Re: Calculus XOR Probability
- From: "Mike Kelly" <mk4284@xxxxxxxxxx>
- Date: 9 May 2006 15:21:57 -0700
Tony Orlow wrote:
Mike Kelly said:
Tony Orlow wrote:
Mike Kelly said:<snip>
Tony Orlow wrote:
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.
But it's /not/ my opinion that whether you want to call it "size" or
not is totally irrelevant to the validity of standard set theory.
Standard set theory doesn't use the word "size" so you're free to use
it or not and it changes nothing.
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.
No, it's not my opinion. Set theory isn't natural language. It doesn't
talk about "actual infinities".
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.
What do you mean so what? Your whole argument there was about sets of
the form {1,2,3, ..., n} - sets with maximal elements. The naturals
don't have a maximal element so your argument doesn't apply to the
naturals.
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.
You showed this for sets of the form {1, 2, 3, ..., n}. If you think
the naturals are a set of that form, we have problems.
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?
The union of sets need not have the same properties as those sets.
You'd have to show that it does.
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.
But the naturals don't have a maximal element.
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.
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.
What is "a specific number"?
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.
It doesn't make me feel better. Maybe when you realise that the dozens
of mathematicians saying that you don't understand standard theory are
right you might learn something about it?
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.
Gibberish.
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.
You mean it's axiomatic?
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.
So? It is still not such a 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.
What is the "value range" of the naturals? I can think of only two
answers : undefined or infinite(UNBOUNDED).
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.
What about 0.000.... 001/2?
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.
No, they aren't. The links you provided don't mention "number lines"
(actually I can't think of any actual mathematics that talks about
number lines, it's just a visual aid, right?) but they're very clear
that are not real numbers. Why then would they be on the ***real
number*** line?
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.
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?
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.
And everyone else on the planet? Weird.
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 does "the standard length" mean?
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.
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.
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.
What is "a standard infinite point"?
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 can count more than the counting numbers? With what?
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.
But these are sequences not sets. What does "value range" mean here?
And why "between two given values"? I don't care about sequences that
end, here. I'm interested in the ones that go on forever.
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.
What line? These are sequences, not sets.I'm not asking about density.
I don't care about density. If I wanted to determine the density
withing a value range I could do so. Density doesn't mean anything
without a "value range", so it seems completely fruitless to demand
that sequences without an end or sets without a value range should be
comapred in terms of things they are not.
Yes, Tony, IF these sequences ended at some certain value then one of
them would be longer than the other. But they don't. They go on
*forever*. Can you really not get your head around that? That sometimes
we want to talk about things that *don't* have a value range?
"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.
Who mentioned density? Not me. I asked about set sizes. What does
density have to do with set sizes? Isn't set size the number of
elements?
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.
Hahahahaha. And your analysis is super rigorous? Not contradictory at
all? Doesn't rely on "magically declaring things to exist"? Hahaha. At
least standard theories explain all of their terms without having to
resort to natural language handwaving whenever probed too deeply.
--
Smiles,
Tony
mike.
.
- Follow-Ups:
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- References:
- Re: Calculus XOR Probability
- From: Mike Kelly
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- Prev by Date: Re: Infinite Meet or no-meets
- Next by Date: Re: Calculus XOR Probability
- Previous by thread: Re: Calculus XOR Probability
- Next by thread: Re: Calculus XOR Probability
- Index(es):
Relevant Pages
|
