How do you construct the set of real numbers from integers?

How do you construct the set of real numbers from integers?

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arxiv.org/abs/math/0405454
youtube.com/watch?v=BBp0bEczCNg
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R is the metric closure of the field of fractions of Z

It's weird user, usually if you have the knowledge to ask this question you already know the answer

metwic cwosure!!!! i wike topowogy!!!!

define "metric closure"

The word is "completion"

then it should have said "a metric closure" instead of "the metric closure"

arxiv.org/abs/math/0405454

>For any metric space M, one can construct a complete metric space M (which is also denoted as M), which contains M as a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f from M to N, which extends f. The space M' is determined up to isometry by this property, and is called the completion of M.

What does that have to do with my post?

The completion is unique. It should have said "the" not "a".

>The completion is unique.
The metric is not unique. It should have said "a" not "the".

>The completion is unique.
up to isometry

>construct

Why do you need to?

therefore unique

Continued fraction representation

By drawing a line and imagining reasonable things about it.

t. geometer

>constructible numbers

even simple lines can be unreasonable... this was really good N*

youtube.com/watch?v=BBp0bEczCNg

Let (X,d) be a metric space and let A,B be closed sets of X with A∩B = empty. Prove there exist two open sets, G,H with G∩H = empty (A included in G, B included in H)

First construct rationals from integers. Then use cauchy sequences of rationals to define real numbers. EZPZ

Float precision.

R ⊊ O

I like it like this:

-Field
-Ordered Field
-Supreme Axiom

But I guess you can use Dedekind for a simple, conceptual approach.

2^N

An uncountably infinite set of sequences over {0,1}

You don't. Take the axiomatic approach instead.

Every Dedekind-complete ordered field is isomorphic to the real numbers. Hence the real numbers are *the* Dedekind-complete ordered field.

This is why nobody takes set theory seriously