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I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory.

But can a structure with the properties of a (possibly uncountably infinite dimensional) vector space be derived from ZFC or is it axiomatic statement that a vector space is a set (and that therefore the axioms of set theory apply to it) ?

I think the question becomes relevant in the proof that every vector space has a basis (from Zorn's lemma = axiom of choice). It's not just a matter of whether one accepts the axiom of choice in set theory, but how one gets to believe that a vector space complies with set theory axioms.

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  • $\begingroup$ I am not sure I understand the question. In ZFC everything is a set. And you can derive the existence of structures satisfying the vector space axioms within ZFC. I have no idea if this answers your question or not. $\endgroup$ – Harald Hanche-Olsen Oct 20 '13 at 12:12
  • $\begingroup$ A vector space $V$ is an $L$ structure where the language $L$ consist of a symbol of binary function $+$ and unary function symbols, one for each scalar, this way you realize scalar multiplication as functions from $V$ to $V$. From this point of view it is a set (plus some structure) $\endgroup$ – leo Oct 20 '13 at 22:06
  • $\begingroup$ The surreal numbers are a field, but are not a set in ZFC. So you can consider them to be a $1$-dimensional vector space that is not a set according to the ZFC axioms. $\endgroup$ – Cheerful Parsnip Jun 27 '16 at 21:03
  • $\begingroup$ The surreal numbers are a Field, not a field. (Notation from On Numbers and Games). $\endgroup$ – GEdgar Jun 27 '16 at 21:17
  • $\begingroup$ @GEdgar: They are not a field in the usual (ZFC) sense, yes. That's why I brought them up. $\endgroup$ – Cheerful Parsnip Jun 27 '16 at 22:05
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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.

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  • $\begingroup$ I'm happy (just) up to the point of constructing real numbers and their properties as the complete ordered field, and can envisage the construction of a countably infinite dimensional vector space and its basis from a cartesian product of the reals. BUT, starting from the other end with an arbirary vector space (and later analysing dimensionality as an aspect of it) it seems one has to make an axiomatic assertion that the vector space is a set. Maybe I'm missing something. $\endgroup$ – Tom Collinge Oct 21 '13 at 7:03
  • $\begingroup$ Probably something which is not really taught. Given, say, a geometry problem, one "analyzes" by considering how it could be true. Using back and forth methods between "analysis" and "synthesis", one arrives at a "synthetic" proof from premises. Problems are first, our foundational theories are the synthetic outcomes of our analysis. What feels unsatisfactory here is precisely what leads to constructive mathematics. However, that is a different logical paradigm with its own regimentations. My own resolution comes from understanding en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma $\endgroup$ – fom Oct 21 '13 at 14:46
  • $\begingroup$ There is also this: math.stackexchange.com/questions/526335/… I will be adding a different answer soon, but that is irrelevant. I direct your attention to the fact that a consistent theory re-introduces topological structure. A separated proximity over terms in a countable language will be dense in another hypothetical space just as the rationals are dense in the reals. Your question is insightful and seems closely related to the indefinability of truth. Hopefully, you will find one of these replies helpful. $\endgroup$ – fom Oct 21 '13 at 15:04
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This is just a continuation from the definition of $\Bbb R$ as a set. The vector space $\Bbb R^n$, for example, is just the set of tuples with pointwise addition. And trivially, if $V$ is a real vector space whose basis is $B$, then $V$ can be thought of as the set of all functions from $B$ into $\Bbb R$ such that only finitely many elements are mapped to a non-zero element.

The axioms of set theory do not stop, or even begin to stop, at the real numbers or the complex numbers. They go well beyond your wildest dreams1. And then some.


Footnotes:

  1. At least if you dream about mathematics and set theory.
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One can define

Definition. A vector space over a field $F$ is an abelian group $V$ together with an action $F\times V\to V$ of the field on it.

Definition. An abelian group is a group where the compostion is abelian.

Definition. A group is a set with a binary operation satisfying ...

Ultimately, this makes a vector space essentially a set - that is: A vector space is a tuple of several things and structures, the most important one being the set of vectors of the space.

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Vector spaces are sets, Cartesian products of fields, and therefore by definition rely on the set axioms.

An other question is how well vector spaces model the real space that surround us all. In the real space there are no dots or lines. In real space there are only relative distances. The real space isn't isomorphic with $\mathbb R^n$ at all. And is no set either.

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  • $\begingroup$ Thanks for the response. I find it hard to agree with this approach as I generally see a vector space defined by its properties of being an Abelian group (of something) with homogeneous scalar multiplication. It's existence as (being isomorphic to) a (possibly infinite) Cartesian product of the underlying field emerges from this definition. $\endgroup$ – Tom Collinge Jun 28 '16 at 8:02

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