What is the role of 'unlimited' numbers in non-standard analysis?

[Suni]

Hi all
I've seen the term 'unlimited' and 'infinity' used interchangeably in non-standard analysis. However when reading a scientific american magazine (Nov 1994 - "Resolving Zeno's Paradoxes") the author says this:

Because an infinitesimal is very small, its inverse will be very large (in the standard realm, the inverse of one millionth is one million). This type of nonstandard number is called an unlimited number. The unlimited numbers, though large, are finite and hence smaller than the truly infinite numbers created in mathematics. These unlimited numbers live in a kind of twilight zone between the familiar standard numbers, which are finite, and the in-finite ones.

I am slightly confused. Which is it? I can't see why the inverse of an infinitesimal would be infinite if 'unlimited' and 'infinite' actually had the same meaning.

 

[matt grime]

All of the words seem scrambled. I have no idea what the author even means by 'truly infinite numbers' that are 'created in mathematics'.

He can't possibly mean infinite cardinals can he? In any case, since the reicprocals of hyperreals are not cardinals the comparison doesn't make sense.

(Conway's surreals may be different)

 

[cogito²]

The writers use the axioms and results of Internal Set Theory (developed by Edward Nelson; link) so you have to take that into account. The results of the theory are only true when using the same axiom system. Since you've probably never worked with IST (I never have; in fact, I hadn't even heard of it until I read that article after seeing your post) the concepts probably seem as odd to you as they do to me, but it all works out logically (I presume) inside IST. So basically, your questions cannot really be answered without actually using IST. If you want to understand it a little better, the wikipedia link I gave is probably a good place to start.

 

[cogito²]

Here is Edward Nelson's homepage and here is a chapter on Internal Set Theory (linked on his homepage) that he has not finished.

 

[Suni]

Thanks for the replies guys!

Yes cogito² i think you're right.. I've mistaken this IST business with the hyperreal set and its implications. It actually looks like the whole process is undertaken in the Real set itself. Looks like i have a lot of reading to do.

The first half of the wikipedia article actually appears to be borderline philosophy.

 

[Hurkyl]

Even in the ordinary framework, there is a sense in which the author is correct... every hyperreal is internally finite.

However, this is confusing, because "finite" has a specific meaning in our external analysis. Therefore, we refer to "internally finite" as "hyperfinite".

The notion of hyperfiniteness arises from the hypernaturals: something is hyperfinite if there exists a hypernatural with greater magnitude. (Remove "hyper" from everything and you have the usual meaning of "finite")


Maybe this picture can help explain the "twilight zone":

When doing real analysis, we like to consider the extended reals. (And the extended naturals). The set of extended reals are defined by $\bar{\mathbb{R}} := \mathbb{R} \cup \{-\infty, +\infty\}$ -- in other words, we simply add two new symbols to the reals. These symbols are incorporated into the ordering <, by saying that $-\infty$ is the smallest, and $+\infty$ is the biggest.

While in your elementary calc classes, you were taught that when these symbols appeared in formulas, it was just a symbol and didn't really mean anything -- well, if you so choose, you can generally reinterpret all of those formulas so that it works with these two infinities that live in the extended reals. (So, for example, saying the limit of something is $+\infty$ means essentially the same thing as saying the limit is $0$)


Allow me to draw a picture of the extended naturals in order:

$$0, 1, 2, \ldots | +\infty$$

Where I've used the pipe (|) for "grouping" -- everything to the left forms one group (the naturals).

Well, all of the unlimited hypernaturals are inserted "before infinity". Here is a picture of the hypernaturals, in order:

$$0, 1, 2, \ldots | \mbox{ unlimited hypernaturals } | +\infty$$

 

[KOSS]

A few quick (late) comments:

(1) It's important to be aware that use of certain words in a technical context is not always what our intuition would expect based on the normal use of the same words in conventional contexts. Thus I too would expect the reciprocal of any infinitesimal hyperreal to yield a transfinite hyperreal. This would be a cardinal of course, because all infinite sets have a cardinality (which might be unknowable given one's axioms, e.g., the continuum hypothesis within ZFC).

So depending upon how Nelson defines infinitesimals in IST we might not get this expectation guaranteed! That's the crazy world of mathematics for you! As long as IST is consistent then we can't complain too much about the results, but you might want to complain a lot about the unfortunate use of words.

(2) It would be interesting if someone followed up and actually read & understood IST and could then tell us what the relationship is in that formal system between it's "infinitesimals" and it's transfinite numbers, or what, if anything, do the IST infinitesimals correspond to in terms of hyperreals in ZFC, or Conways surreals. Maybe they are totally different ghosts.

(3) I believe in Robinson's Non-Standard Analysis we do find the reciprocal of an infinitesimal is indeed infinite. See e.g., page 3 of Goldblatt, R.A., "Lectures on the Hyperreals", Springer GTM-188 1998.

