Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:
A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted $\mathbb{R}$. The underlying set is the completion of the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the real line also known as the continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
Moved the Pavlovic–Vaughan reference to the reference section, cited it, gave link to the published version, and put proper TeX syntax on the additions.
But I’m not sure of this sentence:
\ldots the result is simply the stream of $\mathbb{N}$: $\mathbb{N}\times \mathbb{N}\times \mathbb{N}\times \ldots$, which obviously can be identified with the real number set.
Really? Is it so obvious that this infinite product can be so identified?
Looks more like Baire space to me.
I agree.
1 to 5 of 5