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Dec 18, 2004, 1:49:02 PM12/18/04

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Dear readers,

I have been having fun reading Robert Goldblatt's book on the hyperreal

numbers. Although I am older, and am a geneticist now, I might be

considered at the level of an advanced undergraduate. As such, I have

a question which I hope some of you can help me with.

I have been thinking of the hypernatural numbers as constructible

models for some of the smaller ordinal numbers, in a very elementary

fashion as might be explained to advanced high school students, perhaps

at the level of an article in the MAA's monthly, or an exercise.

Thus, omega = {1,2,3,...}, i.e. the sequence of natural numbers, as he

described.

Then, as a first grader looking for a bigger hypernatural number would

say

omega+1 = {2,3,4, ...}

2*omega = {2,4,6, ...}

2*omega +1 ={3,5,7, ...}

omega^2 = (1,4,9,16, ...}, omega^m = {1,2^m,3^m, ...}.

Now comes my first question. I want to get to omega^omega.

Suppose I first look at {2,4,8,16, ...,2^n, ...}.

What is this? 2^omega in the hypernaturals?

If so, then form 3^omega, 4^omega, ... .

What about {1^1, 2^2, 3^3, ...} ?

Is this omega^omega?

Then as usual {2^(2^1), 2^(2^2),2^(2^3), ...}

Is this 2^(2^omega) ?

Then {1^(1^1), 2^(2^2), 3^(3^3), ...}

Is this omega^(omega^omega)) ?

Then {1, 2^2, 3^(3^3), 4^(4^(4^4)), ...}

Is this omega^(omega^(omega^( ...))) in the hypernaturals?

Second question:

Suppose I look at {Ack(1), Ack(2), Ack(3), ...} where Ack is the

unitary Ackerman function.

What ordinal does this correspond to?

Are there any very elementary references in this regard?

Thank you,

Dec 18, 2004, 1:57:22 PM12/18/04

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sigol...@gmail.com writes:

>Dear readers,

>

>I have been having fun reading Robert Goldblatt's book on the hyperreal

>numbers.

...

>Thus, omega = {1,2,3,...}, i.e. the sequence of natural numbers, as he

>described.

Okay, that's standard enough.

>Then, as a first grader looking for a bigger hypernatural number would

>say

>omega+1 = {2,3,4, ...}

>2*omega = {2,4,6, ...}

>2*omega +1 ={3,5,7, ...}

>omega^2 = (1,4,9,16, ...}, omega^m = {1,2^m,3^m, ...}.

>

>Now comes my first question.

No, before it comes *my* question: are these "first grader" encodings

of "bigger hypernatural number"s than omega being done according to

a scheme from Goldblatt's book (which I've never seen, nor even heard

of before)? If so, could you explain his scheme to those of us who

haven't seen it? If not, can you provide some meaningful explanation

of your own? Not a one of those equations makes sense to me, from

what I learned (long ago, from a book by Szierpinski, I think) about

ordinals. (Since your Subject: header uses the word "ordinals", and

"omega+1",..., "omega^2" are the sorts of names ordinals have, I

assume you are trying to write equations in ordinals. If not--if,

indeed, you're writing equations for some kind of "hypernatural numbers"

that don't purport to be ordinals--then please clarify further.)

Lee Rudolph

Dec 18, 2004, 2:37:28 PM12/18/04

to

Robert Goldblatt's book "Lectures on the Hyperreals" (Springer 1998),

describes aspects of the theory of nonstandard analysis. It is easy

and fun to read.

describes aspects of the theory of nonstandard analysis. It is easy

and fun to read.

Roughly speaking,

consider a system in which numbers are replaced by equivalence classes

of infinite sequences, two sequences being equivalent iff they differ

at only a finite number of positions.

Drastically oversimplifying, this forms a ring under pointwise addition

and multiplication of sequence elements. However, the product of two

non-zero elements can equal zero.

(e.g. {0,1,0,1, ...} * {1,0,1,0,, ...} = 0)

Ordinary numbers correspond to constant sequences.

Infinite numbers correspond to divergent sequences.

Eventually the ordinary reals embed in an extension of such a system,

as well as infinitesimals

(e.g. {0,0,0, ...} < {1,1/2, 1/3, ...} < epsilon

for any ordinary real number epsilon > 0.)

and infininite numbers.

Hope this helps

Ordinary

Dec 18, 2004, 3:13:33 PM12/18/04

to

Well, this makes a little more sense after your second post

on the topic. Enough sense that it's clear that your

questions about which ordinal various sequences "correspond"

to do not have answers, because you have not _specified_

what the correspondence _is_.

It's true that you've defined a few hyperreals that are

ordered in the same way as a few infinite ordinals. But

there are many other ways you could have done this -

there's nothing unique about your choice of which

ordinal corresponds to omega+1, so there's really no

answer to the question of which ordinals those

other hyperreals correspond to until you _define_

the correspondence.

