There is a general construction beginning with the natural numbers.
I prefer to go from the non-zero natural numbers to the positive rationals. Ordered pairs of naturals inherit the order relation of the naturals through the arithmetic of ratios. Hence,
these ordered pairs can be collected into equivalence classes and called positive rational numbers.
By a similar process of using ordered pairs, one goes from the positive rationals to the full set of rationals. In this case, ordered pairs are interpreted by signed differences. If an ordered pair has the same positive rational for both elements, it will be in the equivalence class designated as 0.
From the rationals, you perform the Dedekind cut construction. By the power set operation applied to omega in the set universe, you know that there is set in the universe of the proper cardinality. The set of all countable sequences of natural numbers is uncountable. If I recall correctly, it referred to as the Baire space. It relates to the order relations of the Dedekind cuts through the arithmetic of continued fractions. So, you have a set in the universe corresponding to your construction.
By the constructions thus far, you already have many of your needed operations. You must consider anything more you may need to satisfy the definitions of a vector space or an inner product space on your construction. This is all algebraic. And, this is also where you accommodate dimensionality.
To obtain a Euclidean manifold, you must use a Cartesian product of your construction and an individual copy of your construction. An affine space (vector space) or a Euclidean point space (inner product space) is obtained by a mapping from the Cartesian product into
the individual copy. The Cartesian product in this case is "just a set". The individual copy carries the vector space structure or the inner product structure.
As a disclaimer, let me say that serious mathematicians will see every mistake I have made in describing this. But, I think they would concede that this description has met the general sense of how such a construction ought to proceed.