than sentence components [Potter]
22282 'Direct compositonality' says the components wholly explain a sentence meaning [Potter]
22284 'Greater than', which is the ancestral of 'successor', strictly orders the natural numbers [Potter]
22285 Impredicative definitions are circular, but fine for picking out, rather than creating something [Potter]
22291 Deductivism can't explain how the world supports unconditional conclusions [Potter]
22287 If 'concrete' is the negative of 'abstract', that means desires and hallucinations are concrete [Potter]
22290 The phrase 'the concept "horse"' can't refer to a concept, because it is saturated [Potter]
22295 Modern logical truths are true under all interpretations of the non-logical words [Potter]
22296 Compositionality is more welcome in logic than in linguistics (which is more contextual) [Potter]
22298 Why is fictional arithmetic applicable to the real world? [Potter]
22301 The Identity Theory says a proposition is true if it coincides with what makes it true [Potter]
22310 The formalist defence against Gödel is to reject his metalinguistic concept of truth [Potter]
22324 It has been unfortunate that externalism about truth is equated with correspondence [Potter]
22327 Knowledge from a drunken schoolteacher is from a reliable and unreliable process [Potter]
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]
10707 Mereology elides the distinction between the cards in a pack and the suits [Potter]
13041 Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
10708 Nowadays we derive our conception of collections from the dependence between them [Potter]
10709 Priority is a modality, arising from collections and members [Potter]
13042 If dependence is well-founded, with no infinite backward chains, this implies substances [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]