Infinities and infinitesimals

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Norman Wildberger

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Jun 8, 2007, 5:00:07 PM6/8/07
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Last year I posted the paper `Set theory: Should You Believe?' at my UNSW
website http://web.maths.unsw.edu.au/~norman under Views. It caused a bit of
a stir in certain circles, and some asked me for an alternative framework to
deal with the difficulties with modern set theory. In response, I have now
posted a new paper, also at this site, called `Numbers, Infinities and
Infinitesimals'.

Here are the main points: 1) An `infinity' is a growth rate of a function
from natural numbers to natural numbers. 2) An `infinitesimal' is the
reciprocal of an infinity, namely a decay rate of a function from natural
numbers to rational numbers.

The paper shows how the usual language of ordinals may be partly recast in
this purely finite, computer friendly form, and how nonstandard analysis may
be initiated in a much simpler way than using the axiom of choice and
ultrafilters. It also suggests how this approach allows portions of calculus
to be done purely over the rational numbers.

Norman Wildberger


Aatu Koskensilta

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Jun 9, 2007, 9:30:03 AM6/9/07
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On 2007-06-08, in sci.math.research, Norman Wildberger wrote:
> The paper shows how the usual language of ordinals may be partly recast in
> this purely finite, computer friendly form, and how nonstandard analysis may
> be initiated in a much simpler way than using the axiom of choice and
> ultrafilters. It also suggests how this approach allows portions of calculus
> to be done purely over the rational numbers.

You are aware, surely, of the existence of the thriving field of proof theory
and recursion theory, where ordinal notations, fast growing recursive
functions in subrecursive hierarchies, computable analysis, and that sort of
stuff in general, are studied?

--
Aatu Koskensilta (aatu.kos...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

tc...@lsa.umich.edu

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Jun 10, 2007, 9:30:06 PM6/10/07
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In article <f4ea0r$ur7$1...@news.ks.uiuc.edu>,

Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
>You are aware, surely, of the existence of the thriving field of proof
>theory and recursion theory, where ordinal notations, fast growing
>recursive functions in subrecursive hierarchies, computable analysis,
>and that sort of stuff in general, are studied?

Of course, he probably isn't. But I'm not sure that that is the right
direction to point him in.

As you know, it's pretty common nowadays to run into folks who are
suspicious of infinite sets and uncomputable reals and so forth. They
typically complain that such concepts are "metaphysical" (though they
have no qualms about metaphysical concepts such as symbols, strings,
computation, etc., and usually aren't philosophically sophisticated
enough to notice their own metaphysical assumptions). They have also
not heard the news about reverse mathematics and the development of
large portions of mathematics without infinitary set theory.

To some extent, this is the fault of the mathematics community at large,
which looks down on foundational studies and doesn't typically give good
jobs or other kudos to people who work in that area. But to some extent
it is also the fault of the experts in foundations, for not making these
results accessible enough. For example, Simpson's "Subsystems of Second-
Order Arithmetic" is a magnificent book (which I hope will reappear in
the bookstores soon---I'm not sure why the ASL is taking so long to come
out with the 2nd edition), and presumably Wildberger would be fascinated
to know that, for example, Brouwer's fixed-point theorem can't be proved
without recourse to non-computable reals. But for all its merits,
Simpson's book is too advanced for the beginner.

The expert knows how to eliminate unnecessary uses of infinite sets, how
to translate finite-set-speak into integer-speak, and how to translate
integer-speak into syntax-speak. And so he can look at a set-theoretic
proof and see how to formalize it in, say, RCA_0, and know that in
principle he can develop it in terms acceptable to a skeptic who insists
that he knows what a string is but doesn't know what an integer or a set
is. But where can we find the tutorial that helps the beginner or the
skeptic acquire these translational skills for himself? Nowhere that I
know of.

It may be exasperating to see uninformed renegades think that they are
saying something new when they are reinventing the wheel, and a rather
crude and clumsy wheel at that. But surely we should try to channel
that exasperation into writing that badly-needed expository book that
doesn't seem to exist yet.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

WM

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Jun 11, 2007, 10:00:08 AM6/11/07
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On 11 Jun., 03:30, t...@lsa.umich.edu wrote:
> In article <f4ea0r$ur...@news.ks.uiuc.edu>,

> Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote:
>
> >You are aware, surely, of the existence of the thriving field of proof
> >theory and recursion theory, where ordinal notations, fast growing
> >recursive functions in subrecursive hierarchies, computable analysis,
> >and that sort of stuff in general, are studied?
>
> Of course, he probably isn't. But I'm not sure that that is the right
> direction to point him in.
>
> As you know, it's pretty common nowadays to run into folks who are
> suspicious of infinite sets and uncomputable reals and so forth. They
> typically complain that such concepts are "metaphysical" (though they
> have no qualms about metaphysical concepts such as symbols, strings,
> computation, etc., and usually aren't philosophically sophisticated
> enough to notice their own metaphysical assumptions).

