Re: An uncountable countable set


Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
*** T. Winter wrote:

<snip>

As for "tying together", well if you take the axioms of a ring (even I
think the absolute minimal set of axioms: group under addition,
semigroup under multiplication, distributive law), you have (for any x,
y, in the ring):

x . 0 = x . ( y - y ) = x . y - x . y = 0

Note that we only rely on the *additive* properties of inverses, plus
the distributive law to get this. Therefore there cannot be a nonzero p
such that x . 0 = p, which is what we would need to have a
multiplicative inverse of zero.

That is all very true of absolute 0 and oo.

Huh? I didn't mention "oo", which is an undefined symbol in a ring in
general. I also didn't mention "absolute 0", and have _no_ idea what
such a thing would mean, unless it is simply zero.

.... Where you instead substitute
a measurable infinity for oo, and its inverse for 0, then that
infinitesimal value is greater than 0, and the equation doesn't hold.

I don't understand what you mean by "substitute". Look, I find it very
difficult to keep track of what you're trying to show at any particular
stage, since you seem to leap around tinkering with this and that,
hoping to make your intuitions come true. What are we doing now? You
have been told that a ring cannot have an inverse of zero unless it is
the trivial single-element ring. Several proofs of this have been
given, none more than about two lines long. Are you trying to argue
that this is "wrong" in some sense, or are you trying to claim that you
have something "better", in which zero does have an inverse (an
"O-ring" I suppose).

We could consider something much simpler than an infinite set here.
Suppose Z5 is the field of order 5. It contains five elements (I think
of it as the "five number circle"), we'll call 0, 1, 2, 3, 4, and here
are the addition and multiplication tables:

+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3

x 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1

Of course 0 has no inverse (that is, no nonzero element appears in the
multiplication row or column for 0). Suppose you see this as a problem;
please explain (very slowly, and you should be able to be explicit I
hope) how you would modify Z5 to make an O-ring in which 0 does have an
inverse.

In particular, if this starts by "substituting" something, you could
show the elements in the O-ring after the substitution.



e=(y+e)-y

What's this equation? What is 'e'?


Of course, you can append an object to a ring and call it Bigun (or
anything else) and investigate the resulting structure (see javascript
and my lens calculators for a practical example), but this structure
will not be a ring.

Big'un already exists in 2's complement as 1000... It's its own additive
inverse, and not 0.

Please explain how something "existing in 2's complement" affects the
elements of (for example) Z5. I've given up hoping you'll try to
explain how "Bigun" has a 1 at the left end of a string that extends
leftward without end.


Brian Chandler
http://imaginatorium.org

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