So if the SciAm article is referring to NSA and not IST, then it is wrong or it is a typo as quoted. See http://en.wikipedia.org/wiki/Hypernatural"

The set $^*\mathbb{Z}$ of all hyperintegers is an internal subset of the hyperreal line $^*\mathbb{R}$. The set of all finite hyperintegers (i.e. $\mathbb{Z}$ itself) is not an internal subset. Elements of the complement
$^*\mathbb{Z}\setminus\mathbb{Z}$
are called, depending on the author, non-standard, unlimited, or infinite hyperintegers. The reciprocal of an infinite hyperinteger is an infinitesimal.

So there you go. $1/\epsilon \geq \infty$, contra to the quote from Suni. That's in NSA, I do not know about IST.

(4) So if IST 'unlimited numbers' are finite, then if the statement quoted by Suni from Sci.Am. 1994 is correct, this implies to me that IST 'infinitesimals' do not correspond to the conventional infinitesimals of the hyperreal universe (NSA) or those in the surreal number universe.

So I checked the article, and I think Suni has actually spotted an error in the article McLaughlin elsewhere writes,

Unlimited nonstandard numbers, represented as N and N + 1, are the inverses of infinitesimal nonstandard numbers. Each unlimited number is greater than every standard number and yet less than the infinite real numbers. The nonstandard real numbers prove useful in resolving Zeno’s paradoxes.

---McLaughlin, Sci.Am, 1994, page 87.

Which is not the same as the statement on the same page 87, as quoted by Suni.

But again, I stress that the author (McLaughlin) there is using Nelson's IST, which is not the same as Robinson's NSA. So I'll repeat for emphasis, in Non-Standard Analysis the reciprocal of in infinitesimal is infinite. So whatever Nelson's IST comes up with if McLauglhin interprets it correctly it is pretty kooky stuff. Since Ed Nelson is a , I suspect McLaughlin wrote falsely.

So I read Nelson's book draft a bit, and found the definition of "unlimited",

$x\cong\infty$ case $x \geq 0$ and x is unlimited (i.e., not limited)

where $\cong$ means "infinitesimally close to". Well, so are any finite numbers in IST infinitesimally close to $\infty$?

(5) These "hypernaturals" are very weird. But even in Cantorian set theory we strike the puzzle that $\omega$ is the least ordinal with no finite predecessor (there is no $n: n+1=\omega$ , and yet $\omega \setminus\{0\}$ is an ordinal less then $\omega$. Although perhaps this is not shockingly weird, since at least $\omega \setminus\{0\}$ has the same cardinality as $\omega$. It means you have to be clear about the distinction between ordinal and cardinal, if you do that then there is no paradox.

Still, it makes me wonder that the IST unlimited hypernaturals are perhaps more peculiar than even Suni might have feared! But as I wrote above, in point (3) I think the quote is wrong.

 

[Hurkyl]

(note: this was written before KOSS finished editing his post)

A few quick comments:

It's over 5 years to late to be a quick comment.

Thus I too would expect the reciprocal of any infinitesimal hyperreal to yield a transfinite hyperreal. This would be a cardinal of course, because all infinite sets have a cardinality

Actually, infinite hyperreals have no obvious relationship with cardinal numbers (or ordinal numbers). I'm not aware of anyone having attempted to use cardinal numbers to form a model of hyperreals, nor do I see any non-trivial way to do so.

(One trivial way would be to well-order your favorite model of the hyperreals, and then make a new model using the n-th cardinal number to represent the n-th hyperreal from the original model)

So there you go. $1/\epsilon \geq \infty$, contra to the quote from Suni. That's in NSA, I do not know about IST.

"$1/\epsilon$ is positive and infinite" does not mean "$1/\epsilon > +\infty$"! Via the transfer principle, every hyperreal number is indeed smaller than the extended real number $+\infty$.

I suppose you could mean something different with $\infty$, e.g. creating some new ordered set by taking the hyperreals and inserting a new number between the standard numbers and the positive infinite numbers. But that would be... strange... I'm not sure what use it would be.

(5) These "hypernaturals" are very weird. But even in Cantorian set theory we strike the puzzle that $\omega$ is the least ordinal with no finite predecessor (there is no $n: n+1=\omega$ , and yet $\omega \setminus\{0\}$ is an ordinal less then $\omega$.

Er, no. $\omega \setminus \{0\}$ isn't an ordinal number (in the usual model of them). However, it is a well-order type, so it is reasonable to treat it as if it was naming an ordinal. But it has the same well-order type of $\omega$, so we would have
$$\omega \setminus \{0\} = \omega$$




IST is just a set theory that builds non-standard methods directly into the theory. Sufficiently large non-standard numbers are still infinite by the standard measure and finite by the non-standard measure.


