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D_Malik comments on Open thread, Mar. 23 - Mar. 31, 2015 - Less Wrong Discussion

6 Post author: MrMind 23 March 2015 08:38AM

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Comment author: D_Malik 23 March 2015 07:20:03PM *  1 point [-]

You're right that there is no greatest cardinal number. The number of ordinals is greater than any ordinal; I'm not sure whether that's true for cardinal numbers.

You can sorta get around the arbitrarity by postulating the mathematical universe hypothesis, that all mathematical objects are real.

"Discrete Euclidean space" Z^n would be countably infinite, and the usual continuous Euclidean space R^n would be continuum infinite, but I'm not sure what a world whose space is more infinite than the continuum would look like.

Comment author: [deleted] 23 March 2015 11:42:02PM *  2 points [-]

It is also true that the number of cardinals is greater than any cardinal, leading to Cantor's Paradox.

... Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

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