2 ms·
Dedekind cuts and Cauchy sequences don't result in different reals. They're different methods of proof, but the resulting sets are isomorphic to one another, so
by throwawaymath 8y ago
Dedekind cuts and Cauchy sequences don't result in different reals. They're different methods of proof, but the resulting sets are isomorphic to one another, so they're structurally the same even if a real from one looks a bit different from a real in the other.
All the Dedekind cuts and Cauchy sequences do is tell us, in essence: "yes, the axiomatized reals you're using are correct and fine because there are constructive proofs of their existence." Moreover, the point of those proofs (and analysis more generally) is to demonstrate mathematical continuity and uncountability. No matter how you slice this problem, if you construct analysis you will lose computability along the way.
- pure-awesome 8y agoI think there's perhaps a misunderstanding. You are correct that under classical assumptions of logic ("normal" mathematics), the Dedekind Cuts and Cauchy Sequences indeed result in the construction of isomorphic objects. However, under constructive assumptions of the logical axioms of mathematics, such as intuitionism (not accepting the law of excluded middle, i.e., assuming that the statement "P and NOT P" is not necessarily true for all P), the construction of the Dedekind Cuts and the Cauchy Sequences may result in different objects, i.e., it may no longer provable that they are equivalent.