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I was going through these notes and had the following:

enter image description here

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?

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    $\begingroup$ 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? $\endgroup$ – spaceisdarkgreen Oct 17 '18 at 23:43
  • $\begingroup$ @spaceisdarkgreen are infinite vector spaces the ones that have infinite basis? $\endgroup$ – Pinocchio Oct 17 '18 at 23:46
  • $\begingroup$ @spaceisdarkgreen I guess I don't know why that true and what the second condition is really adding or doing... $\endgroup$ – Pinocchio Oct 17 '18 at 23:47
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    $\begingroup$ 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. $\endgroup$ – spaceisdarkgreen Oct 17 '18 at 23:50
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    $\begingroup$ 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. $\endgroup$ – Rob Arthan Oct 18 '18 at 18:29
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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.

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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:

enter image description here

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):

enter image description here

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!)

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