Is R^3 Equivalent to E^3 in Euclidean Space?

[BHL 20]

If you take an ordered field of numbers with the operations of addition and multiplication, endowed with the completeness axiom, represent in as an infinite series of points constituting a line, then put three such lines orthogonal to each other, it does not seem obvious to me that this is the exact three-dimensional space satisfying Euclid's five axioms. Is there a formal proof of the equivalence of these two spaces, and if there is where can I find it ?

 

[Mark44]

I'm not aware of any differences between R³ and E³. AFAIK, these are just different notations for the same space, so no proof would be needed.

 

[BHL 20]

I'm not aware of any differences between R³ and E³. AFAIK, these are just different notations for the same space, so no proof would be needed.

So the properties of both R³ and E³ are based on the definition of the real numbers. But we still use the results of Euclidean geometry in these spaces so even if we mean the same thing by these notations we're assuming them to be equivalent to the space defined by Euclid's axioms, so a proof is needed. Unless of course all those results can also be proved in a different way, directly from the properties of the R³ i.e. from the definition of the real numbers. Which is it ?

 

[Ben Niehoff]

My understanding is that $\mathbb{R}^3$ is a topological space, whereas $\mathbb{E}^3$ is $\mathbb{R}^3$ with the usual Euclidean metric tensor. That is, $\mathbb{E}^3$ assumes a geometrical structure that $\mathbb{R}^3$ does not.

For example, the hyperbolic plane $H^3$ is $\mathbb{R}^3$ endowed with the metric tensor

$$ds^2 = \frac{dx^2 + dy^2 + dz^2}{z^2}$$

However, authors are inconsistent about the notation of $\mathbb{R}^3$ vs $\mathbb{E}^3$.

 

[D H]

You can't prove it. You can't even prove that the reals correspond to points on a line. That's an axiom, called either the Cantor Dedekind axiom or the Dedekind Cantor axiom.

There is an ambiguity in "E³". Is it the group E³ or the three dimensional space also known as R³?

 

[micromass]

Depending on the definitions, the result is either trivial or quite deep. I would define $\mathbb{R}^3$ as the set of 3-tuples of real numbers. This has an obvious geometric structure where you can define lines and planes.

I don't know how you define $E^3$, since it depends on the author. Now, one possible definition is that this is the geometry defined by the Hilbert axioms: http://www.gutenberg.org/ebooks/17384
In that case it is possible to prove $\mathbb{R}^3 = E^3$ but this is nontrivial. The book by Hilbert should contain the result. Otherwise, you should check Hartshorne's "Geometry: Euclid and Beyond", but this is only the 2 dimensional case. You need to do some (not so difficult) adjusting for the 1-dimensional case.

 

[Mark44]

You can't prove it. You can't even prove that the reals correspond to points on a line. That's an axiom, called either the Cantor Dedekind axiom or the Dedekind Cantor axiom.

There is an ambiguity in "E³". Is it the group E³ or the three dimensional space also known as R³?

I took the context of the question as meaning three-dimensional Euclidean space. I think I remember seeing E³ notation at one time to represent this space. I couldn't tell you where I saw it, as it was a long time ago.