Re: Calculus XOR Probability
- From: Tony Orlow <aeo6@xxxxxxxxxxx>
- Date: Tue, 9 May 2006 11:35:34 -0400
Mike Kelly said:
Tony Orlow wrote:
Mike Kelly said:<snip>
Tony Orlow wrote:
What does "acheive infinity" mean? Sets are either infinite or they are
not. Are externally infinite sets infinite?
Both definitions of infinite set produce potentially infinite sets. If you
limit the number of iterations of the generating routine to only finite values,
you also limit the number of elements to only finite values. So, The interval
infinity within the unit interval is achieved when the intervals are shorter
than any real length, and the same infinite number of unit intervals as reals
in the unit interval covers the real line.
0 is a real length.
Sorry, positive real length.
There is no shortest positive real length.
And there is no largest finite, but if there are infinite numbers greater than
any finite positive real, then there are infinitesimal numbers less than any
finite positive real. They go hand in hand.
<snip>
What you call infinite is
literally "endless", but it includes "endless due to the hazy upper boundary of
finiteness".
Hazy upper bound? No, there is *no* upper bound.
But, there is some distinction made between finite values, and any infinite
values that may lie outside the range of finites. That "boundary" between them
is what's hazy, and declaring some bogus "smallest infinity" and calling it
omega doesn't change that. It just sweeps it under the rug and leaves a lump.
But the "smallest infinity" of standard theories isn't an "infintie
value" like you've been talking about. It describes an equivalence
class.
Yes, I know, but it's called the size of the set, and yet, doesn't satisfy very
many intuitions about what a set size is. You can add an infinite number of
additional elements to an infinite set without changing the cardinality. I find
that objectionable. And, it's referred to as the smallest infinite ordinal
"number", as if it's some kind of quantity, but it's really not. So, with all
due respect to von Neumann, I wouldn't give his ordinals such a central place
in the theory.
It's not called the size of the set in set theory. Set theory is
written in symbols and refers to "bijectability", not "size". People
call it the "size" when discussing it in natural language because it
satisfies more intuitions about what a set "size" is for infinite sets
than anything else anyone (including you!) has come up with.
That's your opinion. It is commonly referred to as the size of the set, and
commonly receives objection because it DOESN'T satisfy a lot of intuitions
about what size means.
From an internal perspective, with all element values and
positions within the set finite, there are no two elements with an infinite
number of elements between them. So, despite the boundlessness which makes
injection into a proper subset possible, I can't consider this set to be
actually infinite in size.
What's the difference between "not finite" and "actually infinite"?
The way I see it, a dimensionless quantity can be finite, greater than finite
and infinite, or less than finite and infinitesimal. I see "countably
infinite" as unboundedly large but finite.
The english-language definition of "finite" is something like "bounded
or limited". The idea that the naturals are bounded but unbounded is
probably causing some of my confusion here.
Well, that's understandable. "Infinite" means literally without end, and the
standard set theoretical deifninition classifies the set of finite naturals as
"countably infinite". My issue with that is that, when no two elements in the
ordered set can ever have any infinite number of other elements between them,
then I don't see as there is an infinite number of elements in the set. There's
no infinite _number_ of elements.
So you don't think "countably infinite" means "really actually
infinite". I'm not sure that matters. Mathematical theories don't use
natural language like "countably infinite is really actually infinite".
Okay, that's your opinion.
Are you saying that sets cannot have properties their elements do not?
So you cannot have an infinite set of finite values? I think that's
faulty logic.
Question : The rationals in [0,1] are countably infinite. Yet there are
only internally infinite, if I understand you right?
Okay, you're new to this discussion. When I see this question I often get
rather irked, because it's usually from someone I have explained this to half a
dozen times. So, I'll try to patiently explain my position without getting
annoyed. :)
We're talking about whether the set of finite naturals is infinite, right? From
the set theoretical definition we would say yes, since it is possible to form
an injection from the set into a proper subset, like the evens. However, from
the perspective of the set having an actually infinite quantity of elements
within it, I would say it does not qualify. COnsider this inductive argument.
As a base case of a set of consecutive naturals with a given size, consider the
set {1}. It contains 1 element, 1, so it has size 1, which is also its maximal
element. As we add successive elements, with each one we increment the set size
by adding an element, and simultaneously increment the maximal element by
adding the incremented value of the previous maximal element. So, with each
additional element incrementing both the set size and the maximal value, both
properties of the set remain forever equal. This means that, if one's
definition of "infinite" is consistently applied to both of these values, one
cannot become infinite unless the other does. So, the set size cannot be
infinite without an infinite maximal element. There's really no way to avoid
this logic except to bury one's head in the sand.
Once again, your argument applies to sets of the form {1, 2, 3, ..., n
} and the naturals are */not/* a set of that form. Can I repeat that
again? The naturals are ***NOT*** of the form {1, 2, 3, ..., n }. There
is no largest natural. The sequence of naturals *is* endless.
So what? There is no element in the set at which point the set has any infinite
measure.
If you really disagree with this I have to assume you mean something
else by "the naturals" and even "the standard naturals" than everyone
else does..
I never said there was a largest finite. I said that no finite in the set
allows for an infinite set, and that the size, if there is any specific one, is
equal to the largest element, and vice versa.
The standard alternative, the von Neumann ordinals, rely on a model starting
with 0 as the empty set, and defines each "natural" as the set of all its
predecessors. It then defines the first limit ordinal, omega, as the set of all
finite naturals, and declares it infinite, since it's larger than all finites.
The problem with this is that, in every finite case, the maximal element is
always one less than the set size, and there is nothing that changes that in
the infinite case. Essentially, omega is the successor to the largest finite, a
bogus concept which is only covered up using ordinal "arithmetic" where omega
has no predecessor. It's much flimsier logic than a direct equality proven
inductively in such a way that there is no possible way the two values could
ever be different.
No, omega is not the successor to any ordinal. I think you know this.
So why say it? It's the union of all finite ordinals.
Because in all finite cases the value of the set is the succesor to the largest
value in it, since each natural is the set of all predecessors. If you want to
apply this to the infinite case, then why should that change? If you start at
1, it's obvious the set size can never be larger than the maximal value, and
adding 0 as an element only increases the set size by one. So, starting at 0,
the set size can never be more than 1 larger than the maximal element. To say
all elements are finite but the set is infinite implies that there is some
lartgest finite with an infinite successor.
So, it's nothing so simple as "infinite sets all have infinite values". The
reals, or rationals, in [0,1] are a trivial counterexample. This has
specifically to do with the naturals, and generally to do with sets that are
everywhere sparse. Such sets require infinite value range to contain an
infinite number of elements.
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.
I hope that cleared up the mystery of my position.
It highlighted a couple of areas where you just plain don't seem to
understand the standard theory. It didn't explain your position much,
it attacked the standard definitions.
Haha. I understand the standard theory. I disagree with some core concepts. If
it makes you feel better to say I just don't understand, go ahead, but it's not
the case.
But I don't think there is a countable sequence of infinite elements
that starts at, say 0 and steps through an infinite number of reals in
[0,1] and then ends with 1. I think sequences either have a start and
an end in which case they have a finite length or they lack an end and
then are infinite sequences.
That's the standard treatment. It appears that the concept of an uncountable
sequence with beginning and end is foreign to most people, but if you define
infinitesimal units, you can certainly conceive of such a sequence in [0,1].
When I look at a bit string of all 0's with a 1 in the x position, I don't care
whether x is finite or infinite. That number is 2^x. I don't need to be able to
count up to that bit from the 0 bit to calculate that. So, that also
constitutes a completed infinity with distinct ends.
No, I really can't conceive of how this makes any sense to you. What
does "2^x" mean to you when x is not a real number? How do you evaluate
it?
First of all I evaluate it compared to other values finitely close in
representation, such as 2(x+1)=2*2^x. When it comes to comparing it with values
indicated by bits infinitely far away, one must choose some unit infinity, and
express that relationship formulaically using that value as a variable.
Yes, it is impossible for a set of naturals of the form { 1, 2, ... , N
} to have a non-finite set size. The naturals are not a set of that
form so I don't see you've said anything that applies to the set of all
naturals.
The fact is true for all n in N, that the set up to and including any one of
them is finite, so there is no point where the addition of any elements
produces a set of infinite size.
True, but so what? The naturals are not "produced" by adding elements.
Successor() doesn't generate new elements?
No. It does't allocate a block of memory and then return a pointer to
that memory. All the pointers are already defined, before you look at
them.
Function Successor(x)
{ Successor(x+1)
}
Are you saying this recursive function will generate the naturals all
immediately, or immediately run out of memory? I've heard that before. "All the
elements spring from the axioms all at once". That's not very analytical.
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.
The identity relation between element count and value characterizes the
naturals.
I'm not sure I know what that means. I thought the Peano axioms
characterised the naturals? "Identity relation between element count
and value" characterises sets of naturals of the form { 1, 2, ..., N }.