May I ask whether you would consider the following comparison as
philosophically unsophisticated too? At least there is no metaphysics
it.

(1) According to Skolem's theorem there are countable models of
consistent first order theories. There are (or at least should be)
externally countable but internally uncountable models of ZF. Such
models are assumed to accomplish the required internal uncountability
by the lack of a mapping the domain of which is their omega. This
mapping is simply not contained in that model.

(2) A finite model of arithmetic (think of Hilbert's famous one, for
instance) with only L bits will not be capable of addressing or
representing an element which has an information contents of L + 1 or
more bits. As a very simple example take a model with only one bit,
which is capable of representing the binary numbers 0 and 1 but not
the binary number 10 and larger numbers. Those numbers are simply not
contained in that model.

I think models like (1) and (2) and the due constraints are not in
question.

Now consider as a model of information processing a surrounding S of
an intelligent being B. S is chosen so large that only the bits
contained in S can contribute to any kind of information processing
including all mathematics which B will be ever be able to perform. But
S is certainly finite and so is its information contents L (even the
accessible part of the universe contains a finite amount of
information processing capacity). Therefore the mathematics of B is
restricted to theorems and numbers which can be represented by L bits
or less. This excludes nearly all natural numbers of ZF from the
model, namely such which can only be represented by more than L bits.

Is this last exclusion less valid than those imposed on (1) and (2)?

Regards, WM


tc...@lsa.umich.edu

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Jun 11, 2007, 10:00:10 PM6/11/07
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In article <f4jkh8$h0d$1...@news.ks.uiuc.edu>,

WM <muec...@rz.fh-augsburg.de> wrote:
>May I ask whether you would consider the following comparison as
>philosophically unsophisticated too? At least there is no metaphysics it.

Unless I'm missing some subtle nuance in your presentation, you're
simply presenting a standard skeptical argument, based on Skolem's
paradox, about completed infinities.

I don't think that sci.math.research is the place to rehash the
standard counterarguments and countercounterarguments to this standard
argument. The purpose of my note was not to take a particular side on
philosophical debates about whether infinite sets (or whatever) are
legitimate, but to point out that there is a considerable body of
technical knowledge that is relevant to such debates, but that for
lack of good exposition seems to remain unknown to many people who
engage in such debates.

Marc Nardmann

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Jun 12, 2007, 9:30:03 AM6/12/07
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tc...@lsa.umich.edu wrote:
> In article <f4jkh8$h0d$1...@news.ks.uiuc.edu>,
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>> May I ask whether you would consider the following comparison as
>> philosophically unsophisticated too? At least there is no metaphysics it.
>>
>
> Unless I'm missing some subtle nuance in your presentation, you're
> simply presenting a standard skeptical argument, based on Skolem's
> paradox, about completed infinities.
>
> I don't think that sci.math.research is the place to rehash the
> standard counterarguments and countercounterarguments to this standard
> argument. The purpose of my note was not to take a particular side on
> philosophical debates about whether infinite sets (or whatever) are
> legitimate, but to point out that there is a considerable body of
> technical knowledge that is relevant to such debates, but that for
> lack of good exposition seems to remain unknown to many people who
> engage in such debates.
>

As a layman in mathematical logics, I am not familiar with those
standard (counter^n)arguments. Where could I read about them?

Let me also ask a question related to WM's point of view. The question
is a bit vague and not stated in technical jargon, but tries to avoid
philosophical debates: Consider a mathematical structure (given in
Zermelo/Fraenkel set theory, say) which models a toy universe with
inhabitants. The structure is such that the inhabitants can only store a
finite number L of bits. (Judging from the models used in current
physical theories, the universe we live in might have a lot in common
with such a toy universe.) What kind of logics would/should the
inhabitants use, given that statements which are too long do not fit
into the toy universe? (As a vague analogue, their logics might be
related to standard logics roughly like finite fields are related to the
field of rational numbers.) Which constraints does this place on the
accuracy with which they can describe the structure they live in, i.e.,
how much does their "frog perspective" on the toy universe differ from
the "bird perspective" of an outsider who can store an infinite amount
of information and thus can use standard logics and set theory to
describe the mathematical structure?

