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Single Idea 13655

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L ]

Full Idea

Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible.

Gist of Idea

The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate

Source

report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.109

A Reaction

[He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal.

The 6 ideas with the same theme [possible axiom saying all sets are constructible]:

9942 Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam]
10211 Quine wants V = L for a cleaner theory, despite the scepticism of most theorists [Quine, by Shapiro]
13655 The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro]
9915 V = L just says all sets are constructible [Putnam]
13040 Constructibility: V = L (all sets are constructible) [Kunen]
13516 If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD]