[FOM] Paleolithic Model Theory and Non-standard models.

[steve newberry]

[The following is a reply to Neil Tennant's crushing refutation of my 
inadequately formulated problem in a previous post. I present it here, 
somewhat corrected and extended beyond its earlier posting on FOM, in hopes 
that it may draw responses from others.]

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Of course you are right. I WAS thinking in terms of General Models-FOL 
semantics, and should have made that explicit. Clearly, I  *must* be more 
explicit, and herewith shall do my best in that regard.

[read '@' as the epsilon of membership, ' /@ ' as its negation, and  ' /= ' 
as "not equal".]

The logic is classical, simply-typed, of finite order, FOL-General Models 
syntax and semantics, and the ground domain is fixed, discrete and 
countable.  It either  IS omega, or its elements are indexed by omega. 
Given those preconditions, I invite you to consider the partition of all 
cwffs [of the above stipulated logic] into two negation-invariant blocks: U 
and  NK, where a cwff B  is said to be *absolute* iff it is either a 
contradiction or a tautology  [and hence in  U]; and  B  is said to be 
*contingent* iff it is not  absolute, and hence in NK, the contingent 
block. But NK splits very nicely in two, to give N + K, so we end up with 
three negation-invariant blocks:  U + N + K, [where by "negation-invariant" 
I mean just that if  B  @ a block, then ~B  also @ that block.] To summarize:

B @ U <=> B  is absolute, i.e., ~B is a contradiction and has no models of any
cardinality, and  B  is the negation of a contradiction, which, for lack of 
a better
word, we'll call a tautology. B  is  "u-valid" [universally valid].
          B @ N <=> ~B has an infinite, but no finite models, and B is 
valid on all and
          only finite domains; ~B  is asymptotically close to being a 
contradiction, and  B
          is asymptotically close to being a  tautology. B  is  "n-valid" 
[valid on all and
          only domains of cardinality n, where n @ omega.] N is not r.e., 
and is the
          reason for the "undecidability" of classical logic. Tarski said 
it better: Classical
          logic is  omega-incomplete.  [N is a very interesting class of 
propositions!]
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".]
I claim that this partition , negation invariant and defined by the 
cardinality of domains of realization, is complete, i.e., every cwff is 
accounted for, and falls into one and only one of the blocks. [I call this 
*view* of  the universe, and the theorems which follow from it, 
"Paleolithic Model Theory", or PMT.]

Are we okay on that? [GM-FOL semantics being now understood.] If not, shoot 
me down. If so, then where do you stand on the paradox?

The "Paradox":  [This is not really a paradox, but merely, to myself, a bit 
surprising.]

  Suppose we have a cwff ~B  which has an infinite, but no finite models. 
[It can be the  conjunction of the axioms for any theory that has no finite 
realizations.]

Then  B  is n-valid.  If  B  is  *n-valid* then B  *has models on all 
finite domains*. This class of models comprise what Keisler ["Fundamentals 
of Model Theory", in HANDBOOK OF MATHEMATICAL LOGIC, ed. Barwise] terms an 
'Elementary Chain' and by the associated 'Elementary Chain Theorem', the 
union of this chain is an elementary extension of  each structure in the 
chain. "It follows that for any theory T , the union of an elementary chain 
of models of  T  is a model of  T ." [Up to this point I would include the 
foregoing in  PMT. Note that no reference is being made to compactness 
here, merely the existence of unions.]

In my earlier post on FOM, I remarked that the union of models on an 
n-valid cwff "would have to be a model", and you responded:

NT Question: The union of those models would have to be an infinite model 
of *what*, exactly?

SN Answer: An infinite model of the n-valid cwff B  of which I was 
speaking.  I'm suggesting that an n-valid cwff is a  cwff which, *by merely 
existing*, in effect says, "the cwff of which I am the negation is 
satisfiable-but-not-valid on omega, hence by the  *existence* of the union 
of all my finite models, taken together with the fact that such a union is 
necessarily infinite, I  *must* have an infinite model, despite all my 
protestations to the contrary."

A reasonable reaction might be, "OK, sure, but why should I be interested 
?" To which my response would be, "Perhaps it is NOT interesting, but it 
brings up a question which is interesting to me, because I have only a very 
rudimentary [if that!] understanding of the concept of  "non-standard" 
models. In the scenario which I have outlined above, 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