Combining Texts

Ideas for 'Mahaprajnaparamitashastra', 'Philosophy of Mathematics' and 'Thus Spake Zarathustra'

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5 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
18114 There is no single agreed structure for set theory [Bostock]
     Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
18107 A 'proper class' cannot be a member of anything [Bostock]
     Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class.
     From: David Bostock (Philosophy of Mathematics [2009], 5.4)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
18115 We could add axioms to make sets either as small or as large as possible [Bostock]
     Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to.
     From: David Bostock (Philosophy of Mathematics [2009], 6.4)
     A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
18139 The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
     Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice.
     From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36)
     A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
18105 Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
     Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist.
     From: David Bostock (Philosophy of Mathematics [2009], 5.4)