I would like to learn something about the (empty? tiny? considerable?)
subfield of mathematical logics which deals with such questions. What is
the technical term for such "finite logics"? Where can I find them in
the literature? I apologise if these questions should be too imprecise
or uninformed to make sense.


-- Marc Nardmann

tc...@lsa.umich.edu

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Jun 12, 2007, 11:28:58 AM6/12/07
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In article <f4m74r$i4n$1...@news.ks.uiuc.edu>,

Marc Nardmann <Marc.N...@bigfoot.de> wrote:
>As a layman in mathematical logics, I am not familiar with those
>standard (counter^n)arguments. Where could I read about them?

One starting point would be Putnam's "Models and Reality" and the myriad
followup responses by other philosophers. Though this is just one
particular angle, and there are others, depending on exactly what
argument you're trying to make.

>I would like to learn something about the (empty? tiny? considerable?)
>subfield of mathematical logics which deals with such questions. What is
>the technical term for such "finite logics"? Where can I find them in
>the literature? I apologise if these questions should be too imprecise
>or uninformed to make sense.

If you're envisioning worlds that are finite but where the upper bound is
"vague," so that you can't point to a specific, concrete integer that is the
absolute largest possible integer, then one keyword is "feasible mathematics."
I don't know a lot about this area, but the topic arises occasionally on the
Foundations of Mathematics mailing list, e.g.,

http://cs.nyu.edu/pipermail/fom/2006-August/010676.html
http://cs.nyu.edu/pipermail/fom/2006-August/010747.html

If there's a hard upper bound on the size of the universe, then you can
probably just work with classical logics and finite model theory, but my
sense is that this is not what you're looking for.

WM

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Jun 12, 2007, 4:51:22 PM6/12/07
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On 12 Jun., 04:00, t...@lsa.umich.edu wrote:
> In article <f4jkh8$h0...@news.ks.uiuc.edu>,

>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> >May I ask whether you would consider the following comparison as
> >philosophically unsophisticated too? At least there is no metaphysics in it.

>
> Unless I'm missing some subtle nuance in your presentation, you're
> simply presenting a standard skeptical argument, based on Skolem's
> paradox, about completed infinities.

I am not presenting or supporting Skolem's scepticism here. By the
way, it was Skolem himself who pointed out that the absence of an
internal mapping with domain omega might accomplish the internal
uncountability of an externally countable model. I am only asking why
in this model the absence of a provably absent notion is accepted
whereas in another model the absence of a provably absent entity is
not accepted.

> The purpose of my note was not to take a particular side on
> philosophical debates about whether infinite sets (or whatever) are
> legitimate, but to point out that there is a considerable body of
> technical knowledge that is relevant to such debates, but that for
> lack of good exposition seems to remain unknown to many people who
> engage in such debates.

So let me present a model which is nearly as simple as Hilbert's
famous model of arithmetic and which does not require any technical
knowledge. I will explain it in few lines:

Consider a tiny universe in which only one statement can exist which
is limited to seven symbols (which can be chosen arbitrarily from the
key board of a usual type writer). Of course, contrary to the real
universe, the meaning of the symbols has to be formulated outside. But
I think the line of thought is not disrupted by this disadvantage,
unless we admit infinitely many ad hoc definitions. For the sake of
simplicity let us assume the decimal system.

In this model we can represent every natural number from 0 to 9999999
by choosing seven symbols from the type writer's set of digits. By
means of abbreviations like
123(9*) which is to represent our number 12333333333 with nine 3's, or
1(999*) which is to represent our natural number consisting of 999
1's,
But we cannot express every number between them. So we cannot express
any number between
123(9*) and 124(9*), i.e., between 12333333333 and 12444444444 by this
kind of abbreviation.

Some advanced (nevertheless familiar) abbreviations may enlarge the
set of numbers of the model including what Gauss called "the
measurable infinity" 9^9^9^9 or even larger numbers like 9^9^9! or
9^9^9!! and so on. So there can be very large numbers, but it is
impossible to represent numbers like 9^9^9^9-1 and most others which
are less than this.