(If that's not clear:

Even if we restrict to just sequences of positive

integers, as you seem to be doing, the hyperreal

you say "is" omega is not the smallest hyperreal

larger than every natural number. For example

(1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,...) is larger than

every natural number but smaller than the thing

you're saying corresponds to omega. Why does

the sequence above correspond to omega, why

isn't it this one?)

>Thank you,

************************

David C. Ullrich

Dec 18, 2004, 3:32:35 PM12/18/04

to

Yes, you are right; I didn't think of that.

Thanks.

Thanks.

But suppose we do everything modulo a particular choice of a

representative sequence for omega.

Since the integer sequences are constant, the first few examples seem

to go through OK.

Then, for the, more advanced examples of larger ordinals

(i.e. each time we exponentiate) maybe we have to choose

a representative again.

Admittedly, this only gives a branching tree of models,

but still interesting, if true.

Dec 18, 2004, 3:34:21 PM12/18/04

to

Lee Rudolph wrote:

>

> sigol...@gmail.com writes:

>

> >Dear readers,

> >

> >I have been having fun reading Robert Goldblatt's book on the hyperreal

> >numbers.

> ...

> >Thus, omega = {1,2,3,...}, i.e. the sequence of natural numbers, as he

> >described.

>

> Okay, that's standard enough.

>

> sigol...@gmail.com writes:

>

> >Dear readers,

> >

> >I have been having fun reading Robert Goldblatt's book on the hyperreal

> >numbers.

> ...

> >Thus, omega = {1,2,3,...}, i.e. the sequence of natural numbers, as he

> >described.

>

> Okay, that's standard enough.

The notation is confusing. The op means the sequence (1,2,3,...) not

the set {1,2,3,...}. So omega is not the smallest infinite ordinal.

Dec 18, 2004, 3:44:55 PM12/18/04

to

You can visualize those numbers in that way, they are names that you

assign to those constructions.

assign to those constructions.

>From a set-theoretic angle, one thing that you are doing is to say {0,

1, 2, 3, ...} = omega, thus any algebaric operation on omega, the

number or ordinal, applies to each element of the set that comprises

omega.

Then, for example with the case 2 * omega, a resulting aspect of the

set, now being the even numbers, is that the asymptotic density has

been halved, it's the reciprocal and in this case inverse of the scalar

multiplier of the ordinal.

Thus, in a way you have the same number, infinity is immutable.

While that is so, infinite sets are equivalent, while that is so,

there's still the notion that you can manipulate the symbol omega,

little Greek w, just as if it were the scalar unity.

For the finite scalar multiplier the asymptotic density is only the

inverse, but with the infinite scalar multiplier, or finite

exponential, then the asymptotic density of the resulting set goes to

zero. It is yet positive: it's infinitesimal.

In another way, the density of the prime number is known, leading to an

infinite ordinal that is the ordinal formed by that set of natural

integers.

Thus, while infinity is in a way immutable, you can consider performing

operations on it as if it were a number, where it automatically

equalizes to itself with the accounting in the second column.

Don't mind me too much, I'm well-known for discussing infinity.

Basically, the notion is that infinity is in various ways usable and

conincident and coexistent with zero, one and after the negative

numbers negative one.

That's where at the deep foundation of infinity, it is the same as the

void, and the difference subsumes all inbetween.

I, and sometimes we, can consider infinity and null as the same or

indistinguishable things, because they share common properties of the

necessarily unique and existent non-entity (entities). That enables

the mathematical discussion to neatly dovetail with the logical and

even philosophical, and for some, the epistemological and existential.

Happy Holidays, and warm regards,

Ross Finlayson

Dec 18, 2004, 11:17:03 PM12/18/04

to

Your comments have all been helpful,

and I now I can say things a little differently.

and I now I can say things a little differently.

A few points.

First, two apologies.

A. For now, as George Cox mentioned above,

I was unfortunately using brackets to designate sequences not sets;

I will try to make it clearer in the future and less confusing.

(I borrowed the notation from the book.)

B. A correction: Professor Goldblatt never actually wrote or described

that omega,

the smallest infinite ordinal of set theory = sequence{1,2,3, ...}.

This was an overenthusiatic error of my own,

and as pointed out by Prof. Ullrich above, cannot be correct in the

naive form in which it was written,

Second, that out of the way, what am I trying to say?

Suppose, inspired by the construction of the hyperreal numbers in non

standard analysis,

we choose a particular integer sequence as a name for the smallest

infinite ordinal of set theory,

which is usually called omega. This sequence name is not uniquely

specified,

as pointed out by Prof. Ullrich above, but my hunch is that

the structure of what follows will be invariant with respect to

different

initial choices of otherwise equally good one sided sequences as names

for omega.

