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by mabbo
3076 days ago
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> But this is just a flaw in our current definition of real numbers. The distance between discrete infinite sets (countable) and continuous infinite sets (uncountable) is a pretty large gap. Are you saying that there is no gap? There are pretty well studied proofs showing that uncountable sets are much bigger than countable. What's your take on this? Got a counter proof to Cantor? |
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As much as I am a fan of all things about cardinals, measure and category (Oxtoby's booklet on this stuff has been on my nightstand for several months, to a great pleasure), I still fail to recognize its technological and physical implications.
My take on this is modelled after the famous article "The dawning of the age of stochasticity" by David Mumford, where he argues that the current definition of real numbers leads to infuriating contradictions, especially in the light of probability theory. This article left me with the impression that in a near future, we will see a clever definition of "real-valued random variable" that does not depend on the definition of real numbers and avoids all kind of disgusting paradoxes.