By introducing the symbol "..." (to be understood as a single place
symbol), we can even represent infinite strings like 0.1212... or
1,2,3,... .

It is clear, however, that there are no infinite sets of numbers
existing in our model.

Regards, WM

WM

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Jun 12, 2007, 5:19:58 PM6/12/07
to
On 12 Jun., 15:30, Marc Nardmann <Marc.Nardm...@bigfoot.de> wrote:

> As a layman in mathematical logics, I am not familiar with those
> standard (counter^n)arguments. Where could I read about them?
>
> Let me also ask a question related to WM's point of view. The question
> is a bit vague and not stated in technical jargon, but tries to avoid
> philosophical debates: Consider a mathematical structure (given in
> Zermelo/Fraenkel set theory, say) which models a toy universe with
> inhabitants. The structure is such that the inhabitants can only store a
> finite number L of bits. (Judging from the models used in current
> physical theories, the universe we live in might have a lot in common
> with such a toy universe.)

According to Skolem's advice, internal uncountability of an externally
countable model might be accomplished by the absence of an internal
mapping with domain omega. Perhaps this absence can be enforced by
definition. But is it possible, vice versa, to enforce the presence of
an infinite set in a finite model?

In order to construct a simple model which also non-experts in
advanced logic should be able to follow consider a tiny universe in


which only one statement can exist which is limited to seven symbols
(which can be chosen arbitrarily from the key board of a usual type
writer).

In this model we can represent every natural number from 0 to 9999999


by choosing seven symbols from the type writer's set of digits. By

means of abbreviations we can represent
123(9*) which is to represent our natural number 12333333333 with nine


3's, or
1(999*) which is to represent our natural number consisting of 999

1's.


But we cannot express every number between them. So we cannot express
any number between
123(9*) and 124(9*), i.e., between 12333333333 and 12444444444 by this
kind of abbreviation.

Introducing the symbol "..." (to be understood as a single place


symbol), we can even represent infinite strings like 0.1212... or
1,2,3,... . It is clear, however, that there are no infinite sets of
numbers existing in our model.

> What kind of logics would/should the


> inhabitants use, given that statements which are too long do not fit
> into the toy universe? (As a vague analogue, their logics might be
> related to standard logics roughly like finite fields are related to the
> field of rational numbers.) Which constraints does this place on the
> accuracy with which they can describe the structure they live in, i.e.,
> how much does their "frog perspective" on the toy universe differ from
> the "bird perspective" of an outsider who can store an infinite amount
> of information and thus can use standard logics and set theory to
> describe the mathematical structure?

Which outsider can store an infinite amount of information? Where and
when would who be able of doing so?

Regards, WM

Marc Nardmann

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Jun 13, 2007, 10:32:02 AM6/13/07
to
Thank you to Tim Chow for the key words and references. (I am not sure
that finite model theory can tell me what I would like to know, but I
should learn more before asking further questions.)

WM wrote:

If you feel better that way, let us say that the frog can store 10^9
bits in his toy universe, while the bird can store 10^40 bits in her
meta-universe. As one (but not the only) aspect of what I was alluding
to, consider the following situation.

The frog and the bird are given a description/definition of the toy
universe (formulated, say, in terms of standard logics and ZF set
theory), the description consisting of 10^5 bits, say. (The frog would
probably call such a description a physical "theory of everything".)
They start to deduce theorems about the toy universe. (The bird does it
just for fun, whereas the frog tries to deduce whether his pond will run
dry in the near future.) There will be sentences that the bird can
prove, but the frog cannot, having not enough memory to store the lines
of the bird proof. So there are more facts about the toy universe which
are undecidable for the frog than facts which are undecidable for the
bird. (Here I am using the word "undecidable" not in the standard
technical way, but in a related way which is hopefully intuitively
understandable.)

Can we quantify/estimate in some sense how much more? How would such a
quantification change when we replace the 10^40-bird by a 10^60-bird?
Can we prove some asymptotic law?

Note that this question can be discussed even by the frog. He can write
down decimal expressions of the numbers 10^40 and 10^60, and he should
be able to prove theorems about frog undecidability vs. bird
undecidability. And the frog could also discuss the situation for a bird
which can store an infinite amount of information; this has nothing to
do with whether he himself has access to infinitely many bits (and/or
whether such a bird "exists" in some metaphysical sense). In fact, human
mathematicians in our own universe do just this: they prove theorems
about infinite structures, and that makes perfect sense even though the
mathematicians have, according to current physical theories, access to
at most L bits, where L is something between 10^70 and 10^200 (or so;
physicists might want to improve or correct my crude estimate).