The naturals are not such a set.
Sure they are. You can start anywhere, and 1 is the natural place to start,
especially since the element value is identical with its position in the set.
It's as valid as starting at 0, and demonstrates clearly how the infinitude of
the set size depends on the element values.
No, the naturals are *not* a set of the form { 1, 2, ..., N }. The
naturals do *not* have a largest member.
Eveyr element is of the form "n", and no matter which of those n's you look at,
there is no infinite set up to that point.
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.
This in sharp contrast with the real
interval. No matter what distinct point you choose from any other point, there
is an infinite number of points between them.
So the natural numbers don't form a continuum, agreed.
Right, they're everywhere sparse, and if they contain an infinite number of
elements, it's because they have infinite value range, not because there's any
point of condensation on the line.
The finite naturals are like all
the points a finite number of points from 0, compared to the infinite real line
as the interval [0,1]. Any finite number of infinitesimal points can only
create an infinitesimal distance. An infinite number of such infinitesimals
produces a finite distance. Likewise, an infinite number of finites produces an
infinite distance, meaning that not all points can be in finite locations.
Seeing as the reals form a continuum I'm not sure what "a finite
*number* of points from 0 means". What's the first point after 0? If
it's "x" isn't "x/2" closer to 0? Shouldn't x/2 have been the next
point, after all? What about x/4?
That concept depends on the concept of the point as infinitesimal segment on
the line, as I said. Any infinitesimal difference, multiplied by any finite
number of times, is still infinitesimal.
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.
It does not follow that there is some number you can call the size of
the set of reals in [0,1) or the size of the set of naturals unless you
rigorously define some way of assigning "sizes" to infinite sets and
decide to call these sizes "numbers".
That's the plan.
Cardinality is one system for doing this. The "numbers" used in
cardinality aren't anything much like the real numbers or the natural
numbers. You can't do any of the standard arithmetic with them. They
don't feel like "numbers" to me, but they are rigorously defined and
can be looked at mathematically.
Yes, but they're really not numbers, and you can't really do any math with
them.
Well, you can do math with them. Just not all of the arithmetic you'd
like to.
Well, I don't like that kind of thing. :)
Not everything in maths is arithmetic.
No, but if you can't draw any usable conclusions, then what good is it?
You can draw useable conclusions from cardinality. Maybe you don't know
about them, your loss.
I haven't seen any of the utility drawn from transfinite notions of
cardinality.
Then I can only reccomend you take an introductory analysis course.
But you're not working in the Hyperreals. What's *your* system?
I told you. I have Big'un unit intervals on the real line, and Big'un
infinitesimal reals per unit interval, each occupying Lil'un units on the real
line. Not very axiomatically stated, but clear nonetheless. The T-riffic
numbers enable us to represent such numbers by defining digital points in
addition to the one at the 0 bit, which may be at infinite bit positions
defined as a formula on Big'un, each with a countable neighborhood of bits, and
each countable neighborhood connected to the next and last by finitely
repeating bit strings of uncountable length, which may be interpreted as
rational fractions of a given infinity.
I think you're going to have to work on this. Even loosely stated as it
is I just don't understand it. It certainly isn't clear.
Just a random point of confusion : what is a rational fraction of an
infinity?
A finite rational times the infinity. 2/3 Big'un, or 0:666...666.666...666 in
decimal, is a fine example, or 1/7 Big'un, which is
0:142857...142857.142857...142857. Any finite rational fraction will produce
the kind of repeating pattern that is required for specifying the possibly
uncountably long strings between the finite neighborhoods of the limit points.
See, this still makes zero sense. Going to give up on this bit for
now..
It doesn't make sense that rational fractions are the same as repeating digital
fractions? What does make sense to you?
http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.htmlWell, then, I guess we have different intuitions. Picture the unit interval
[0,1] as a little shrunk-down real line from 0 to oo.
"to" oo? The real line doesn't have endpoints, surely? But [0,1] surely
does.
http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html
-oo and oo aren't endpoints to the real line in either of those.
Affinely Extended - "Note that these improper elements oo and -oo are
not real numbers, and that this system of extended real numbers is not
a field."
How can something which isn't a real number be on the real line?
Uh, well, they're included, aren't they, on the affinely extended number line?
No. They aren't real numbers.
They're included on the line. So, you explain to em how something that can;t be
on the line is there. Oy.
But they aren't included "on the real number line". They aren't real
numbers.
