Re: Galileo's Paradox and the Project of the Reals
- From: Tony Orlow <tony@xxxxxxxxxxxxx>
- Date: Mon, 01 Jan 2007 15:01:34 -0500
Robert Maas, see http://tinyurl.com/uh3t wrote:
From: Tony Orlow <t...@xxxxxxxxxxxxx>
In any linear ordering each of those sets, the countable sets
will have each element finitely before or after each other,
What does that mean?? Look at the lexicographic p-adic ordering on
the positive integers which I defined the other day. Please tell me
what you mean that is true for that particular ordering that is not
true for any ordering of an uncountable set.
If you have countably infinitely many digits in a p-adic number, does the set of such p-adics not contain p^aleph_0, or uncountably many strings? It would appear to be an uncountable set of p-adics, even though each can be compared to each other for size, down to finite differences. Hmmm....
whereas in the uncountable set, any linear enumeration will
result in elements infinitely past other elements.
What does that mean??
For instance, in the p-adics, you will encounter not only differences such as ...888823-...888816=7 or 75-11=66, but also infinite distances between elements, such as ...88832-...22111=...66721. Is ...666666721 infinite? Countably so, anyway... :)
The standard definition of a finite natural suffices, as the size of a set which cannot be bijected with any subset. A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. By the way, the handwaving is just to distract you from the fancy footwork. ;)That's what I meant by "finitely distant" - in terms of
intervening elements.
Please supply a precise definition of "finitely distant" in terms
of intervening elements. Just mumbling "in terms of" isn't a
definition, it's just a "handwave".
In reality, if you want to talk about sets of points, as you
have, any two distinct points have an infinite number of
intermediate points.
If you're talking about sets, there's no meaning to the word
"intermediate". Intermediate applies only to ordered sets (sets
plus one particular order relation), not to sets. Given a set with
two different orderings on it, what's intermediate per one of the
orderings isn't necessarily intermediate per the other. And as I
understand it, for *any* non-finite set, there are at least two
different orderings which are *not* order-isomorphic.
Sure, that's true. But, when we discuss real or even hyperreal values, we can place them all on the same line, and establish quantitative order as a common attribute. Where we have more than one dimension, we've found what we can simulate linearity using a "diagonal" approach. We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. The point here is that, in ANY ordering which one may choose for an uncountable set, if the elements are indexed using a finite-base digital number system, there will necessarily be elements indexed with infinite strings. In a countably infinite set, all elements will be indexed with finite strings, such that the strings cannot be bijected with any substrings. :)
If you combine order and set size, then you are not comparing sets,Then the naturals are not a set, nor the reals.
you are comparing something more complicated than just sets.
That's a really stupid thing for you to say.
I disagree. :)
There's a set per se,
and there's that same set plus additional structure such as an
order relation. The two things aren't the same. One is a set, the
other is an ordered set. The set of naturals is a set. The set of
naturals plus the Peano ordering is an ordered set.
Define the naturals without the successor() function. I dare you! :)
Successor(x) = {yeN | y>x ^ (~E zeN | z>x ^ y>z)}
Successor depends on order.
The set of
naturals plus the lexicographic-p-adic order is another ordered set
different from the Peano-ordered naturals. Likewise after
constructing the reals as Dedekind cuts or equivalence classes of
Cauchy sequences or equivalence classes of nested-interval
sequences, we have the ordered field of the reals which is also a
metric space. If we discard the order relation and metric, we have
just a field of reals. If we discard the arithmetic properties but
keep the order and metric, we have a metric space. If we discard
both arithmetic and metric, we have just an ordered set. If we
discard all of arithmetic and metric and order, we have just a set.
"The reals" can mean any of those versions depending on which
information from the construction from rationals you keep and which
you discard.
Um, why do you "discard" information that is readily available? Isn't that kind of like handing out rice to fat people with gonorrhea?
there are quantitative sets
What do you mean by the term "quantitative set"?
R or n-tuples of R, like the Cartesian plane or the complex numbers. Continuous sets, really, as opposed to discrete.
and discrete sets
What do you mean by the term "discrete set"?
One in which successor() has meaning.
Successor(x) = {yeN | y>x ^ (~E zeN | z>x ^ y>z)}
of symbols or values. A common approach, I think, can handle both
better than cardinality,
Cardinality is designed to compare only sets, not whatever you mean
by "quantitative sets" or "discrete sets" which presumably have
additional structure added to the base set.
