# How does one show a set of sentences are models of infinite vectors spaces over F?

I was going through these notes and had the following:

where $$F$$ is a field and $$V$$ is a group. Note that $$\Sigma_{F}$$ is the set of sentences whose models are exactly the vector spaces over $$F$$ (I have no diea what this is though).

I was wondering why $$\Sigma^{\infty}_F$$ are exactly the infinite vector spaces over $$F$$. Why is that true? How do we prove this? What does that even mean? Does it mean that every L-structure that is a vector space (over the language $$L_F$$) satisfies:

$$\mathcal A \models \Sigma^{\infty}_F$$

is that true?

• It is saying a structure is a model of $\Sigma_F^\infty$ iff it is an infinite vector space over $F$. A structure satisfies $\Sigma_F$ iff it is a vector space over $F$, and satisfies the sentences $\{\exists x_1\ldots\mid n=2,3,4\ldots\}$ iff it is infinite. I can't tell what is confusing you... do you know what an infinite vector space is? – spaceisdarkgreen Oct 17 '18 at 23:43
• @spaceisdarkgreen are infinite vector spaces the ones that have infinite basis? – Pinocchio Oct 17 '18 at 23:46
• @spaceisdarkgreen I guess I don't know why that true and what the second condition is really adding or doing... – Pinocchio Oct 17 '18 at 23:47
• No, just an infinite number of elements. So $\mathbb R^2$ is an infinite vector space over $\mathbb R$ that is finite dimensional (and any nontrivial vector space over $\mathbb R$ is infinite). The second collection of sentences say "the structure has two or more elements," "the structure has three or more elements", and so on. – spaceisdarkgreen Oct 17 '18 at 23:50
• I doubt the link actually says exactly what you say it does about $\Sigma_F$. However the link is a document with over 100 pages and I don't see why we should have to read them all to answer your question. Please at least provide a page number. – Rob Arthan Oct 18 '18 at 18:29

The source you cite says that $$\Sigma_F$$ is intended to be some set of sentences whose models are exactly the vector spaces over $$F$$ and not the set of sentences whose models are exactly the vector spaces over $$F$$. I.e., it says that $$\Sigma_F$$ is some axiomatisation of vector spaces over $$F$$. $$\Sigma_F^{\infty}$$ adds to $$\Sigma_F$$ sentences $$\phi_n$$ asserting for each $$n \in \{2, 3, \ldots\}$$ that the model has at least $$n$$ elements. A model of these axioms is a vector space over $$F$$ because it satisfies $$\Sigma_F$$ and cannot be finite because it satisfies each $$\phi_n$$. Conversely an infinite vector space over $$F$$ will satisfy $$\Sigma_F$$ and each $$\phi_n$$. To find a suitable $$\Sigma_F$$ just take the universal closures of the equations defining a vector space given on that page in your citation.

The notation is really bad but:

$$\Sigma_{F}$$

is the set of sentences satisfies by vector spaces over field $$F$$ and NOT sentences satisfied by fields. Not these are different because fields satisfy the sentences the abelian sentences plus some more:

basically adding commutativity to * and multiplicative inverse to non-zero elements (which makes both ops abelian groups btw, especially when multiplication doesn't include zero).

In this case $$\Sigma_F$$ is the sentence satisfied by vector spaces, which are abelian operations (since we can only do addition reliably between vectors, not dot products are not always multiplication I assume so only addition) plus the nice operations with scalars (which are elements in the field F):

which it would mean just defining the above in sentence and L-language notatation.

Last but not least the infinity sigma just says that there exists 2 pairs of different vectors AND 3 different pairs...up to infinity. So infinite vector spaces (NOT infinite dimensional!)