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what would be a good language for the theory of vector space?There are two different varieties of objects , scalars and vectors.

zahra
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  • I've always found that Mathematics is a language particularly suited for dealing with such objects ;) – MPW Oct 17 '14 at 19:00
  • but i could not undrstand any thing – zahra Oct 17 '14 at 19:00
  • @MPW If you want to treat vector spaces formally as models for a first-order theory, you need to specify the language, i.e. the set of symbols used to write down axioms. You're not going to be able to prove anything new about vector spaces by viewing them formally like this, but vector spaces are a useful example to have in mind when thinking about first-order logic. – Alex Kruckman Oct 17 '14 at 19:03
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    @zahra What is it, specifically, that you don't understand? – Alex Kruckman Oct 17 '14 at 19:04
  • could tell {0,1,+,.} – zahra Oct 17 '14 at 19:07
  • no.i want to specify the language of vector space. – zahra Oct 17 '14 at 19:09

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Usually, one talks about, for each field $K$, the "first-order theory of vector spaces over $K$: it's a single-sorted theory, where all objects are vectors. The language includes, for each element $a \in K$, a unary operator "multiplication by $a$".

But if you want the field to be part of the theory, the obvious approach should work. You just take the language of fields as applied to the sort of scalars and the language of a vector space over a field as applied to the sort of vectors, except rather than using the collection of unary scalar multiplication operators, you would would use a binary scalar multiplication operator (scalar $\times$ vector $\to$ vector).

Same thing for the axiomatization: you just take the usual axioms of fields and vector spaces.

(for the sake of clarity, note there should be two different addition operators: scalar addition and vector addition. Of course, we usually use the same symbol for both because it's easier to read and write and understand)

  • Or that you can declare that adding or multiplying things which makes no sense to have added or multiplied by one another gives $0$. – Asaf Karagila Oct 17 '14 at 19:47