User:IssaRice/Construction of the real numbers: Difference between revisions
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something that i have seen clarified ''exactly once'' is why constructing the real numbers (for instance via dedekind cuts or equivalence classes of cauchy sequences of rationals) constitutes a proof that <math>\mathbf R</math> (or a complete ordered fields or whatever you want to call it) exists. the reason is this (?): to show that something exists, one way to do this is to show that a model (in the model theory sense) of it exists. | something that i have seen clarified ''exactly once'' is why constructing the real numbers (for instance via dedekind cuts or equivalence classes of cauchy sequences of rationals) constitutes a proof that <math>\mathbf R</math> (or a complete ordered fields or whatever you want to call it) exists. the reason is this (?): to show that something exists, one way to do this is to show that a model (in the model theory sense) of it exists. | ||
"It takes a good deal of mathematical sophistication to even appreciate why someone would want to prove that the theory of the real numbers is consistent, and even more sophistication to appreciate why we can do so by making a “model” of the theory." [https://arxiv.org/pdf/1204.4483.pdf] | |||
Revision as of 07:12, 6 July 2019
something that i have seen clarified exactly once is why constructing the real numbers (for instance via dedekind cuts or equivalence classes of cauchy sequences of rationals) constitutes a proof that (or a complete ordered fields or whatever you want to call it) exists. the reason is this (?): to show that something exists, one way to do this is to show that a model (in the model theory sense) of it exists.
"It takes a good deal of mathematical sophistication to even appreciate why someone would want to prove that the theory of the real numbers is consistent, and even more sophistication to appreciate why we can do so by making a “model” of the theory." [1]