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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L

[possible axiom saying all sets are constructible]

6 ideas
Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam]
     Full Idea: Gödel proved the classical relative consistency of the axiom V = L (which implies the axiom of choice and the generalized continuum hypothesis). This established the full independence of the continuum hypothesis from the other axioms.
     From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Hilary Putnam - Mathematics without Foundations
     A reaction: Gödel initially wanted to make V = L an axiom, but the changed his mind. Maddy has lots to say on the subject.
Quine wants V = L for a cleaner theory, despite the scepticism of most theorists [Quine, by Shapiro]
     Full Idea: Quine suggests that V = L be accepted in set theory because it makes for a cleaner theory, even though most set theorists are skeptical of V = L.
     From: report of Willard Quine (works [1961]) by Stewart Shapiro - Philosophy of Mathematics Ch.1
     A reaction: Shapiro cites it as a case of a philosopher trying to make recommendations to mathematicians. Maddy supports Quine.
The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro]
     Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible.
     From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism
     A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal.
V = L just says all sets are constructible [Putnam]
     Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets.
     From: Hilary Putnam (Models and Reality [1977], p.425)
Constructibility: V = L (all sets are constructible) [Kunen]
     Full Idea: Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom.
     From: Kenneth Kunen (Set Theory [1980], §6.3)
If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD]
     Full Idea: It has been said (by Burt Dreben) that the only reason set theorists do not generally buy the view that V = L is that it would put them out of business by settling their open questions.
     From: William D. Hart (The Evolution of Logic [2010], 10)
     A reaction: Hart says V=L breaks with the interative conception of sets at level ω+1, which is countable is the constructible view, but has continuum many in the cumulative (iterative) hierarch. The constructible V=L view is anti-platonist.