Re: Calculus XOR Probability
- From: Virgil <Virgil@xxxxxxxxxxx>
- Date: Mon, 15 May 2006 15:01:16 -0600
In article <MPG.1ed26289d50e00ca98acd3@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
Mike Kelly said:
Tony Orlow wrote:<snips>
To say
all elements are finite but the set is infinite implies that there is
some
lartgest finite with an infinite successor.
No, it doesn't. It implies there are an infinite(UNBOUNDED) number of
finites, each with one successor.
Unboundedly large but finite number.
That oxymoron won't fly.
None is infinitely after any other. They
are all within a finite number of steps from one another.
That depends a bit on how one interprets "all". The set of all
differences is also unbounded, as is the set of all differences of
differences, and so on ad infinitum.
What is "a specific number"?
A point on the line, on the complex plane, or whatever quantitative space you
have defined for your system.
The only "quantitative space" possible at the start is that provided by
the set of naturals themselves, as nothing else has any a priori
existence in standard mathematics.
If TO wants to rely on having anything prior to the naturals, then he
must present us with his system including all its basic axioms and
definitions.
Absent TO's presentation of such a system, he is just talking balls.
Consider
...0001000.... and
...0010000....
We don't know where the digital point is on either one. It could be
infinitely
to the right or left, for all we care. If we know the digital point is in the
same place in both, then we can say without any reservations that the second
is
twice the value of the first.
Not at all. One must already have a system of numbers and of binary
representation of those numbers to say any such thing, but TO does not
have any such system and continues not to have any such system, so he is
still talking balls.
It means its not a mechanical explanation of how the set arises,
but a sort of magical *poof* it exists. If you want to gauge the
size of the set in some infinite case, then you look at how the
set grows.
You mean it's axiomatic?
I mean it uses axiomatic method as an excuse not to examine the
underlying logic any more than necessary. Axiomatic method has the
drawback of giving the illusion that a rule once declared applies in
absolutely every circumstance equally. It's very easy to miss hidden
assumptions in sets of axioms, and worth examining them in the event
of any untoward results.
The "assumptions" in the various sets of axioms used to create the set
theories that TO so objects to have been examined much more carefully
than TO can possibly imagine by those with much more talent for such
things than TO will ever have.
If TO were to have examined his own assumptions with anything like as
much care, he would never have made them.
Yes, so any set of the form {1, 2, ..., N } is not the set of all
naturals. I don't disagree.
And every n has such a set associated with it.
And the set of naturals is not such a set.
It is the union of all such sets, which are not disjoint, and none of
which
contain an element marking an infinite set.
So? It is still not such a set.
But they are still such elements with such sets of predecessors, none of
which
are infinite, and each a subset of the next.
So why does TO insist that such an endless sequence have an end? A
union, yes, as guaranteed by an axiom, but a last member to the
sequence, no! That union is NOT a member of the sequence of which it is
the union.
What is the "value range" of the naturals? I can think of only two
answers : undefined or infinite(UNBOUNDED).
I consider it unboundedly large but finite
TO plays the red queen again.
Then that's not a "number" in any sense that most people use the word.
I guess I'll remind you that "number" has no mathematical definition I
know of.
It does for me. A number is a symbolic representation of a quantity.
That's a natural language definition of "numeral", not a mathmatical
definition of "number". If you don't know what a mathematical
definition is, we have a problem. How do I determine conclusivley if
some object is or is not a number?
In your sense of number, or in mine?
In the mathematical sense, of course. TO hasn't any.
Well if BigUn is "the count of unit intervals on the real line" then
what else is it but the number of counting numbers? You seem to think
it should be obvious what your BigUn is. It's not. You're the only
person you've ever explained it to. And I'm not so sure about that.
I told you. It's the unit infinity. It's the number of reals in the unit
interval and the number of unit intervals on the standard infinite line. It's
a
primitive.
If it is the number of unit non-overlapping intervals in the real line,
then it is the number of naturals, which is countable and simultaneously
the number of reals in the unit interval which is uncountable.
TO seems to be having some trouble with his definitions.
What is "a standard infinite point"?
It's a point on the number line declared to be infinitely many finite units
in
distance from the origin.
Except that there are no such points on standard "the number line", and
TO has produced no system in which there are any.
You can count more than the counting numbers? With what?
More counting numbers.
When you have used up all the counting numbers, where do you get more,
TO? Do you have a secret factory making more? In area 51, perhaps?
I'm trying to prove we can do
better than what we have.
So far, with no success whatsoever.
.
- References:
- Re: Calculus XOR Probability
- From: Mike Kelly
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- From: Mike Kelly
- Re: Calculus XOR Probability
- From: Tony Orlow
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