They are included on the Affinely Extende real number line idividually, and on
the Projectively Extended Reals as a mutual point.
If x is the size of a set of cosecutive naturals starting at 1, then it's the
maximal element. Aleph_0 is not the maximal element, therefore it's not the
size of the set. It's the "cardinality". That's something different.
No, if x is the size of a set of consecutive naturals starting at 1 and
finishing at x then it's the maximal element. Aleph_0 is not the
maximal element, it's something called "cardinality" which is a way to
decribe the size of an infinite set in terms of bijections /
equivelence classes.
By your logic : If x is the size of a set of cosecutive naturals
starting at 1, then it's the maximal element. BigUn is not the maximal
element, as BigUn+1 is also a natural, therefore it's not the size of
the set. Hmm...
No, there is no reason why one cannot have a count higher than the set of
intervals on the line. If the number of T-nats is Big'un, the number of
multiples of 1/2 in that set is Big'un*2. There are an infinite number of
infinite values between Big'un and the infinite dimensional continuum, and an
infinite number smaller than Big'un. It's the length of the real line, but not
the largest possible infinity.
But the "set of intervals on the line" is the set of all possible
counts?
Right. You can count point within each interval, as well as the intervals
themselves. Then you can take power sets, etc, and create ever larger
infinities.
<snip>
Aleph_0 sucks because it offends your intuition by not being subject to
arithmetic? OK...
Yeah, it's not subject to consistent logic of any sort. I was very impressed by
yet another novice coming up with the hard questions.
What consistent logic is it not subject to? It's just a label
describing the "size" of the set of naturals. It's not a number in the
sense of a natural number or a real number so it doesn't have their
sort of arithmetic. I wasn't expecting it to.
Then you weren't expecting to be able to actually conclude anything about
anything. Well, that's one way to avoid disappointment.
You can't do arithemtic with everything in mathematics.
No, only with numbers that make sense can you calculate anything that makes
sense.
So you cannot perform *any* calculations of *any* kind without numbers?
Can you?
Sure. Check out Group Theory.
Think of the number of points in a finite line segment, the reals in [0,1].
It's a larger number than any finite number, because any finite number leaves
points or reals unincluded. That's why it's defined as larger than any finite.
Or there's no "number" that says how many points there are?
There are points. You can try to count them. You will go through all sorts of
numbers, but the number of points to count is more than you can ever count.
There is SOME number of points there. It's an infinite number of points. We
declare the number of points in 1 unit of space to be Big'un.
"There is SOME number of points in there". This is an interesting
assertion, but not one I agree with. You can declare the number that
denotes the size of an uncountably infinite number of points as
"BigUn". I'm then entitled to ask, what about the BigUn+1th element? I
think there is no "number" that satisfies the idea of "you can count
all the points and end with this number".
Add pi the set of T-nats. DO you have Big'un+1 elements now? I do.
No. What if you start the counting with pi then go through all the
naturals?
Same. The set of naturals, plus pi.
Just so I'm clear - I don't get what an "infinite n" is. By your
description of "infinite n" I think there aren't any "infinite n" at
all.
So we're back to finite numbers of points in the line segment? (sigh)
No, we're back to no "number" of points in the line segment. There's
not finitely many points. So I say in-finitely many points. Infinite
precisely because there is no "number" of points.
No finite number.
Why should there be a number, at all?
Because there are points, and you can count their number. It's just that you'll
never finish counting, so the number is infinite.
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.
At least I'm not just in hand waving denial.
Oh, that made me laugh. You've been doing nothing *but* handwaving.
And you've been questioning the very idea that a set has a size. What do you
call that?
Pretty reasonable, without a definition of size? The standard theories
have notions of bijectability which match up exactly with the notion of
"size" for finite sets and work pretty well for infinite sets. I can
agree that any set has a size using the definition of bijectability.
Your definition which is something like "count the elements in the set
and the last number you get to is the set size" only gives a size to
finite sets. To declare "oh, but you just keep counting until you get
to the end of the infinite sequence, and then you've counted to an
infinite number, obviously"... that, that is handwaving, Tony.
If you say so, but some of the simpler ideas involved seem to elude you.
Segments resulting from subdivision an infinite number of times until their
endpoints are no longer distinguishable from each other. They no longer have
standard real length and can each be considered to contain one standard real.
You can subdivide reals for as long as you like and you never get an
infinitesimal distance. Just like you can follow the Successor() trail
as long as you like and never find an infinite natural.
That's true for any finite number of iterations.
And an "infinite number" of iterations will either have a limit of a
real number or no limit at all.
