Re: Skolem paradoxical state of affairs

zuhair wrote:
On Sep 17, 12:42 pm, Nam Nguyen <namducngu...@xxxxxxx> wrote:
zuhair wrote:
And it is obvious that the two predicates "UNCOUNTABLE WITHIN M"
and "COUNTABLE OUTSIDE M" are not necessarily equivalent!
I see now what is the problem.
But still we have what I call a criterion problem, or a measure
problem, from what you said it is clear that the set of all injections
between omega and its power is not the set of all injections between
them (outside and inside the model).
This leads us to what I call the inadequacy of the set of all
injections within the model to be a sufficient criterion to determine
the cardinal relation between two infinite set if there is no
bijection between them within the model.
This is my take on the issue.

This arguably is the prime reason why some would find it difficult
to understand LST. "Within the model" is in the realm of truth-
expressibility of 1st order language, while "outside ... the model"
is the meta language counterpart. The 2 languages and their
expressibilities aren't the same, as you've alluded to above with
"...are not necessarily equivalent!".

For instance, let T be the formal system with the lone axiom:

T = {Ex(~(x=a))}

"Within a model" which L(T) can express - for the lone-axiom T - we could
count no more than 2 elements (i.e. a T's model would have a finite
cardinality of 2). "Outside" the model - which we could observe in
the *meta* level, a model of T could certainly have an infinite
cardinality. And nothing LST states would be contradictory.

Yes, I understand the argument, only that it is in the opposite
direction, the discrepancy between what is provable by T, and what is
actually their in T is not a big problem, that is not the matter with
LST.

Not sure what you tried to say here. The issue here is that problems
would arise when one tries mixes up between what a 1st order formula
expresses and what a meta statement says.


Let's take another formal system with the following axiom (in FOL with
identity).

E!x (x=x)

Now the Model of T contain only one x. now if you tell me that LST
states that there is infinity model of T, then this would be strange
if not a paradox.

Nothing strange. LST basically says: "*IF* .....".

.

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