Axiomatizable Things & First Order Logic Stuff

[Jerbearrrrrr]

First order logic stuff.
A lot of things don't seem to be axiomatizable. I have like a few remarks that need explaining if anyone could:

-There is no first order theory whose models are precisely the finite fields of characteristic 2.

-The theory of infinite fields of characteristic 2 has no finite axiomatization

-The class of finite groups cannot be axiomatized.

Seems to be related to completeness. Not sure D:

thanks

 

[Landau]

This follows from the Compactness theorem: given a theory T in some language L, if every finite subtheory of T is consistent, then so is T itself (where 'consistent' means 'has a model').

I could try to explain how it follows from this, but you can probably look it up somewhere without too much effort.

 

[Jerbearrrrrr]

I don't really know what I'm looking for so I'm not sure what to search :\

What kind of contradictions do the proofs of these things end up with?
I have a non-example of an attempt to axiomatize the fields of characteristic "2 or 3", but I can't decode it. I must have missed something or miscopied.

Do you know of any sites or material I could look for in the library?

 

[JSuarez]

The compacteness theorem is on pratically all logic textbooks, but if you want a quick look, see here (corollary 22):

http://plato.stanford.edu/entries/logic-classical/#5"

There are a lot of things that aren't axiomatizable in first-order logic: the real field $\mathbb R$ and second-order Peano arithmetic aren't; also for the concept of "finite" set or class (this is actually a direct application of compactness). And this is not restricted to Mathematics: the Kaplan-Geach sentence "Some critics admire only each other" cannot even be written in first-order logic.

In current mathematical practice, it's usual to work with well behaved fragments of second-order logic to circunvent these limitations (full second order logic is a mess).

 

[Jerbearrrrrr]

I think I've got it now (the aim is to construct, say, a group that satisfies conditions to be finite, but have it to have an infinite number of elements), but thanks. I'll definitely have a read to make sure.

I knew about compactness, but I just couldn't figure out why it made certain things not-axiomatizable. Cheers.

 

[SW VandeCarr]

I think I've got it now (the aim is to construct, say, a group that satisfies conditions to be finite, but have it to have an infinite number of elements)

I'm not sure if what you're saying is what the compactness corollary (22) says:

"A set G of formulas is satisfiable iff every finite subset of G is satisfiable." It says nothing about whether G is (denumerably)infinite or finite. If at least one finite subset of G is not satisfiable, the set G is not satisfiable.