FOM: CH and 2nd-order validity

[John Mayberry]

On Fri, 13 Oct 2000 17:45:59 +0100 Thomas Forster 
<T.Forster at dpmms.cam.ac.uk> wrote:

> This mailing list is for foundations of mathematics, not theology!
> 
> >Is it known whether CH is independent of 2nd-order validity?
> 
Dear Thomas,
	The fact that all complete ordered fields are isomorphic is 
hardly theology. It is a perfectly straightforward mathematical 
theorem that ought to be taught to all mathematicians in their first 
course in real analysis. It is a trivial observation that every 
model of the second order theory of complete ordered fields is  . . . 
a complete ordered field. From these two facts it follows immediately 
that either CH is true in all models of the axioms or CH is false in 
all such models. Now let COF be the conjunction of the finitely many 
axioms of that theory. Then, on the natural assumption that there is 
a complete ordered field, CH is equivalent to " 'COF -> CH' is 
univerally valid in 2nd order logic".

John Mayberry
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John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at Bristol.ac.uk
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