Models and consistency

Hello,
I have some conflicts about terminlogies related to First order
Logic. The way I understand things, it seems that consistency is same
as having a model. So, if L is a model of ZF then ZF is consistent
isn't it?.

I am just checking on my undertstanding of the terminologies, that
are
often interchanged in books.

I am writing below my understanding of the terms. Could you please let
me know if I am going wrong in any of these below :


1) First of all the terms soundness and syntactic consistency mean
exactly the same always. Consistency and soundness have nothing to do
with the truthhood or falsehood of a statement.
By consistency we always mean syntactic consistency :
--- Both P and not P, will not be proved.


2) "Semantic consequence" means, if p is a sentence and S is a set of
sentences, then p is a semantic consequence of S, iff every model of
S
is also a model of p.


3) Completeness theorem says, Semantic consequence and syntactic
consistency mean the same.


4) Unsatisfiable - No model exists. Unsunstasfiable means the same as
being inconsistent.
If a model exists it is satsfiable. If no model exists it is
unsatisfiable. Also, satisfiability is semi-decidable. If a theory is
unsatsfiable (inconsistent) there will be a proof for it. If it is
satsfiable then there is no proof for it.


Are all of the above true?.


Questions :


a) Now, given this, if ZF were inconsistent, then there will be a
model for ZF which is also a model for some p and ~p. Is it possible
at all. Doesn't make sense to me.
So, even if there is one model for ZF, does that mean that ZF is
consistent?.

( I have read that consistency means the same as having a
model (Drake & Singh, p22). )


b) It looks to me like, Satisfiablity <==> consistent. Unsatisfiable
<==> Inconsistent.
Is it true?.


b) I heard a discussion where it was mentioned, if we come up with a
model for a theory then we can prove its consistent?. Is that true?.
There is something basic I am missing here.

If that is so, since we know that L is a model for ZF, Hence we know
ZF is consistent. So,
where am I going wrong?.
Thanks


.

Relevant Pages

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