Re: Tarski's metatheorem & VNSA-L digest 326

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Dear Aiviet, Vanvu and other academics, 

Thanks that a "beauty" for me is a beauty for other guys here.  
First of all I'd like to denote that Tarski is his real familly name and
(probably) in English Tarski is known as Tarsky.
 
I did not give you a definition of ordered field which is closed in the
"real sense" in my previous message, because short and
simple introduction to famous Tarsky's theorem can not contain 
all definitions.  Technical details usually cover a contents, so I 
tried avoid it in my communications (under name "small talks") with
friends of many diff. specializations here.
 
Aiviet denoted a clasifying theorem of ordered closed fields, but it's 
important for algebra and not for logics.  We don't want to know about the
above clasifying theorem for proving Tarsky's theorem. Ordered fields are
well-understood by you and problem is only what does mean exactly the term
"closed in the real sense". Let me now define that.
Ordered field is called closed in real sense if
Two following properties are hold:

1)For each polynomial f: if (a<b and a,b \in K) f(a).f(b)<0, then 
  there exists c\in K: a<c<b such that  f(c)=0;  (continous property)
2)For each polynomial f: if f(c)=f(d)=0, then there exists t\in K
  such that f'(t)=0.

(Here f is a polynomial with coeff. from field K.  We don't want to
provide here any topology, (of course, we can define a good topology
here, but it's unecessary!) and we do only algebraic operations.)


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PS. 1)Tarsky's theorem is not a consequence of Frobenius theorem.
      Two theorems are independent.  Aiviet's question about why 
      Tarsky's theorem is not valid for un-ordered fields is very
      interesting.  I understand technically why, because I have a proof.
      I putted myself two questions, you can see it in the next message.
    2)I've said only about "no direct applications" and I haven't said
      "no applications".  Tarsky's theorem, like other mathematical
      theorems,  is not constructive (it does not give us an algorithm 
      how to kill quantifiers).  But remind that for theoreticians
      problem of existence (very phylosophical problem ) is usually more
      important. 
    3)Proof of Tarsky's theorem is very hard and not clear.  It's very
      typical for proofs of greatest mathematical theorems.  If someone
      study a proof, he can easyly understand each line of a proof, but
      it's not understable for him why the proof is done by this way. 

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Open Problems:
In the next my message.

enjoys,
SN
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|  Sonnet Nguyen,  Polish Acad. of Scie.                |
|  Tel. (Office) (48-22) 43-70-01 ext. 1313             |
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