I have stated the situation in deliberately vague and colloquial terms
in order to make clear that it is directly relevant to foundational
questions of physics. I think, however, that with a bit more effort one
could translate most of the discussion into precise questions in
mathematical logics. Because I know next to nothing about mathematical
logics, I suspect that some of these questions have been, or still are,
discussed by logicians, in some jargon that makes it hard for me to
recognise them. That motivated my original question on this thread.

But sci.math.research is probably not the right place to continue this
discussion. At least not for me, because this is far from my area of
research. Thanks for all replies.


-- Marc Nardmann


WM

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Jun 15, 2007, 1:07:13 AM6/15/07
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On 13 Jun., 16:32, Marc Nardmann <Marc.Nardm...@bigfoot.de> wrote:

I have stated the situation in deliberately vague and colloquial
terms
in order to make clear that it is directly relevant to foundational
questions of physics. I think, however, that with a bit more effort
one
could translate most of the discussion into precise questions in
mathematical logics. Because I know next to nothing about
mathematical
logics, I suspect that some of these questions have been, or still
are,
discussed by logicians, in some jargon that makes it hard for me to
recognise them. That motivated my original question on this thread.

But sci.math.research is probably not the right place to continue
this
discussion.

Let me add a word by Zermelo from his first lecture about
Mathematische Logik (Vorlesungen gehalten von Prof. Dr. E. Zermelo zu
Göttingen im S.S. 1908, lecture notes by Kurt Grelling). This
statement has been published only very recently in a comprehensive
Zermelo-Biography (by H.-D. Ebbinghaus: Ernst Zermelo, Springer,
Berlin Heidelberg 2007) and it seems to fit the present discussion
extremely well:

"It has been argued that mathematics is not or, at least, not
exclusively an end in itself; after all it should also be applied to
reality. [...] One can argue here that of course one first has to
convince oneself whether the axioms of a theory are valid in the area
of reality to which the theory should be applied. In any case, such a
statement requires a procedure which is outside logic."

And let me slightly adapt this quote to the present discussion:

It has been argued that mathematics is not or, at least, not
exclusively an end in itself; after all it should also be applied _in_
reality. [...] One can argue here that of course one first has to
convince oneself whether the axioms of a theory are valid _in the
reality_ in which the theory should be applied. In any case, such a
statement requires a procedure which is outside logic.

It is certainly outside of logic to look whether mathematics can be
applied in some field. But it is not outside of mathematics to look
for a model in which mathematics can be done at all! In set theory we
have to observe the basic condition that elements cannot form a set
unless we _can_ distinguish them. And further we have to consider the
fact that this requirement cannot be satisfied, even in the greatest
possible model, for any infinite set. This is, according to my
opinion, not outside of mathematics.

Regards, WM


mark...@yahoo.com

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Jun 19, 2007, 5:42:05 PM6/19/07
to

On Jun 8, 4:00 pm, "Norman Wildberger" <wildber...@pacific.net.au>
wrote:

> The paper shows how the usual language of ordinals may be partly recast in
> this purely finite, computer friendly form, and how nonstandard analysis may
> be initiated in a much simpler way than using the axiom of choice and
> ultrafilters. It also suggests how this approach allows portions of calculus
> to be done purely over the rational numbers.

That's a misconception. The point of the usual construction in terms
of ultrafilters is *not* to define non-standard analysis, but merely
to establish its equipollence; particularly that it can be constructed
with the Axiom of Choice, but appears to require something less than
it. So, on the equipollence hierarchy, it (or any other means of
explicating non-standard analysis) lies somewhere between the full
power of Choice and the lesser power of Ultrafilters.

All an ultrafilter is, is just a fancy way of saying that you're
assigning a Boolean value to infinite Boolean sequences that respects
(1) constant sequences, (2) bit-wise AND, OR and NOT. Once you
understand that, then everything you're doing is just a transparent
restatement of the standard construction.