In particular, suppose we choose (my hunch above is without loss of

generality)

the particular sequence (not set) {1,2,3, ...}

as the new fixed name for omega, the smallest infinite ordinal of set

theory.

Then, in the ring of hyperinteger sequence names, other names suggest

themselves for other ordinals, as indicated in my original post, above.

For example,there is of course in set theory an ordinal sucessor to

omega, often called omega+1.

The sequence (not set) {2,3,4, ...} suggests itself as a name for this

ordinal,

under this naming convention, and so on as suggested above.

What's the point? If the guess of invariance with respect to naming

convention is right,

then the set of names under fixed convention have the structure one

might have guessed when one was 9 years old and wondering about

infinity,

i.e. a commutative ring, just like the integers.

So in the age old discussion, one could have said,

"No, infinity is not a number, but its name is a hypernumber", or

something.

E.g seq{1,2,3, ...} - seq {2,4,6, ...} = - seq {1,2,3, ...}, etc.

This seems cool. The question is, is it ever at variance with the

actual structure of the ordinary ordinals,

or as I assumed, can the ordinals be embedded in the hyperintegers with

naming convention,

in the same way that the reals can be embedded in the hyperreals?

P.S. Why the guess of invariance with respect to the initial choice of

names for omega?

Because whatever name we choose, the ring structure still applies.

Dec 19, 2004, 6:06:12 AM12/19/04

to

In article <1103429823.9...@f14g2000cwb.googlegroups.com>, <sigol...@gmail.com> wrote:

> Because whatever name we choose, the ring structure still applies.

You know, addition and multiplication in the hyperreals satisfy the usual

identities, addition and multiplication in the ordinals do not...

not commutative, for example.

There is a scheme used by logicians where rapidly-growing sequences if

natural numbers correspond to ordinals; but they do not bring in nonstandard

analysis as an intermediary.

--

G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Dec 19, 2004, 9:55:09 AM12/19/04

to

sigol...@gmail.com wrote:

>

>

> For example,there is of course in set theory an ordinal sucessor to

> omega, often called omega+1.

> The sequence (not set) {2,3,4, ...} suggests itself as a name for this

> ordinal,

> under this naming convention, and so on as suggested above.

>

>

> For example,there is of course in set theory an ordinal sucessor to

> omega, often called omega+1.

> The sequence (not set) {2,3,4, ...} suggests itself as a name for this

> ordinal,

> under this naming convention, and so on as suggested above.

No, if omega is the smallest infinite ordinal, omega + 1 does not equal

{2,3,4, ...} (sequence or set).

If omega is 1,2,3,... then omega + 1 is 1,2,3,...,1.

Dec 19, 2004, 11:16:16 PM12/19/04

to

In article <1103429823.9...@f14g2000cwb.googlegroups.com>,

sigol...@gmail.com writes:

(I shortened the lines here.)

|In particular, suppose we choose (my hunch above is without loss

|of generality) the particular sequence (not set) {1,2,3, ...}

Probably (1,2,3,...) would be better.

|as the new fixed name for omega, the smallest infinite ordinal

|of set theory. Then, in the ring of hyperinteger sequence names,

|other names suggest themselves for other ordinals, as indicated

|in my original post, above.

Ordinals correlate to well-orderings. So if you want to relate them

to the hyperintegers, what you want to do is find a well-ordered

subset of the hyperintegers. I think this is what you want to do. :-)

The hyperintegers >0 are not well-ordered. You have for example an

infinite descending sequence (1,2,3,...)>(1,1,2,3,...)>

(1,1,1,2,3,...)>.... So if you want to have a well-ordered subset

of the hyperintegers, you have to just decide to take only some of

them.

Someone pointed out already that the operations on the hyperintegers

and the operations on the ordinals don't correspond. I'll give you

another example of that. As ordinals, 2^omega=omega. This is different

from how hyperintegers work, where 2^omega>omega, and is also different

from how operations on infinite cardinals work. So you need to be

careful. If I were you, I'd pick a different name for (1,2,3,...),

actually, than "omega" just to be sure I didn't confuse the two.

The sum, product, or exponential of two ordinals will not generally

map to the sum, product, or exponential of the hyperintegers you

mapped them to. But that's okay as long as you remember it.

Given a countable collection of hyperintegers, there exists a

hyperinteger greater than all of them. (Exercise!) So any countable

ordinal can be embedded in them, and even aleph_1. I don't know

what ordinals can be embedded in them; I don't see offhand whether

aleph_1+1 embeds in them.

Keith Ramsay

Dec 20, 2004, 7:28:55 AM12/20/04

to

Isn't everything defined componentwise?

