Re: JSH: Critique means slow, and thorough

William Hughes wrote:
> jst...@xxxxxxx wrote:
> >>
> > Adjoining such an element gives you the field of reals.
> >
> > Now this issue has come up before, where I've noted that even doing
> > something as simple as adjoining 1/2 to the ring of integers will
> give
> > you reals.
> >
>
> And it was immediately noted that this is nonsense as
> rings are not closed under limits (you don't even
> need a definition of limit). Rings do not have
> infinite sums.
>
> - "William Hughes"

That is an unprovable assertion, as you cannot block the infinite sums.

And that's what I think breaks the theory of ideals as well, the false
assumption that you can simply proclaim that infinite sums are blocked,
when they cannot be blocked.

For those who wonder, if you append 1/2 to the ring of integers, and
allow infinite sums, then you get reals.

The generally accepted opinion has been that you can choose whether or
not you can have infinite sums, but I'm saying that mathematically
there is no way to block them in rings of infinite size, like if you
append 1/2 to the ring of integers.


James Harris

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