Re: The shocking truth about the naturals

On Aug 29, 12:20 pm, "J. Burse" <janbu...@xxxxxxxxxxx> wrote:
Aha, you mean by the outside the set { x | x in y & ~A(x) },

Right.

but this is nothing else than the
set { x | x in y & A'(x) }, with the new formula being A'(x) = ~A(x).

Right.

And this set also exists according to the
axiom of separation.

Of course.
Oobvioulsy an axiom of "separation" must separate one thing
FROM ANOTHER, and therefore must produce TWO parts.
But the point is, if you make EITHER of these sets
THE DOMAIN OF THE MODEL into which you are interpreting
your theory, described in your meta-theory, then the OTHER set
contains
things that DO NOT EXIST *in* your model, OR IN YOUR THEORY EITHER!
Every thing in the set derived from A' or ~A is in a strange dual
state:
it both does AND DOESN'T exist. It DOES exist "outside", in the model
theory and the meta-mathematical domain, AND DOESN'T exist "inside"
the theory. This proves that the model's take on what exists is
essentially
fraudulent.

Not really II. Exists according to the theory,

But that's the whole point.
The theory doesn't always MATTER.
The theory is not ALWAYS entitled to an OPINION.
Sometimes, there is a FACT of the matter.
Uncountable sets either exist or they don't.
If the theory says the wrong way then the theory is WRONG.

means that such a set { x | x in y & A(x) } is found in each object
model of the theory.

You're hopelessly confused here.
That's an axiom. Sets like that exist all the time IN SET theory.
Whether they exist in other theories in other languages is more
complicated and depends on how the terms of the object theory
got INTERPRETED into these sets in the meta-theory.

The other exists is just an object of the given object model.

Of course -- this is MY point!

Such a distinction is simple,
but I don't see what it has substantially to do with
löwenheim skolem.

Whether a set is or isn't countable depends on the existence of
a bijection. If the bijection in question is in the half of the set
that
separated into A, then the theory knows the set is countable.
If the bijection is in the OUTSIDE half of the set related to A',
then theory doesn't know the set is countable, BUT IT STILL *IS*
ANYWAY,
because the bijection STILL EXISTS. The model+Theory's "combined"
opinion that the bijection doesn't exist IS SIMPLY WRONG.
The allegedly "uncountable" set IS countable.
This proves that the theory by itself simply can't answer this
question.

.

Relevant Pages

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