I.e. (A,B,...)^(a,b,...)=(A^a,B^b,...)

(when you wrote (A,B,...)^2 this is to be interpreted as

(A,B,...)^(2,2,...), because 2 corresponds to (2,2,...) (or the

equivalence class containing (2,2,...) anyway ) )

Dec 20, 2004, 1:30:16 PM12/20/04

to

If you wish, or if someone wishes. But if you use "ordinal" and "omega"

near one another, people[1] will think that omega is the ordinal of that

name, and ordinal arithmetic is not defined componentwise (there are no

components).

[1: I did, sorry if I got you wrong.]

Dec 20, 2004, 3:54:02 PM12/20/04

to

> If you wish, or if someone wishes. But if you use "ordinal" and

"omega"

> near one another, people[1] will think that omega is the ordinal of

that

> name, and ordinal arithmetic is not defined componentwise (there are

no

> components).

Dec 23, 2004, 11:08:37 PM12/23/04

to

Thanks, this has been a big help, a little like having a lunch

together.

together.

Summarizing, what I have learned so far:

1. I should not have been talking about the ordinals, but simply about

the hyperintegers as algebraic representations of infinities. Any

relation to any ordinal numbers is yet to be shown. My mention of the

ordinal omega above was an error in the above context.

2. The positive hyperintegers are not well ordered ,but rather

partially ordered. There is no smallest infinite hyperinteger.

3. Athough the hyperintegers form a commutative ring, the ordinals are

not commutative under addition, multiplication, or exponentiation.

Note that some additions, multiplications, and exponentiations in

the ordinals appear consistent with what might be expected in the

positive hyperintegers, namely if a larger limit number is to the left

of a smaller number in the binary operation. The incompatibility

appears when the larger limit ordinal is to the the right of the

smaller one.

Speculation

4. Limit ordinals are limits. It is not clear how the hyperintegers I

was mentioning are limits, although it might be interesting to try to

define sucessors.

5. "Ordinals correlate to well-orderings. So if you want to relate them

to the hyperintegers, what you want to do is find a well-ordered

subset of the hyperintegers. I think this is what you want to do. :-)

The hyperintegers >0 are not well-ordered. You have for example an

infinite descending sequence (1,2,3,...)>(1,1,2,3,...)>

(1,1,1,2,3,...)>.... So if you want to have a well-ordered subset

of the hyperintegers, you have to just decide to take only some of

them." -from KRamsay (above)

Yes, I think this is precisely what I want to do. Intuitively, I feel

that if I look at the branching tree of the partially ordered

hyperintegers > (0,0,0, ...), some of the branches will contain

constructible representations of apparently very big entities, maybe

even something like Cantor's epsilon0.

Then, if we choose just the paths leading to this representation of

something apparently large, we will get an ordered seguence, which

might, somehow, be compared with the ordinals. The non commutativity

remains an issue, of course.

Thanks again,

Message has been deleted

Dec 23, 2004, 11:20:57 PM12/23/04

to

PS Note:

" 4. Limit ordinals are limits. It is not clear how the hyperintegers I

was mentioning are limits, although it might be interesting to try to

define sucessors. "

By "successors" I meant nonunique partially ordered analogs of

successors,.

" 4. Limit ordinals are limits. It is not clear how the hyperintegers I

was mentioning are limits, although it might be interesting to try to

define sucessors. "

successors,.

Dec 24, 2004, 2:39:15 PM12/24/04

to

sigol...@gmail.com wrote:

>

> constructible representations of apparently very big entities, maybe

> even something like Cantor's epsilon0.

>

> constructible representations of apparently very big entities, maybe

> even something like Cantor's epsilon0.

Does epsilon_0 count as very big? It's countable.

Dec 27, 2004, 10:32:11 PM12/27/04

to

That was surprising and at first shocking news to me. I'll look for

the bijection. It just goes to show that if you start out ignorant,

you can learn a relatively large amount quickly.

the bijection. It just goes to show that if you start out ignorant,

you can learn a relatively large amount quickly.

Two ideas occur in response.

First, there appear to be an uncountable number of hypersequences,

which as above, can be arranged in a branching partially ordered

infinite tree.

Second, the question arises, if the whole program outined above works,

can one say that no totally ordered path (chain) ever reaches the

representation of an uncountable ordinal?

This would say that however rapidly a sequence grows, considered as a

hypersquence, it always corresponds to something countable, i.e. the

length of the chain going back to the sequence (1,2,3, ...) is always

countable I am specifically thinking of sequences (perhaps like the

"busy beaver" sequence?), which I have read are not recursively

computable by a Turing machine, and grow very rapidly.

This seems like an interesting question.

Dec 27, 2004, 10:39:07 PM12/27/04

to

PS. I mean here integer hypersequences > (0,0,0, ...)

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