Re: Calculus XOR Probability
- From: "MoeBlee" <jazzmobe@xxxxxxxxxxx>
- Date: 9 May 2006 11:33:37 -0700
Tony Orlow wrote:
To say
all elements are finite but the set is infinite implies that there is some
lartgest finite with an infinite successor.
In what theory? In set theory? No, what you said is not true of set
theory. So in some other theory? If in some other theory, then what are
its axioms and primitives and what are its definitions of 'infinite',
'largest', 'finite', and 'successor'?
So the (standard) naturals have a finite value range? What is it?
The value of the largest of them minus the smallest, but that's not a specific
number in the finite naturals, is it? However, since all values are finite, all
differences between values are finite, and there is no x-y that is infinite.
So, while the value range is unboundedly large, it cannot be infinite in this
sense.
In what theory? IF you're talking about set theory, then what you
concluded is incorrect. "The value range" must be defined as an object.
IF you define "the value range" as the set of all possible differences,
then that set is PROVEN to be infinite.
Huh? The naturals aren't generated by axions, they are described by the
axioms. Succesor describes a relationship between a given natural and
another.
What does "not very analytical" mean?
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.
There is NO "magical proof". And as to existence, "generated", etc.,
the context needs always to be clear: first order PA or set theory
(which also has models of consistent first order theories) or some
other theory.
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.
What's not disjoint? The empty set is one of them, and it is disjoint
from all sets whatsoever. Otherwise, yes, all other natural numbers are
not disjoint with one another since they all have the empty set as a
member.
So what's the first point after 0?
The next nonstandard first level point is Lil'un, or 1/Big'un. In T-riffics,
it's 0.000...001.
So now we're in your theory. Okay, what are the axioms and the
non-"magical" definition and proof of the existence of Lil'un?
Big'un is the standard length of the real line, but not the rgeatest number.
If Big'un is a whole number, Big'un^2 is too, and bigger than Big'un, being the
number of reals on the line.
Still in your theory now. As I recall, you had said that the existence
of Big'un is axiomatic. In set theory, the existence of the set of
natural numbers is axiomatic, but, as I understand, you object to that
as some kind of magic wand having. But then why wouldn't the axiomatic
existence of Big'un be magic wand waving?
Also, you say it is the "standard" length of the real line. Standard?
Where do you get such a thing in standard set theory? If standard set
theory has such a thing as the length of the real line, and your Big'un
is exactly that thing, then your Big'un is just part of standard set
theory.
So what point is BigUn at? The end of the line? The end of the
*unending line*? The end of the line which you can nevertheless carry
on past just fine?
It's at a standard infinite point which is considered the end of the infinite
unit interval.
"Standard"? A unit interval has infinte points in it, but what STANDARD
"infinite point" is at the end of a UNIT interval?
Show me where I said Big'un is the largest counting number. It's the number of
unit intervals on the line, but you can count more points than the whole
numbers. It's a unit infinity.
What "line"? Some line in your theory or the standard real numbers? And
your "unit intervals" are intervals of what? Intervals of your T-real
numbers between one of your T-naturals and its T-succesor? Of intervals
of your T-real numbers between two T-real numbers that are a distance
of one T-natural apart? Or what?
Right, that's the standard treatement. You look only at corresponding elements
without regard to how quickly each progresses along the line. What I am
suggesting is that if you look at the number of elements within a given value
range, that gives a measure of the density of the set on the line. Unless one
of the sets has greater range than the other, the set with the greater density
should be considered the larger set.
'Density' defined in what theory? Density is relative to an ordering.
You seem to overlook that a set can have more than one ordering, and
different orderings will make different subsets dense depending on the
ordering. Thus, cardinality based on density is not firm, since we
don't have a definition for an arbitrary set of its "standard ordering"
or "canonical ordering" or whatever you want to call it.
"In the infinite case" the "inductivley proven equality" between "the
number of terms" of sequence A and sequence B still holds!
Yes, in the infinite case of the set index, but not when set density is taken
into account.
In what theory? What sets? Sets don't just have density. ORDERINGS are
dense or not. And a set might have MANY different orderings.
MoeBlee
.
- References:
- Re: Calculus XOR Probability
- From: Mike Kelly
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
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