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Sewing-Machine comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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I would be interested in knowing if there is any second-order system which is strong enough to talk about continuity, but not to prove the existence of a first uncountable ordinal.
van den Dries, "Tame topology and o-minimal structures," Cambridge U Press 1998
develops a lot of 20th century geometry in a first order theory of real numbers. You can do enough differential geometry in this setting to do e.g. general relativity.