Yes, true. It's meant to be a generalization for use with limited information. As such, it shouldn't directly contradict conclusions that are merited with further information. If two sets have the same cardinality, but different Lebesgue measure, or Schnirelmann density or whatever, then they should not be considered the same "size" exactly. Set theory makes claims like, "if I stick my fingers in my ears and bury my head in the pillow, then it's just like watching Beethoven perform." Lack of information does not produce more solid conclusions that information.
when order is taken into account.
That's the topic of ordered sets, not sets. There's already a
theory of ordered sets, and in particular Cantor's ordinal theory
of well-ordered sets.
vN ordinals are bunk. Sorry. Start at 1. The size of the set of naturals is the largest one. Is the size aleph_0? That's not included in the set, so there's a contradiction. There is no "size" to the finite realm. There is no smallest infinity, despite the contortion of logic that leads to its inception as an idea, any more than there is a largest finite, or a smallest nonzero finite, or a largest or smallest infinitesimal, or a largest infinity besides the incalculable absolute oo. Where you can count forever from ...222 to ...333, you can never pinpoint where you stop having a most significant digit or 2. It's been called the Twilight Zone, and the Tunnel of Love, and maybe the "end of the ditty that never ends". It's something we all just have to deal with.
You cannot explicitly well order the reals. I got as close as it gets, but it's an infinite regression. So well ordering isn't a top priority of mine.
If we have a notion of order on a set, should we ignore it?
No, that's why there's a separate theory of ordered sets.
Should the theory of sets in general contradict the theory of sets with order?
You can't do order-dependent operations on sets unless you have
a particular order fixed for the duration of the discussion.
You can't claim generalizations on sets, if they don't apply to all sets. It's best not to call cardinality any sort of set size measure, but only a rough classification of types of sets.
If all you have is a set, with no particular order on it, then you
can't assert statements that would be true for one order but not
for another order. But if you *do* have a particular order, agreed
upon at the start of the discussion, *then* you can assert
statements that depend on that order.
That goes back to my assertion for which you requested clarification. It doesn't matter which order you choose, if you use a finite alphabet to produce a language, an uncountable set will require infinite strings for most elements.
Supposed you asked me to add 4+4, and tell me what color the sum
was? That would not be possible, because you haven't told me what
color the two 4's are that go into the addition nor what the rules
are for combining colors.
I'd guess a reddish brown, but what are you getting at....
Note that as ordered sets, using the usual Peano ordering, the even
integers and *all* the integers are isomorphic, because the mapping
2*n <-> n preserves order. But as arithmetic systems, the two are
not isomorphic. In particular the integers has a multiplicative
identity but the even integers doesn't. But if you keep only order
and addition, not multiplication, once again the two are
isomorphic.
Well, sure. That's why I advocate starting with the real, geometric, line, and space, as a foundation. It includes the universe. We define the basic arithmetic operations geometrically, base our axioms on them, and build onward, from what we live on, into abstraction, don't we then, eh? I think that's a worthy approach.
In the future, please don't pretend you are defining something in
set theory when in fact you are actually defining something in
ordered-set theory or algebraic-ring theory or some other field
that involves the study of a set of elements plus operations and/or
relations on those elements.
I'm sorry. I didn't mean to be "pretending", but only "contending" that pertinent information should not be ignored when available, and that when speaking of the naturals or reals, we have additional information available regarding order. I'll try to be more careful in the future. Again, my apologies. ;)
For the **set** of integers (or positive or non-negative integers),
there are only two things to know, whether some particular element
is in the set (and compoundly whether some other specified set is a
subset of the given set, or vice versa), and the cardinality of the
set which is Aleph-null.
You don't define that, by any chance, using the primitive successor() relation, as opposed to the primitive includes() relation, do you?
For the *ordered* set of integers (or positive or non-negative
integers), per the usual Peano ordering (sucessor of any element is
greater than the original element), there's a whole lot more to
know about it, such as number of other elements between two given
elements, whether an element has a predecessor or not, the
cardinality of the initial segment ending at that element, etc.
It's also possible to define additional properties such as
arithmetic directly in terms of the ordering. For example using
Peano ordering of non-negative integers, if x has no predecessor,
then x + y = y, otherwise x + y = predecessor(x) + successor(y),
which is a valid recursive definition because it bottoms out when
predecessor(pred...essor(x)...) has no further predecessor.
That's right. That's the way the set is defined, through "successor". That's different from "element of". Two separate primitives in play here. Let's make sure they play nice.
For the set of integers (or positive or non-negative integers) with
a weighting on each element such that every weight is positive the
the sum of all the weights equals exactly 1, i.e. a denumerable
probability space, there's even more to know about it, including
matters of "density" of subsets. But there's no standard weighting
of elements that is preferred over the gazillions of other possible
weightings. For example, you could pick any random eumeration of
the integers (or positive integers etc.), and define weight of
first-enumerated integer as 1/2, weight of second-enumerated
integer as 1/4, etc.
See Han's "Calculus XOR Probability".
There are only a finite number of finite naturals between any two
others.
That depends entirely on which order relation you impose on the
naturals. With the lexicographic-p-adic geometry, there are an
infinite number of naturals between any two naturals. For example,
between 2 and 4 there are all odd multiples of 2 (except 2 itself
of course).
p-adics are apparently not considered naturals, even though they have successors and can be proven to represent only finite values if all digit positions are finite. You are superimposing some infinite extension into the unit intervals between the naturals, but none of those superimposed elements are naturals. In the natural quantitative order of finite naturals, no more than any finite number lie between any two others. They are all a finite number of units from each other on the line.
This is "countably" or "potentially" infinite.
Countably infinite is a pure set property which Cantor defined.
It's nice that you agree with his definition.
But "potentially" infinite has no meaning AFAIK. Please tell me
what you mean by that term, but first tell me whether it's a
property of sets per se, or only a property of ordered sets (sets
per a particular ordering).
It means there is always one after every other, but that none is infinitely past any other. Finite internal distances.
Personally, I don't consider that "actually" infinite.
Please tell me what you mean by "actually" infinite.
Uncountable is actually infinite.
Until you tell me what you mean by "potentially" and "actually"
infinite, and tell me whether they are properties of sets or only
properties of ordered sets, your sentence is meaningless.
I am telling you that you can equate, when speaking to me, "countably infinite" with "potentially infinite", and "actually" with "uncountably". I believe that's the proper correspondence between the ancient and "modern" terms.
I defined the lexicographic-p-adic geometry on the positive
integers quite clearly in an earlier article (circa yesterday).
Please be so kind with any personal jargon you wish to use in
discussions in this newsgroup.
Ah, yes, I see it now. I'll print it out and peruse. Not a lot to say right now about that. Sorry. I'm juggling neurons and hormones. :)
It doesn't matter whether somebody *allows* something or not. PerThat really depends on the application of the von Neumann ordinals.
the Peano axioms, there *are* more than any finite number of
integers, and more than any finite number of rationals.
I looked up von Neumann ordinals. It's a system of constructing the
ordinals by setting 0 = {} and n = {k | k<n}. You are incorrect.
What I posted is *not* dependent on that in any way.
Then present your proof.
If you begin the naturals at 1, then the count of n consecutive
naturals ends at n.
Correct.
If there are aleph_0 naturals, then aleph_0 is a natural.
No. Aleph-null is a limit ordinal.
No kidding. But the size of the set of natural numbers greater than or equal to 1 is aleph_0. So that must be the last element of the set. Not.
It is not the immediate
successor of any other ordinal. Every ordinal less than Aleph-null
is a non-negative integer, and every non-negative integer is an
ordinal less than Aleph-null, per that construction. The
non-negative integers via that construction are order-isomorphic to
any ordered-set of non-negative integers per Peano's postulates.
yada yada yada. Heard it all before, and it don't wash, Chico.
That makes no sense.
What you said just previously makes no sense, and is flat out
wrong, what a coincidence!!
It's not wrong. It's non-standard. There's more to space than the line you toe. Okay, that was a joke. No offense intended. :)
The notion of a smallest infinity violates the basic concept of
subtraction as producing a smaller result.
Subtraction producing a smaller result is not a basic concept, it's
a specific property of some specific numerical systems which is not
a property of other numerical and non-numerical systems. For
example, consider the integers modulo 7, and the Peano ordering on
residues (0 < 1 < 2 < 3 < 4 < 5 < 6). Subtracting 1 from any
nonzero residue makes it smaller, but subtracting 1 from 0 makes it
**larger**, it runs it all the way from the smallest to the very
largest!!
Yeah, that's a ring, not the number circle, uh, I mean, line. But anyway, you take some away, the thing got smaller. Yeah, that's pretty basic. Then you put some, and it gets bigger. Yep, confirmed through laborious experimentation....
So, I reject the von Neumann model of the naturals as bunk.
