Re: Skolem paradoxical state of affairs

zuhair wrote:
On Sep 18, 2:18 am, Herman Jurjus <hjm...@xxxxxxxxx> wrote:

Actually, if you remove the axiom of infinity from ZFC and replace it
with its negation, then you can still concoct non-standard models that
contain a set like the one above.
What? you mean I state the axiom: for all x: x is finite, and then I
can find a set IN THE MODEL that is infinite, but this is a clear
contradiction? I am really confused here.
The standard trick is as follows.

Let T be the theory mentioned.
Step 1: extend (temporarily) the language of ZFC with one extra constant
name, say, 'F'. Consider the theory T' = T union {0 in F, 1 in F, ...}
If T is consistent, then so is T', by the compactness theorem.

(Sketchy: if T' is inconsistent, then there exists a proof of p & -p
from T'. Since in FOL, proofs always contain only finitely many steps,
and each step only uses finitely many assumptions, we can conclude that
p & -p can be proved from some finite subset of T'. In particular, the
proof of p & -p uses only finitely many assumptions from T' \ T.
But for any finite number of sentences of the form 'n in F', there's a
model of T + these finitely many sentences.)

By the completeness theorem, T' has a model.
This model is also a model of T, and remains a model of T when we remove
'F' again from our language; after all, F referred to some element in
the model, and this element remains in existence.
So we have now a model of T that contains an object that contains each
of 0, 1, 2, ...

On second glance , though I am not sure but I think there is something
wrong.

I was talking about ZFC minus infinity + For all x: x is finite
were finite is defined as having a bijection to a natural number.

In ZFC, the notion 'natural number' is defined as 'element of omega', where omega is defined as the smallest set satisfying certain conditions. But in the new theory, there is no such omega, so how would you define 'natural number'?

Now if I add the primitive F to the language of this theory and I have
the axioms
0 in F, 1 in F, 2 in F,.....

then the resulting theory is contradictive, i.e inconsistent, because
F would not have a bijection to a natural number.

How to define 'natural number'? How to define 'finite'?

No matter what definition you choose, as long as it's first order expressible, the above idea gives an F that satisfies the condition, but is infinite in reality ('as appearing from outside the model').

If you were speaking about ZFC and you added F to its language, and
then you removed it, it will make no difference because when I remove
axiom of infinity and replace it with the axiom asserting that every
set in the model must have a bijection to a natural number in the
model, then this F will be thrown out of M.

Nope; you could have non-standard natural numbers that (for the model) are indistinguishable from 'real' natural numbers.

--
Cheers,
Herman Jurjus
.

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