Re: Mathematicians are in deep *** for 2 reasons

rupert says

And I gave you a patient, detailed explanation of exactly what you have to
do to finish off the argument, to show that it actually *is* a
contradiction. I suggest you go back and read it.

I asked you to prove that the Skolem paradox is


i say

lets take subers account

http://www.earlham.edu/~peters/courses/logsys/low-skol.htm

This strange situation is not hypothetical. There are systems of set
theory (or number theory or predicate logic) that contain a theorem which
asserts in the intended interpretation that the cardinality of the real
numbers exceeds the cardinality of the naturals. That's good, because
it's true.

Such systems therefore say that the cardinality of the reals is
uncountable. So the cardinality of the reals must really be uncountable
in
all the models of the system, for a model is an interpretation in which
the theorems come out true (for that interpretation).

Now one would think that if theorems about uncountable cardinalities are
true for a model, then the domain of the model must have uncountably many
members.


HERE IS THE CONTRADICTION

But LST says this is not so. Even these systems, if they have models at
all, have at least one countable model.

HERE IS THE CONTRADICTION

So the cardinality of the reals must really be uncountable in all the
models of the system, but [ in contradiction ]these systems, if they have
models at all, have at least one countable model.



AND YOUR SKOLEM RELATIVISM

IS NOT ACCEPTED AS A SOLUTION OF THAT CONTRADICTION

AS IT GUTS SET THEORY

IE

This means that there simply are no sets whose cardinality is absolutely
uncountable. For many, this view guts set theory, arithmetic, and
analysis.

It is also clearly incompatible with mathematical Platonism which holds
that the real numbers exist, and are really uncountable, independently of
what can be proved about them.

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