How to Make Your Own Non-Standard Model
sci.logic, 1993 February 21
http://groups.google.com/group/sci.logic/msg/0e71aea618621dcc?dmode=source

WM

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Jun 25, 2007, 1:15:01 PM6/25/07
to

On 8 Jun., 23:00, "Norman Wildberger" <wildber...@pacific.net.au>
wrote:

> Last year I posted the paper `Set theory: Should You Believe?' at my UNSW
> websitehttp://web.maths.unsw.edu.au/~normanunder Views. It caused a bit of

> a stir in certain circles, and some asked me for an alternative framework to
> deal with the difficulties with modern set theory. In response, I have now
> posted a new paper, also at this site, called `Numbers, Infinities and
> Infinitesimals'.
>
> Here are the main points: 1) An `infinity' is a growth rate of a function
> from natural numbers to natural numbers. 2) An `infinitesimal' is the
> reciprocal of an infinity, namely a decay rate of a function from natural
> numbers to rational numbers.
>
> The paper shows how the usual language of ordinals may be partly recast in
> this purely finite, computer friendly form, and how nonstandard analysis may
> be initiated in a much simpler way than using the axiom of choice and
> ultrafilters. It also suggests how this approach allows portions of calculus
> to be done purely over the rational numbers.

Dear Norman,

I read your article "Numbers, Infinities and Infinitesimals" with
great interest and pleasure. Let me mention some points which
attracted my attention.

You define the natural numbers by means of heaps instead of sets. This
is certainly the natural approach, also shared by Georg Cantor, when
he described a cardinal number as a set consisting of merely 1's (*).
He did not yet know the word "heap" as a mathematical term.
Nevertheless sets started more or less as heaps (in your sense). When
Bernard Bolzano introduced the German word "Menge" (the English
translation is "set") into mathematics, he felt obliged to excuse
himself for denoting also such sets as Menge which consist of only two
elements (because in German everyday use Menge denotes multitude or
large quantity). And he claimed that a set contains nothing but its
elements [BG, p. 152]. The strict border between "all natural numbers"
and "the set of all natural numbers" had not yet been erected. The set
containing A was just A and nothing else. This becomes clear from the
fact that Cantor called the empty set "strictly speaking not
existing" (**). Surprisingly, for me, I recently learned that even
Zermelo, the first creator of omega from the empty set, in letters to
Fraenkel considered the empty set as "not a genuine set" (#) and
increasingly doubted its justifiability (##). I wonder, however, why
you build the numbers upon the empty set. Isn't that a bit
inconsequent? The "empty heap" would be nothing. In my opinion the
natural number 1 should be represented by something being and not by
something not being. This was also the method by which Cantor defined
his finite cardinal numbers (C, p. 289).

You aim to develop a mathematics that our computers can implement, and
that has real applications. By means of computers "we can never get to
the end. But the beautiful fact is that we don't need to, and we don't
need to pretend to be able to." That is a comforting statement. Based
upon (among others) work of Schmieden and Laugwitz (the latter name is
misprinted on your p. 3) you get rid of uncountable sets (###) and
even of any actual infinity.

So you stay with potential infinity which you call a growth rate. An
even more rigorous name would be "a direction" - it cannot be
intensified or surpassed. "Perhaps the terms get so overwhelmingly
large that the world can no longer hold them past a certain point, or
time itself runs out. Or perhaps there is a three-headed dragon
somewhere down the sequence that devours anyone who gets there". Is
the dragon really required?

Cantor himself was probably the last great scholar who was convinced
that the actual infinite exists in reality (+). Zermelo held the
opposite opinion (++), (+++). Nevertheless, he claims: Something is
presupposed which transcends the perceptual realm. The mathematicians
must have the courage to do this [ZE, p. 156]. Must we? According to
Zermelo' s own confession, the axiom of infinity is "essentially due
to Dedekind" [Z, p 266]. Dedekind, however, gave an intuitive argument
only for the existence of a potentially infinite set (in a manner
already anticipated by Bolzano [BP, p. 14]): Starting with an object t
of one's thoughts, a "thought-thing", one successively gets new
thought-things by the thought of t, the thought of the thought of
t, ...; hence the set of all thought-things of a human being is
infinite [ZE, p. 85]. This infinity is a potential one though, because
no one will ever have thought infinitely many thoughts. Therefore,
even without a dragon, there will never be an actual or finished
infinity.

Your calculus works well and will easily be understood by freshmen. If
they know already some basics, they will hardly notice a big
difference to your approach. The limit symbol is removed and delta x
is replaced by 1/n. I personally prefer the notation lim_[x --> x_0]
or lim_[delta x --> 0], but that is only a matter of habit, if
potential infinity is understood by this notation.

