Kurt Gödel Archives



More Generally: Historical Thinkers (239)

September 15, 2015

Sets and Possible Worlds

This semester I'm directing an independent study on modal logic with a couple of students with strong math background. Yesterday some questions about sets and possible worlds came up, so I wrote up some notes for my students on the subject. This blog post is adapted from those notes. Introduction The development of axiomatic set theory was launched by consideration of Russell's Paradox: let A be the set of all sets that do not contain themselves. Does A contain itself or not? (This was on Existential Comics just yesterday!) The collection of axioms mathematicians developed to avoid paradox has the...
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September 7, 2010

Would a Being With All Positive Properties Be God?

Sobel's final objection to ontological arguments is that, even if they are sound, their conclusion does not mean that God exists. That is, according to Sobel, a necessarily existing 'being than which none greater can be conceived' or 'being with all perfections' or 'being with all positive properties' would not be God. His argument for this is rather confusing and depends (1) on a controversial modal intuition, and (2) on an odd definition of 'worshipfulness'. As far as I can tell, the argument goes like this: it is clear (so Sobel claims) that such properties as consciousness, knowledge, power, love,...
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September 2, 2010

Modal Collapse: Sobel's Objection to Gödel's Ontological Argument

The last ontological argument Sobel discusses is the Leibniz-inspired argument put forward by the famous logician Kurt Gödel. Gödel sets up a formal system in third-order quantified modal logic with equality and abstraction (!) and proves within that system the theorem: □∃xG(x) Where the predicate G is defined as follows: Gx ↔ ∀φ[P(φ) → φ(x)] Where P is primitive. (Sobel includes the complete source texts for Gödel's proof on pp. 144-146.) Now, unsurprisingly, given that the proof was originated by Gödel, everyone agrees that the proof is valid in the formal system. The question is whether there are any interpretations...
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