Since you came up with a totally false and meaningless claim
(subtracting 1 from stuff makes it smaller, but in fact if
subtracting 1 means taking predecessor, then omega has no
predecessor so you can't subtract 1 from it), you have no basis for
rejecting the von Neumann model of ordinals (which includes the von
Neumann model of integers within it).
No predecessor! Ain't that grand! Yep, I've heard it all before. It "springs from the axioms". Ho hum. "Bigger than any finite, and the smallest at that!" It's a bunch of anti-measure pipe fillin's. It ain't math, Feller. Math's over there, by that line....
The set of finite naturals is at least as large as every finite,
but not larger.
That's false. You really are stupid!!! Here's proof you are wrong.
(Denote omega to mean "The set of finite naturals" below.)
Given: The set of finite naturals is at least as large as every finite.
Restated: Omega is at least as large as every finite.
Theorem (per von Neumann ordinals): The set of finite naturals is *larger* than every finite.
Restated, to prove: Omega is *larger* than every finite.
Proof: Let n be any finite.
Let s(n) be the successor of n.
s(n) > n.
s(n) is a finite.
Omega >= s(n). (Per your Given)
Omega >= s(n) > n.
Omega > n. Q.E.D.
You really are stupid if you fall for that crap. Sure, you derived a contradiction. Now, figure out what to blame it on. You assumed a largest finite. That's the problem. There is none. So what? You did not prove that omega is *larger* than *every* finite. You prove there is no largest finite. So?
No countably infinite sequence of points can ever achieve any
finite measure as a line.
That's false. Define weight(n) = 1/(2^n) for each non-negative integer.
Define measure(S) as the sum of all the weight of the elements in S.
Then measure(Omega) = 1 + 1/2 + 1/4 + 1/8 + ... = 2.
Every subset of Omega likewise has positive measure, except the empty set which has measure zero.
Okay, I didn't state that quite correctly. No countable sequence of contiguous points can ever achieve any measure, in any space.
If infinite naturals are allowed as numerators and denominators, ...
There you go again, making up your own jargon and not telling us
what definition you are using. Please define what you mean by an
"infinite natural".
You can think of p-adics, if that's what you're used to, or Google my T-riffics.
Note that rational numbers are ordinarily defined as equivalence
classes of fractions with an integer in the numerator and a
positive (or nonzero) integer in the denominator. If a natural
number is so large you have never in your whole life counted that
far one-by-one, is that what you consider an "infinite" natural?
For example, you've probably never had the patience to count up to
one million, so is that one of your "infinite" naturals?
Oh, say, the size of the set of reals in (0,1]. That's a whole number, larger than any finite. Unless, of course, you allow partial points....?
While it's true that any finite is still finite when incremented,
this only holds for a finite number of increments.
That's correct. But incrementing an infinite number of increments
is a never-ending process, so if there is to be any meaning to what
you say above you need to define it specially.
Sure. It means an uncountable sequence.
Where you increment 0 an infinite number of times, you have
produced that infinite number.
That has no meaning, because you haven't defined what it means to
"increment 0 an infinite number of times". Do you mean you start a
program running and promise never to shut it off? So you have a
running program. What's the point you're trying to make?
If you start the following program.
x := 0;
while true x := sucessor(x);
what exactly is the "infinite number" you see "produced" by that
ever-running program? The set of all values that x has already been
assigned or ever will be assigned in the future (i.e. Omega), or
what? Tell me what you mean by "produced". Have you *already* in
some sense produced Omega just by starting that program and
promising never to shut it off? What if you lied, or there's a
power failure, or the Universe burns out, so the program doesn't
keep running forever? How can you say just by starting that program
you have already **producED** an infinite number? I agree you have
produced the *potential* to build toward an infinite number, but
"...ED" is past tense, so your use of "produced" bothers me. Please
explain what you are claiming, what you mean.
You have the unit interval, (0,1]. Within it you may construct, geometrically, a point midway between 0 and 1, distinct from each, and you can continue this process "forever". That is, for any finite number of subdivisions you perform, there will always be a finite distance between any two neighboring points, allowing a further subdivision, and identification of additional reals as those points on the line. So, no finite number of subdivisions will identify all reals in the interval. So, there are an infinite number, termed Big'un, of reals in the unit interval, and a relation that states that in (x,x+y] on the real line lie y*Big'un reals. That's an actually infinite unit, its inverse being the unit infinitesimal, and the whole system lending itself to IFR. :)
Uh, no. I am saying that if we start the naturals at 1 instead of 0, and assume that the difference of that single element can't change a finite into an infinite number, we can conveniently note that the set of the first n consecutive naturals has as its maximal element n, and that therefore the size of the first aleph_0 elements in the set, the entire set, would have as its maximal element aleph_0. However, aleph_0 is considered infinite and is not "allowed" in the set of naturals, which are all finite. So, we have a contradiction.That's not allowed in the set.
