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

jan says




Let m=x, and r=x1. You will see the l�¶wenheim skolom theorem does not
yield a cardinality contradiction.


i say rubbish

suber disagrees
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.

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


again the contradiction is

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.




--
Message posted using http://www.talkaboutscience.com/group/sci.logic/
More information at http://www.talkaboutscience.com/faq.html

.

Quantcast