9942 | 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. |
10211 | 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. |
13655 | 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. |
9915 | 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) |
13040 | 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) |
13516 | 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. |