Re: JSH: Critique means slow, and thorough



jstevh@xxxxxxx wrote: 
Jesse F. Hughes wrote:

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. 


Well, it'd be nice if it did, but it doesn't.

Think of coprimeness as being like a filled balloon.

If you prick the balloon by introducing something that contradicts any
coprimeness result in the ring of integers, like sticking in 1/2 so
that 2 is no longer coprime to 1, then you pop the balloon.

Now you can *define* a minimal ring all you want, except that the ring
is infinite in size.  Because the ring is infinite in size, what you're
doing is like trying to look at only the foot of an elephant and
*define* that foot to be the entire elephant.

I don't think you understand what is meant by "minimal". A minimal ring is a ring that only has those elements that it must have to satisfy the ring axioms (including closure under its operations). So 1/3 would not have to be in Z[1/2] precisely because there are not *two* elements that add/multiply to give 1/3. 1/8 is there because 1/2 * 1/2 requires 1/4 to be in the ring, and 1/4 * 1/2 requires 1/8 to be in the ring.

Mathematically the *definition* has no meaning, and you get infinite
sums whether you want them or not.

Where in the above did I multiply more than 2 numbers at a time, much less infinitely many of them?

One way to look at it is that to the mathematics either you have
coprimeness rules or you don't.


It's a definition, not a rule.

If 1 and -1 are the only rationals units in the ring, like with
integers, plus my other requirement, then you have a ring where
coprimeness holds.

If you break the coprimeness rules even a little bit, you break them
completely.

You cannot break them halfway.

The idea that you can just stick in 1/2 and get a partial break is just
wrong.

It's a complete break. 

What does it break?

2 and 3 are coprime because 2*(-1) + 3*(1) = 1, adding 1/2 to the ring doesn't change that.

It even sounds bogus when you think about it that arbitrarily you can
remove convergent sums just because they're infinite sums.


Infinite sums aren't defined in a ring.

Hey, they converge, they're in there.

So, mathematically, despite what semantics you use, when you break the
integers, you get reals, whether you want to or not.

James, you will not successfully communicate with mathematicians if you refuse to understand what they mean by various terms. Some of your ideas may be interesting, but redefining or undefining part of the standard terminology is not terribly productive.

--
Will Twentyman
email: wtwentyman at copper dot net
.

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