It will have a limit of 0, obviously, but is not truly zero except for absolute
oo. Given a specific infinity of subdivisions, one can put a specific
infinitesimal number on the lengths of the intervals.
Firstly the phrase "The size is amorphous until declared to be
specific" confuses me. Are set sizes not fixed in your theory? Does the
set of naturals have many sizes?
The boundary between finite and infinite cannot be pinned down in any
substantial sense, so any set defined using finiteness as any criterion suffer
from this issue.
That didn't answer my question. Are set sizes fixed in your theory or
not?
Sizes of infinite sets are comparable and orderable over a common value range.
I don't know what you mean by "fixed". It depends on the value range you're
discussing. When you define an infinite set with a recursive definition, and
then only perform a finite number of iterations, it's not an infinite set, but
a finite recursive set. In general, one will compare two infinite sets over the
range [0,Big'un] to get a standard measure of the set in terms of a formula on
infinite units.
So your T-naturals *don't* have a size?
Yes, over the real line there are Big'un T-nats, since that is the length of
the line in unit intervals, and there's one natural per unit. You remember the
identity relationship between element count and value in the naturals?
So BigUn is the maximal element of the naturals? What about BigUn+1. Is
BigUn+1 beyond the scope of the real line, somehow?
Big'un is the length of the line and the unit infinity. On the line, one may
have as many as Big'un^2 standard reals. Beyond that, one has higher
dimensional spaces with powers of that value. No one said there weren't numbers
greater than Big'un.
So, again "BigUn" is the number of natural numbers and yet you can have
a natural number greater than "BigUn"?
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.
What is a "value range".
The maximal possible difference between any pair in the set. If there is no
maximal difference, but all differences are finite, then the range is
unboundedly large, but finite.
So BigUn is the distance between 0 and BigUn and that's the size of the
real line, BigUn? Firstly, do you not find that circular? Secondy, what
about BigUn+1? And, again, "unboundedly large but finite" appears to me
an oxymoron.
No. Discussed. I know.
What is "an infinite set with a recursive definition"?
A set defined such that the existence of each element implies the existence of
at least one other unique element, with no restriction of finiteness on the
number of such implications inferred.
Does "infinite number of iterations" mean anything?
Yes, it means more than any finite number of iterations. What happens if you do
it forever.
What is "the range [0,Big'un]"?
The positive real number line.
What is the range of [0,BigUn+1]?
Big'un+1, greater than the length of the line in units.
And what is the length of the line? BigUn! So BigUn is the end of the
line? But you can carry on past it? But I thought it was the length of
the line and the end of the line? Well yes, but you can just keep on
going anyway! So in what sense is it either a length or an endpoint?
Well, don't you see?? It's DECLARED to be so!
Sigh.
It's a UNIT infinity. We count Big'un reals in [0,1). Does that mean 1 doesn't
exist, or can we have [00,1] with Big'un+1 elements? We can. I never said one
cannot count beyond Big'un. That may seem contradictory to you, but that's just
from years of dealing with aleph_0. It's not aleph_0.
What is "Big'un" for that matter? The length of the real
line? What does "length" mean, then? What are "infinite units"?
Big'un is the number of reals per unit interval, and the length of the real
line as a count of unit intervals and their associated T-naturals. This
infinite number is used as a unit on the infinite scale, so that we can say
Big'un+Big'un=2*Big'un, just like with finite units.
What does "length" mean to you?
It means distance between two points, measured in some unit intverals end to
end.
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.
undefined1,2,3....Big'un-3, Big'un-2, Big'un-1, Big'un, Big'un+1, ...
What on Earth is this? "BigUn" is the length of the real line, but you
can have "BigUn+1", too? What does any of this even mean? Is this
supposed to be a number system you can do arithmetic with?
Yes, absolutely, and you can certainly have Big'un+1. Big'un is the size of the
set of T-nats, and then, say, wee throw in pi as an additional member. Now we
have Big'un+1 elements in the set.
What are the sizes of the following sets?
{0, 1, 2, ...}
{0, 1, 2, ... BigUn}Big'un+1
{0, 1, 2, ... BigUn+1}Big'un+2
{1, 2, 3, ...}undefined
{1, 2, 3,... BigUn}Big'un
{1, 2, 3,... BigUn+1}Big'un+1
{Pi, 0, 1, 2, ...}undefined
{Pi, 0, 1, 2, ... BigUn}Big'un+2
{Pi, 0, 1, 2, ... BigUn+1}Big'un+3
{Pi, 1, 2, ...}undefined
{Pi, 1, 2, ... BigUn}Big'un+1
{Pi, 1, 2, ... BigUn+1}Big'un+2
So the Tnaturals have a different size depending on whether you start
from 1 or 0? So you can't define any kind of size for sets without a
maximal element? So the largest natural BigUn nevertheless has a
natural larger than it? How quaint.
