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In free variable logic, it is clear what we mean by a follows
from S algebraically. So what does “a follows from S
formally” mean here?
For this general context, the most a.a.g. friendly way to go
is to avoid derivations and use another algebraic notion. 5
We say that a follows from T locally algebraically iff for
every algebra D and assignment t to the variables, if all
elements of T are true under t then a is true under t.
The so called substitution instances of a free variable
statement are obtained by replacing identical variables with
identical terms.
THEOREM 3. (Herbrand’s theorem). Let S be a set of free
variable statements and a be a free variable statement. Then
a follows from S algebraically iff a follows from a finite
set of substitution instances of elements of S locally
algebraically.
We can think of this finite set of substitution instances as
a “Herbrand proof” of a from S, and count the number of
occurrences of function symbols as a measure of its size.
COROLLARY 4. (Tarski compactness). Let S be a set of free
variable statements and a be a free variable statement. Then
a follows from S algebraically iff a follows from a finite
subset of S algebraically.
Let S be a set of free variable statements. There is an
important process of expanding S through the introduction of
new symbols that goes back to Hilbert with his Œcalculus. It
amounts to a relatively a.a.g. friendly treatment of
quantifier logic.
Let j(x1,...,xn+1) be any free variable statement that uses
only function symbols appearing in S.
We then introduce a new function symbol F which is nary, and
add the free variable statement
j(x1,...,xn+1) Æ j(x1,...,xn,F(x1,...,xn))
to S.
We can repeat this process indefinitely, eventually taking
care of all free variable statements in this way involving
any of the function symbols that eventually get introduced.
Any two ways of doing this are essentially equivalent. We
write the result as S*.
THEOREM 5. Let S be a set of free variable statements. Any
algebra in which S holds universally can be made into an 6
algebra in which S* holds universally without changing the
domain and functions of the original algebra.
COROLLARY 6. Let S be a set of free variable statements and a
be a free variable statement using only function symb...
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 Fall '08
 JOSHUA
 Math, Logic

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