Re: JSH: Critique means slow, and thorough

jstevh@xxxxxxx writes:

> Jesse F. Hughes wrote:
>
>> Here's my guess.
>>
>> Z is a subset of Q.
>>
>> Q contains 1/2.
>>
>> Z[1/2] is a superset of Q.
>>
>> Therefore, Z[1/2] is not the minimal ring containing 1/2 and a
>> superset of Z.
>>
>> If you try to add 1/2 to Z, you automatically overshoot Q. Now, if
>> you *start* with Q and "add" 1/2 to Q, probably you stay with Q. So
>> the process of adjoining elements to rings is non-monotonic (and a
>> touch unpredictable).
>>
>> Anyway, that's my guess.
>>
>
> The entire idea of the "field" of rationals is that you have numbers
> a/b where a is an integer and b is a non-zero integers, but you DO NOT
> ALLOW INFINITE SUMS.
>
> However, mathematically, *saying* you do not allow infinite sums
> does not stop them from entering anyway, when the ring is infinite
> in size.
>
> An infinite sized ring will allow convergent infinite sums.

James,

I was really just funning. Honest I was. I didn't really mean that
you thought adjoining 1/2 to Z gave you a bigger ring than Q.

Sorry for any confusion I caused there, but I really think you need a
different approach. It doesn't do your new self-critical stance any
good at all if you make such blatant logical errors that my
four-year-old chuckles.

Let's take it slowly, shall we?

(1) The notation Z[1/2] *means* the least ring R such that Z is a
subset of R and 1/2 is in R.

(2) Z is a subset of Q.

(3) 1/2 is an element of Q.

Therefore:
Z[1/2] *must* be (*by definition*) a subset of Q.

You may ask how we know that there *exists* a least ring containing
both Z and 1/2, but that's a different (and easy) matter. But the
fact is, if Z[1/2] denotes any ring at all, then it denotes a ring
contained in Q.

Honest, it does.

Your admiring archivist and humble servant,
Jesse
--
"I'm starting to absorb information [...] as I find myself more and
more fascinated by my own prime counting function. The rate of
information absorption is on an exponential scale, like it always has
been for me for things I'm interested in." -- James S. Harris
.

Relevant Pages

  • Re: JSH: Critique means slow, and thorough
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  • Re: JSH: Keep it simple
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  • Re: JSH: Critique means slow, and thorough
    ... and the algebraic integers is the ordinary integers Z. ... infinite sums, so what happens is by *saying* they are blocked, you step into a non-math area, which falls apart if you do anything that depends on your arbitrary choice. ... you need to take the step *explicitly* to define what it is you mean by a convergent series. ... you need to prove that any ring must be complete with respect to that definition of convergence. ...
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  • Re: JSH: Critique means slow, and thorough
    ... >> Adjoining such an element gives you the field of reals. ... >> something as simple as adjoining 1/2 to the ring of integers will ... as you cannot block the infinite sums. ...
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  • Re: JSH: Critique means slow, and thorough
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    (sci.math)