Combining Philosophers

Ideas for Douglas Lackey, Lynn Holt and Tim Button

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3 ideas

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

18694	Permutation Theorem: any theory with a decent model has lots of models [Button]
	Full Idea: The Permutation Theorem says that any theory with a non-trivial model has many distinct isomorphic models with the same domain.
	From: Tim Button (The Limits of Reason [2013], 02.1)
	A reaction: This may be the most significant claim of model theory, since Putnam has erected an argument for anti-realism on it. See the ideas of Tim Button.

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox

21554	Sets always exceed terms, so all the sets must exceed all the sets [Lackey]
	Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets.
	From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127)
	A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom.

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox

21553	It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey]
	Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1.
	From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127)
	A reaction: Formulated by Burali-Forti in 1897.