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.
You honestly think there are "more" element in the sequence
pi, 0, 1, 2, ....
Than
0, 1, 2, 3, ....
Without a maximal element ther eis no value range over which to compare the
sets, but given the same value range, yes, the first has one more element.
These are sequences not sets. What does "value range" mean here?
it means that the set conaitns one more element between any two given values
than the other, as long as the value range includes pi.
What aboutOver the same value range, the first has half as many elements.
2, 4, 6, 8, 10, 12, ...
1, 2, 3, 4, 5, 6, ...
What about over the same range of the sequence? First 10 terms? 10
terms each. First 100 terms? 100 terms each. First n terms? n terms
each. First n+1 terms? n+1 terms each! First BigUn terms? BigUn terms
each! First BigUn+1 terms? BigUn+1 terms each!
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.
"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.
Infinitesimals can
be considered discrete, or continuity can be extended to the infinitesimal
level, leading to sub-infinitesimals. Discreteness on the infinitesimal level
does not violate the Archimedean principle on the finite level, since two
consecutive infinitesimals are not distinguishable standard reals, and
therefore do not require an intermediate value.
The real line is a continum of real numbers. There is no "finite level"
and a smaller "infinitesimal level". The reals go allllll the way down,
as far as you like.
But, in standard analysis, if there is no finite difference between two reals,
they are considered equal. If you can't conceive of infinitesimal differences,
then all can suggest is you consider that when you touch something, you don't
really touch it. Archimedes would agree with me. :)
No, I can't concieve of infinitesimal distances. When "you" touch
something forces at the subatomic level are bouncing electrons around.
Where's the infinitesimal distance? You think infinitesimal means "real
small"?
Smaller than can possibly be measured in any finite units. A digital number
with a most significant bit in a negatively infinite position in the string.
Things in real life can be so small that we can't measure them with our
technology. The real numbers have no such restrictions. They can
measure things arbitarily small.
Like the difference between 1 and 0.999...? There is a valid concept there. Not
everyone likes infinitesimals. What did Cantor call them, "bacilli cholera"?
0.999... is a notation for the limit of the summation of (9/10^n). That
limit is 1. There is no difference between 0.999.... and 1.
Not at all. It leads to very intuitive and consistent conclusions. It all
depends on what you consider "size". Cardinality doesn't fit the bill when it
comes to infinite sets.
The idea that the sequences
0, 1, 2, 3, ...
1, 2, 3, 4, ...
2, 3, 6, 8, ...
have diferent "lengths" is neither intuitive not consistent.
Even though the first is a proper superset of the others? Hmmm....
These are sequnces not sets, Tony! 0, 1, 2, 3 is not a supersequence(?)
of 2,4,6,8.
Look tony, can't you see? The inductively proven equality between terms
of sequences holds even in the infinite case. You keep on counting and
never run out of numbers for any of the sequences, even when you reach
the BigUnth term. Then the first sequence is at BigUn-1 the second is
at BigUn and the third has gotten to 2*BigUn. And you can keep on
going!
Yes, *I understand the standard half-baked analysis.
"Number" doesn't have a mathematical definition that I know of. You can
define natural_numbers or real_numbers or rational_numbers or
complex_numbers or hyperreal_numbers but "number" floating around on
its own doesn't have a definition. "An infinite unit which can be used
in any arithmetic formula" is not a definition.
I can't imagine that any definition would be considered sufficient for you,
given your apparent position on the infinite anyway. So, I guess I won't worry
myself too much about it.
You haven't given me *any* definitions about anything which leads me to
think you don't even have any about anything. Do you think there's a
mathematical definition of "number"? What is it?
I think of a number as distinct from a quantity. For me a number is a symbolic
representation of a quantity defined in some kind of language called a number
system. A quantity can be thought of as a point in numberical space, whether
that be the real line, the complex plane, or something else.
So does a "number" have to correspond to a "quantinity" which is a
point in some "space"? If so, what "space" is the point represented by
"BigUn", in?
The real line.
So does that make BigUn a real number?
Being an infinite natural, it's also an infinite real, yes.
mike.
--
Smiles,
Tony
.
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