Why ever stop creating quotient-algebras/algebraic-extensions etc. on $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset ....$

Question

Using inductive reasoning it appears that doing this (forming new algebras through extensions or quotients) helps mathematicians study algebras whose carrier sets are lower down in a chain of inclusions by studying the algebras with carrier sets higher up in said chain of inclusions. For instance here are five quick examples:

Evaluating complex integrals in $\mathbb{R}$ becomes easier using tools which require $\mathbb{C}$

Finding exact formula for roots of polynomials becomes easier using tools which require $\mathbb{C}$

Studying bounds regarding prime numbers in $\mathbb{N}$ becomes easier using tools which require $\mathbb{C}$

Studying Diophantine equations in $\mathbb{Z}$ becomes easier if working in an algebraic field extension of $\mathbb{Q}$

Studying quadratic/ternary forms in $\mathbb{N}$ is easier using field extensions of algebraic number fields.

So my question is given any algebra why not keep taking a variation of algebraic extensions and/or quotient algebras via equivalence classes of pairs, or sequences of elements to form a new algebra and then embed the older one in it. Why not keep doing that indefinitely? Why not define I don't know lets say maybe equivalence classes of sequences of complex numbers and give them a field structure by endowing each block with pointwise analogs of addition/multiplication etc. like one would were they going to construct the reals as cauchy sequences of real numbers. Why not do this again to get another field whose elements are equivilence classes of equivilence classes of sequences of complex numbers, why not another time so we have equivilence classes of equivilence classes of equivilence classes of sequences of complex numbers endowed with a field structure? Why not keep on doing it? Why should one stop at some arbitrary point? Since each previous algebra can be embedded into the one before it, you don't lose any of the tools if working within one of the lower carrier sets, as you would had the new algebra not been constructed. So it seems aside from complexity of notation one can only gain new tools to work with that might be beneficial for proving properties of the algebras lower down in the inclusion chain, thus I'm confused why this isn't done.

Answer 1

We don't stop. And we even do more complicated things than simply continuing to increase in one direction. Examples of algebras that are commonly used are:

• The ring of all continuous real-valued functions on the reals
• Finite fields and $p$-adic fields. (which are basically incompatible with the real numbers)
• Cardinal and ordinal numbers (which are also incompatible with real numbers) which have a useful arithmetic, but don't even satisfy familiar algebraic properties, like $0$ being the only solution to $x = x+1$, or even $a+b = b+a$.

The real numbers have the nice properties to make them very amenable to study and to be useful as a basic building block, and have historical importance due to their relationship to Euclidean geometry (e.g. the number line).

Answer 2

The complex numbers are complete as a metric space and algebraically closed. Since cauchy sequences of complex numbers already converge, nothing new is going to result (every sequence will be identified with one point in C already). Since it's algebraically closed, an algebraic closure is trivial and won't produce anything new.

To get something useful we'd need some new extension beyond these. Not only that but we'd like the extension to be a field and a complete metric space still, which is harder (for example quaternions are not a field since they're not commutative).