What set are you talking about now? We've been talking about an
infinite number of different sets:
0 = {} = The null set
1 = {0} = 0 union {0}
2 = {0,1} = 1 union {1}
3 = {0,1,2} = 2 union {2}
etc.
Omega = {0,1,2,3,... (all the nonnegative integers indicated above)}
Omega+1 = Omega union {Omega}
etc.
Omega+1 most certainly contains an infinite element, in fact it
contains Omega, which is the set of all values assigned to x in
that program.
So I guess Omega+1 isn't the set you're talking about?
if x-1<x is to have any consistent meaning,
Why should that have any consistent meaning? It's universally true
if you're discussing the system of integers per the usual ordering,
but not true for the positive integers (because 1-1 isn't defined),
nor for the non-negative integers (because 0-1 isn't defined), nor
for the system of positive integers per the lexicographic-p-adic
ordering (not defined for 1-1, and defined but not true for any
other odd value of x), and it's not true for the integers modulo 7,
and not true for the ordinals (because it's not defined for "limit
ordinals" such as Omega).
It's defined geometrically on the real line as a translation of the unit interval from any point.
You seem to have a hairbrained idea that if a particular
expression, such as x-1, has a meaning in one particular system you
happen to be most familiar with, in this case the integers (not the
positive integers, where it's not defined when x=1), that somehow
that same meaning can be extended in a consistent way to *all* of
mathematics. The *only* thing that can be extended consistently to
*nearly* all of mathematics is propositional calculus and rules of
inference. Even first-order predicate calculus isn't valid in some
fields of math where propositional calculus is valid, although it
is valid in *nearly* all such, which is *nearly* *nearly* all of
math.
Well, you're off in symbol-land like most math folks, but I recommend a revisitation of geometry. Not that I disrespect the symbolic representation of quantity and relation, but it's only one side of the "equation".
aleph_0 is bunk.
No it isn't. It's the cardinality of the set of positive integers.
With which we can calculate.....what?
It's a phantom, like the largest or smallest finites,
In the set of positive integers, there is a smallest finite, namely
1, but no largest finite. 1 isn't a phantom.
What about the "real" world?
or the largest infinitesimal.
That probably makes no sense, but I'm not an expert on nonstandard
analysis.
You should check it out.
But there's no order-preserving mapping between Omega+2 and Omega.That's all very well and good. I understand the standard position
Try to find one if you don't believe me.
on that. I just disagree that it has any real value,
If you can't see the value in it, then don't use it. Someday
perhaps you will be working on a problem where application of the
theory of 1-1 mappings between ordered sets has some value. Until
then, please feel free to ignore order-preserving mappings between
ordered sets and never mention them again.
Order-preserving mappings between sets? Whyever would I consider order in mappings between unordered, pure, unsoiled sets? That might confuse me with too many...facts. ;)
and recommend a system wherein order is taken into account where
it exists and where proper subsets are smaller sets.
If you do that, you are guaranteed *not* to have a total ordering,
only a partial ordering. Most any two sets you have will be
uncomparable per your way of comparing them. Unless perhaps you
use my idea of lexicographic comparison of well-ordered sets.
Let S and T be sets. Let e be the least element of S xor T.
Then S < T iff e elementOf T.
Lexicographic methods are one side of the coin, the other being quantitative. In other words, discrete vs. continuous. There are alphabets, and then there's the line.
But that works only for well-ordered sets, such as subsets of
non-negative integers per Peano ordering.
Because it's discrete...
For example, the odd positive integers and the even positive
integers have a distinguishing element 1 which is the smallest
number in one set but not in the other (0 isn't in either set), and
it's in the odd integers but not in the even integers,
so even < odd. Is that what you are recommending?
No. I am recommending comparing sets which lie on a common line according to how they occupy that line, whether the real line, or a line of symbols.
Order is imposed on standard sets
Again your private jargon you haven't yet defined: "standard set"
What do you mean by that term??
