User:IssaRice/Construction of the real numbers - Revision history

IssaRice at 18:52, 18 July 2019

2019-07-18T18:52:32Z

← Older revision		Revision as of 18:52, 18 July 2019
Line 5:		Line 5:
	"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]		"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]

	https://math.stackexchange.com/questions/2044775/prove-that-a-theory-gamma-is-consistent-if-and-only-if-there-is-a-structure		https://math.stackexchange.com/questions/2044775/prove-that-a-theory-gamma-is-consistent-if-and-only-if-there-is-a-structure This is actually just a restatement of the soundness and completeness theorems: a set of sentences is satisfiable (has a model) iff it is consistent.

	also why doesn't constructing a [https://en.wikipedia.org/wiki/Herbrand_structure term model] prove existence of R?		also why doesn't constructing a [https://en.wikipedia.org/wiki/Herbrand_structure term model] prove existence of R?

IssaRice at 07:50, 6 July 2019

2019-07-06T07:50:07Z

← Older revision		Revision as of 07:50, 6 July 2019
Line 1:		Line 1:
	something that i have seen clarified ''exactly once'' to me 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'' to me (and never in a textbook!) 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.

	"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525		"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525

IssaRice at 07:46, 6 July 2019

2019-07-06T07:46:12Z

← Older revision		Revision as of 07:46, 6 July 2019
Line 4:		Line 4:

	"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]		"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]

			https://math.stackexchange.com/questions/2044775/prove-that-a-theory-gamma-is-consistent-if-and-only-if-there-is-a-structure

			also why doesn't constructing a [https://en.wikipedia.org/wiki/Herbrand_structure term model] prove existence of R?

IssaRice at 07:15, 6 July 2019

2019-07-06T07:15:13Z

← Older revision		Revision as of 07:15, 6 July 2019
Line 1:		Line 1:
	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'' to me 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.

	"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525		"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525

	"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]		"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]

IssaRice at 07:14, 6 July 2019

2019-07-06T07:14:37Z

← Older revision		Revision as of 07:14, 6 July 2019
Line 1:		Line 1:
	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.

			"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525

	"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]		"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]

IssaRice at 07:12, 6 July 2019

2019-07-06T07:12:33Z

← Older revision		Revision as of 07:12, 6 July 2019
Line 1:		Line 1:
	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]

IssaRice: Created page with "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 ration..."

2019-07-06T07:09:50Z

Created page with "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 ration..."

New page

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.