Re: JSH: Critique means slow, and thorough



jstevh@xxxxxxx wrote: 
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.


It's not any such thing. The definition of the process of adjoining
an element to a ring is that the result is the *minimal* ring that
contains both the original ring and the new object. Since the definition
of *ring* is satisfied by only using finite sums, the restriction to
finite sums is justified by *minimality*.

Have you noticed that the field of rational numbers contains Z[1/2]?

Why haven't you been railing about the "flaw" in the rational numbers?
Pi is rational!


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.


Yes, and no one has contested that. However, it's really not the full
story. The infinite sums you want to admit don't have to yield the
reals. Moving from the rationals to the reals involves the notion of
completion, and that requires some version of metric. Just because
you wish to use the absolute value metric doesn't make that the only
one in town.

The fact is, the only unique construction is the one that's been given,
by restricting sums to being finite sums. Once you add an additional
piece of machinery in specifying a metric, then there are infinitely
many ways to proceed. Google p-adic numbers.

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.


You are clearly in no position to say *what can or cannot* be justified
mathematically.


James Harris


Dale
.

Relevant Pages

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  • Re: JSH: Critique means slow, and thorough
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  • Re: JSH: Critique means slow, and thorough
    ... integers gives you reals, as you end up with numbers that result from ... infinite sums, whether you want them or not. ... My point is that the "field of rationals" is in fact, not a field, as ... Actually, the act of adjoining any fraction like 1/2 or 1/3 to a ring, ...
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  • Re: JSH: Critique means slow, and thorough
    ... My point is that the "field of rationals" is in fact, not a field, as ... if i is in the ring, then you have the field of complex numbers. ... An infinite sized ring will allow convergent infinite sums. ... Blah blah blah. ...
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