[FOM] Correction/Addendum to "Paleolithic Model Theory by Steve Newberry
steve newberry
stevnewb at ix.netcom.com
Tue Mar 11 20:38:40 EST 2003
To: F.O.M. Subject: Correction/Addendum to "Paleolithic Model Theory"
by Steve Newberry
The statement, "B @ K <=> both B, ~B have finite models. The models of cwffs in
K are either finite and co-finite, or infinite and infinite, which I call
"co-infinite".] is misstated:
What I had *intended* to write was: "B @ K <=> both B, ~B have finite models.
The *number* of finite models of any given cwff pair B, ~B is either
finite/co-finite or infinite/infinite [which I call co-infinite]."
/* Read 'D/k' as "D subscripted by k " */
Example of the former: B =df= "There are less-than-or-equal to 137 discrete
elements in the universe". B has models [and is valid] on domains
D/k for 1 =< k =< 137.
The negation, ~B =df= "There are more than 137 discrete elements in
the universe" has models on domains
D/k for k > 137.
B has finitely many finite models, ~B has infinitely many finite models and
has an infinite model, by the elementary chain theorem. Finite/Co-finite.
[Here as elsewhere, B is *valid* on D/k <=> ~B has *no models* on D/k .]
Example of the latter case: B =df= "All integers are of odd parity". Then ~B
=df= "There exist integers of even parity". Both B, ~B have infinitely many
finite models. Co-infinite.
The differentiation between the finite/co-finite and the co-infinite
subclasses
of K induce a further partition of K into K' + K"; but I haven't anything
interesting to say about the two blocks beyond the fact that they are
recursively inseparable. Similarly, N = N' + N" + . . . under a partition
imposed by the complexity of the quantifier prefixes [arithmetical hierarchy]
but I have nothing interesting to say on that either.
In the scenario which I have outlined above [i.e., in the prior post], we find
that *every* n-valid cwff B of classical logic possesses an infinite model,
and by definition, ~B has an infinite model, and I believe that one of these
models is generally held to be "non-standard"; *which one* is the non-standard
model? And if it is the model of ~B which is the non-standard model, then
where does that leave the entire class of theories which have no finite models?
I really *do need* some clarification on this question.
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