Naturals, Reals, etc. Sets commonly used to compare other sets.
through the naturals, with the addition of the primitive order
operator, successor(), or '<'.
That's the Peano ordering on the naturals, the transitive closure
of x < successor(x). That's just one possible ordering of the
naturals. So let's consider not the *set* of naturals, but the
Peano-ordered set of naturals, OK? So how exactly do you extend
this particular ordering of the naturals to other sets which aren't
subsets of the naturals?
All the Peano axioms say is there's a starting element, and a subsequent element to every element, and that if you can prove p(x) and p(x)->p(s(x)), then p can be said to hold for all neN. The particular formula used to calculate s(x) CAN be inc(x) or x+1, but needn't be. However, for the naturals, it is.
Set membership alone does not allow us to compare most infinite
sets with any precision.
That's true. If we have no structure whatsoever, just a pure set,
no order, no arithmetic, no metric, no topology, not anything
except what elements are in each set and what elements aren't in
the set, then there are only three basic ways to compare two sets:
- Cantor definition of cardinality (a total ordering on equivalence classes)
- Subset (a partial ordering directly on sets)
- Whether one special element, such as the Sunnyvale resident
known as Heather Thompson, is a member of the set or not (a
total ordering on just two equivalence classes, any no-Heather
sets < any Heather sets).
Is cardinality the most general rule? If so, why does it violate the subset-is-smaller relation?
Where we have order, we should use it.
Yes, when we have ordered sets, not just sets, we can use the
ordering if we choose.
Should the general conclusions for sets contradict the conclusions for particular sets when order is known? I don't think so.
Call it "sequence size" if you want,
I would never call it that. That's another private term you made up
without defining, so please define how you think that term should
be used as a property of ordered sets.
Successor(x) = {yeN | y>x ^ (~E zeN | z>x ^ y>z)}
There is one element directly fter each other.
Defined only by inclusion, or involving order?but then don't call the naturals a set.
The set of naturals are a set.
The Peano-ordered set of naturals are an ordered set.Using successor()
The lexicographic-p-adic-ordered set of naturals are another ordered set.Using symbol order in alphabet, and string position
The power set of the naturals is a set.Ordered lexicographically by using a binary inclusion code and bit position for each element
The power set of the naturals per the lexicographic-well-ordered
ordering induced by the Peano ordering on the naturals is an
ordered set.
Sure, it's based on an ordered set, represented by an ordered language. It'd be hard to avoid...
Defined how?You can't have it both ways, can you?
You can have it both ways, in fact you can have it all three ways:
You can have the pure set of naturals, no order, just a bunch of
elements not in any particular order.
You can have the ordered set of naturals per the Peano ordering.Naturally, but without measure...
You can have the ordered set of naturals per theWell, that's about language, not values, but sure....
lexicographic-p-adic ordering.
You can have it any of a gazillion other ways too.
But you need to state at the start of any argument which way you
are having it. It's not fair to claim you're talking about the *set*
of naturals, but then try to use some ordering that you presumed
wasn't there to begin with when you used the word "set" by itself.
The naturals are defined by successor, and a notion of measure regarding what that means: ++.
You can have it any way you want, so long as you say "up front"
what system you're talking about at any time.
It's just like the crank who seems to be talking about algebraic
integers but keeps introducing things which are algebraic
non-integers. He's not honest. You haven't been honest either,
preteding like you're talking about sets, but the switching over to
ordered sets from time to time.
Try to be honest from now on, OK?
Define the naturals without the successor operation, and then tell me no order is necessary to the concept. I won't call you dishonest, but will offer you that challenge. Define the naturals as a pure set. N={x| f(x)}. What is f?
Do you know the difference between a CPU and a computer system? A
computer system has a CPU connected to memory and device
controllers and power supply etc. An ordered set has a set
connected to an order relation. Don't confuse the two, just a set,
and a full ordered set. If you say that the *set* of integers has a
least element, that's as stupid as saying that the Intel Pentium 3
CPU chip has 48 megabytes of RAM and Windows NT operating system.
I imagine I know more about that particular subject than you, and find it a lame comparison, but hey! It's prose, right?
Try this one on for size. Take an elephant and a mouse to the post office, and try to convince them it should be the same price to ship each. Point out the two ears, and four legs, and two eyes, and the tail. Show them they both have fur, and make poop and noise. They'll see they are the same set equivalent of cells, with the same countable cardinality, and I'm sure they'll ship your elephant for 4.95 shipping and handling, just like the mouse.
Have fun!
Tony
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