For the sake of more clarity (on your p. 14), I would propose that the
full second derivative of f[2](a) at point a should be written

(f[1](a + 1/n) - f[1](a))/(1/n)
= ((2a + 3/n) - (2a + 1/n))/(1/n)
= 2

because few lines above we see that

f[1](a) = 2a + 1/n
and so
f[1](a + 1/n) = 2(a + 1/n) + 1/n.

On p. 15 1/n^2 is misprinted as n^2, and at the top of your p. 16, 5th
line, the sequence of terms is exchanged. (That does not really matter
but is a bit disturbing.)

As far as I know Cauchy already interpreted his quantités infinement
petites as sequences. Advanced students and colleagues will doubt it
and miss rigorosity which, in the eyes of many, can only be achieved
by an actually infinite uncountable set of real numbers extending your
rationals. So your approach will not be welcome to most. It might be
appreciated only in case it turned out, after all, that relativity of
set theory is as strong as Skolem (§) suspected, and we are in a
countable model which only is lacking a function from omega to
P(omega). Perhaps this is even so if no one ever notices.

Regards, WM

____________________


(*) Da aus jedem einzelnen Elemente m, wenn man von seiner
Beschaffenheit absieht, eine "Eins" wird, so ist die Kardinalzahl
<<M>> selbst eine bestimmte aus lauter Einsen zusammengesetzte Menge,
die als intellektuelles Abbild oder Projektion der gegebenen Menge M
in unserm Geiste Existenz hat [C, p. 283].

(**) Es ist ferner zweckmäßig, ein Zeichen zu haben, welches die
Abwesenheit von Punkten ausdrückt, wir wählen dazu den Buchstaben O; P
== O bedeutet also, daß die Menge P keinen einzigen Punkt enthält,
also streng genommen als solche gar nicht vorhanden ist [C, p. 146].
(#) Letter from Zermelo to Fraenkel of 31 March 1921, p. 2: It [the
empty set] is not a genuine set and was introduced by me only for
formal reasons [ZE, p. 135].
(##) Letter from Zermelo to Fraenkel of 9 Mai 1921: I increasingly
doubt the justifiability of the "null set." Perhaps one can dispense
with it by restricting the axiom of separation in a suitable way.
Indeed, it serves only the purpose of formal simplification [ZE, p.
135].
(###) Laugwitz, in his introduction to non-standard analysis, wrote
about Cantor's diagonal argument: "The [initial segment of the
diagonal number] certainly differs from the first numbers of our list.
It must be among the later ones. This is an example showing how the
same proof can lead to completely different theorems, according to the
notions the theory is based upon" [L, p. 226-227].
(+) Accordingly I distinguish an eternal uncreated infinity or
absolutum which is due to God and his attributes, and a created
infinity or transfinitum, which has to be used wherever in the created
nature an actual infinity has to be noticed, for example, with respect
to, according to my firm conviction, the actually infinite number of
created individuals, in the universe as well as on our earth and, most
probably, even in every arbitrarily small extended piece of space [C,
p. 399].

(++) Infinite domains "can never be given empirically; they are set
ideally and exist only in the sense of a Platonic idea" [ZE, p.
153-154].

(+++) The infinite is neither physically nor psychologically given to
us in the real world, it has to be comprehended and 'set' as an idea
in the Platonic sense" [ZE, p. 154].

(§) There is no possibility of introducing something absolutely
uncountable, but by a pure dogma [S, p. 48].

(Translations of German texts by me.)

BP = Bernard Bolzano: Paradoxien des Unendlichen, Reclam, Leipzig,
1851.
BG = J. BERG (ed.): Bernard Bolzano, Einleitung zur Grössenlehre,
Friedrich Frommann Verlag, Stuttgart (1975)
C = Georg Cantor, Gesammelte Abhandlungen, E. Zermelo (ed), Spinger,
Berlin 1932.
L = Detlef Laugwitz: Zahlen und Kontinuum, BI, Zürich 1986.
S = Thoralf A. Skolem: Ueber einige Grundlagenfragen der Mathematik.
Skrifter utgit av Det Norske Videnskaps-Akademi i Oslo I. Matematisk-
naturvidenskabelig klasse, 4, 1-49.
ZE = Heinz-Dieter Ebbinghaus: Ernst Zermelo, Springer, Berlin
Heidelberg 2007. (in English)
Z = Ernst Zermelo: Untersuchungen über die Grundlagen der Mengenlehre
I, Math. Ann. 65, 261-281.


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