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Axiomatizable things

  1. May 28, 2010 #1
    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
     
  2. jcsd
  3. May 29, 2010 #2

    Landau

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    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.
     
  4. May 29, 2010 #3
    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?
     
  5. May 29, 2010 #4
    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 [itex]\mathbb R[/itex] 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).
     
    Last edited by a moderator: Apr 25, 2017
  6. May 30, 2010 #5
    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.
     
  7. May 30, 2010 #6
    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.
     
    Last edited: May 31, 2010
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