There is a philosophical notion that is often associated with IST (but, strictly speaking, it is not part of IST). That notion is to invert how we conceptualize the relationship between the standard and non-standard models. So rather than think of the non-standard model as an extension of the standard one, we think of the standard model as a restriction of the non-standard one.

But in IST we have the power to build all of the same predicates. Such as "x is 'externally infinitesimal' if |x| is smaller than every positive standard real number", exactly the same as the usual way of doing things. And another thing is the same is that 0 is the only hyperreal number that is 'internally infinitesimal'.

 

[KOSS]

Hmmm,... reading the book draft by Nelson, it seems in IST the notion of a standard natural number is undefined,

The reason for not defining "standard" is that it plays a syntactical, rather than a semantic, role in the theory. It is similar to the use of "fixed" in informal mathematical discourse. One does not define this notion, nor consider the set of all fixed natural numbers. The statement "there is a natural number bigger than any fixed natural number" does not appear paradoxical. The predicate "standard" will be used in much the same way, so that we shall assert "there is a natural number bigger than any standard natural number." But the predicate "standard"—unlike "fixed"—will be part of the formal language of our theory, and
this will allow us to take the further step of saying, "call such a natural number, one that is bigger than any standard natural number, unlimited."

---Nelson, page 1 of "Internal Set Theory",http://www.math.princeton.edu/~nelson/books/1.pdf"

Oh crap! Just read Hurkyl's reply. Didn't think you'd be reading five years later!

Still what seems bizarre to me is that the word "infinite" is never used in Nelson's book. So I'm thinking McLaughlin is interpreting the IST ideas wrongly, the word "unlimited" in IST I believe is semantically interpretable as "infinite".

I've always had a hard time groking those "in-between" transfinite ordinals, like $\omega -1$ (is that a better way of writing it)?...

...and figuring out how/where they fit in the hyperreal system, if indeed it is appropriate to think of them this way? I guess you'd say I'm confounding separate ideas.

It's been informative reading your reply though. Thanks for the corrections and insight.

 

[Hurkyl]

Hmmm,... reading the book draft by Nelson, it seems in IST the notion of a standard natural number is undefined,

It is defined -- it is a natural number that is standard. The quote you gave is explaining reasons why there isn't a set of standard natural numbers.

This corresponds to the classical viewpoint on NSA where we say that the set of standard natural numbers is an "external set".

Roughly speaking, it means that the set cannot be constructed through internal means, and so the set theory internal to the non-standard model does not apply to such things.

Still what seems bizarre to me is that the word "infinite" is never used in Nelson's book. So I'm thinking McLaughlin is interpreting the IST ideas wrongly, the word "unlimited" in IST I believe is semantically interpretable as "infinite".

I think a lot, or even most treatments of NSA do this. The main reasons, I think, are

• There is a lot of potential for confusion due to there being two notions of infinite
• The word "infinite" brings a lot of baggage with it that can be confusing or even inapplicable

I've always had a hard time groking those "in-between" transfinite ordinals, like $\omega -1$ (is that a better way of writing it)?...

There aren't any "in-between" transfinite ordinals -- every ordinal number less than $\omega$ is finite.

Also, $\omega - 1$ and $\omega$ are actually equal, corresponding to the fact that $1 + \omega = \omega$.

This latter fact follows from the order isomorphism between the two rows:

0 0 1 2 3 4 5 6 7 ...
0 1 2 3 4 5 6 7 ...

The first row is the ordinal 1 followed by the ordinal $\omega$. The second row is just the ordinal $\omega$


(However, $\omega + 1 > \omega$. Yes, this means addition of ordinal numbers is not commutative; sometimes $x + y \neq y + x$)

I guess you'd say I'm confounding separate ideas.

I would. The ordinal numbers don't really fit into the hyperreal numbers, or vice versa. You can make ordinal-indexed transfinite increasing sequences of hyperreals, but that isn't really saying anything.

(However, the least upper bound on how long such a sequence can be is an interesting property of various models of the hyperreals)

 

[Hurkyl]

Argh I keep forgetting to make the addendum I wanted.


My impression is that the most useful quality of IST is that it gives rigor to a certain philosophical ideal, and I think I've only ever seen expositions of IST concurrently with this philosophy being detailed.

The basic idea of this philosophy notion is the observation that a mathematician, due to the finite amount of materials around him and his finite lifespan, can only ever involve himself with finitely many numbers. Typical ways to encode this philosophy in mathematics simply don't work -- if you try to make a set of all numbers a mathematician can use, it becomes immediately clear that mathematicians can also use "the first natural number bigger than everything in that set", which leads to a contradiction.

But because of the way IST is set up, we can make this philosophy rigorous by interpreting "standard" to mean "the numbers accessible to mathematicians", and we don't run into any of the contradictions that plague this notion.

(this method isn't completely satisfactory since if x<y and y is standard, then x is standard. But it's better than anything that came before it)