Infinitesimal Arithmetic

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Topic:	Science > Physics
User:	"Lester Zick"
Date:	08 May 2007 11:28:48 AM
Object:	Infinitesimal Arithmetic

Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
.

User: "MoeBlee"
Title: Re: Infinitesimal Arithmetic	26 May 2007 04:16:58 PM

On May 26, 1:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

And in that regard, even as you must
recognize by your own admission that you don't understand the first
chapter of Robinsion's book (in which he summarizes the set theory and
mathematical logic that the book depends on), you need to learn the
basics before taking on such advanced work as Robinson's non-standard
analysis.

P.S.
Here it is, from page 6:
"In this chapter we introduce the reader to some formalisms whose
appreciation is essential for the sequel. The text is written in such
a way that it can be understood by anyone with a rudimentary knowledge
of Mathematical Logic and Abstract Set Theory."
And that rudimentary knowledge, which is PREQUISITE to even
understanding the chapter of the book that itself gives the
prequisites of mathematical logic for the rest of the book, is just
the rudimentary knowledge you do not have. Moreover, Robinson himself
is being too generous as to readers' abilities, as even that chapter
can be some pretty tough going for one who has only a rudimentary
knowledge of mathematical logic and set theory.
So it is indeed a source of wonder to me why you think you are in any
position to continually spout about Robinson when you haven't even
learned a mere whisp of the rudiments required to even barely keep up
with the rudiments in his own chapter on the prequisites for the rest
of the book.
MoeBlee
.

User: "MoeBlee"
Title: Re: Infinitesimal Arithmetic	26 May 2007 09:04:37 PM

On May 26, 1:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

Aside from Robinson's own particular *N, we can look at this even more
simply by this illustration:

First order PA has its standard model. The universe of that model is
w, which is denumerable. But any first order theory that has a model
with a denumerable universe also has models with uncountable
universes.

P.S. I want to emphasize that I am not claiming this is the ordinary
route to a non-standard model (the route may instead be the
compactness theorem or ultrafilters or something else), and also that
there are non-standard models of PA that have a DENUMERABLE universe.
So, my purpose was just to as quickly and easily illustrate the more
general idea of 'infinite members' not being in the sense of
cardinality but rather in the sense of members of the universe that
exceed by a certain ordering (not a cardinality ordering) all of the
members of the "core" of the universe.
It occurred to me that 'Computability And Logic' by Boolos, et. al has
an excellent discussion of non-standard models. Their explanations are
nice and clear. And it may be helpful to keep in mind that Robinson's
own approach is probably not among the more streamlined and easier to
grasp approaches as others have come after his original work.
Moreover, perhaps the piece de resistance is the proof by Kanovei and
Shelah of a DEFINABLE non-standard model of the reals; so that we can
name a particular non-standard model for the reals. When I catch up in
my studies to do some systematic study of non-standard analysis, it
will be the Kanovei and Shelah paper that I will be aiming at.
MoeBlee
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 11:18:44 AM

MoeBlee wrote:

On May 26, 1:44 pm, MoeBlee <jazzm...@hotmail.com> wrote:

P.S. I want to emphasize that I am not claiming this is the ordinary
route to a non-standard model (the route may instead be the
compactness theorem or ultrafilters or something else), and also that
there are non-standard models of PA that have a DENUMERABLE universe.
So, my purpose was just to as quickly and easily illustrate the more
general idea of 'infinite members' not being in the sense of
cardinality but rather in the sense of members of the universe that
exceed by a certain ordering (not a cardinality ordering) all of the
members of the "core" of the universe.

It occurred to me that 'Computability And Logic' by Boolos, et. al has
an excellent discussion of non-standard models. Their explanations are
nice and clear. And it may be helpful to keep in mind that Robinson's
own approach is probably not among the more streamlined and easier to
grasp approaches as others have come after his original work.
Moreover, perhaps the piece de resistance is the proof by Kanovei and
Shelah of a DEFINABLE non-standard model of the reals; so that we can
name a particular non-standard model for the reals. When I catch up in
my studies to do some systematic study of non-standard analysis, it
will be the Kanovei and Shelah paper that I will be aiming at.

MoeBlee

Yes, I started to read that paper, but I think I need to study up on
ultrafilters before I can grasp what they're doing there. MoeBlee, I
want to assure you that I understand the difference between infinite
cardinality and infinite non-standard reals, okay? I get that. I would
prefer it if cardinality did not co-opt the word "infinity", as it is a
term which may refer to other concepts besides bijection with a proper
subset. That's just a matter of terminology, and not mathematically
important, but calling such cardinalities "transfinite" would avoid some
confusion perhaps, and avoiding calling transfinite cardinalities the
"sizes" of sets would probably cause less adverse reaction to the
subject, at least on my part. No big deal.
http://shelah.logic.at/files/825.pdf
TOEknee
.

User: "Aatu Koskensilta"
Title: Re: Infinitesimal Arithmetic	27 May 2007 09:13:23 AM

On 2007-05-27, in sci.logic, MoeBlee wrote:

This much follows from the incompleteness theorem. Interestingly, Godel, in
his review of Skolem's paper in which the construction of a non-standard
model of true arithmetic occurs, quite misses the importance of Skolem's
construction, saying, incorrectly, that Skolem's result is a trivial
consequence of incompleteness; of course it does not follow from the
incompleteness theorem that there is a countable model elementarily equivalent
but not isomorphic to the naturals. Also curiously, it was apparently
realized only in the fifties that compactness easily yields the result --
the imporance of compactness in general was, it seems, only realized quite
some time after the completeness theorem.

Moreover, perhaps the piece de resistance is the proof by Kanovei and
Shelah of a DEFINABLE non-standard model of the reals; so that we can
name a particular non-standard model for the reals. When I catch up in
my studies to do some systematic study of non-standard analysis, it
will be the Kanovei and Shelah paper that I will be aiming at.

It is not the existence of a definable non-standard model of the reals that's
important, but the existence of an omega-saturated definable non-standard
elementary extension of the reals. This is a much stronger claim. Happily,
the Kanovei-Shelah paper is quite readable, untypically for Shelah, who
favours a brute-force approach, brutally manhandling mathematical structures
with powerful mathematical devices, in a way often all but incomprehensible
to lesser mortals.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

User: "Herman Jurjus"
Title: Re: Infinitesimal Arithmetic	29 May 2007 04:15:42 AM

Aatu Koskensilta wrote:

On 2007-05-27, in sci.logic, MoeBlee wrote:

[snip]

(...) Also curiously, it was apparently
realized only in the fifties that compactness easily yields the result --
the imporance of compactness in general was, it seems, only realized quite
some time after the completeness theorem.

Perhaps the appearance of the first Henkin-style completeness proof
(1951) had something to do with it?
--
Cheers,
Herman Jurjus
.

User: "Aatu Koskensilta"
Title: Re: Infinitesimal Arithmetic	29 May 2007 04:37:18 AM

On 2007-05-29, in sci.logic, Herman Jurjus wrote:

Perhaps the appearance of the first Henkin-style completeness proof
(1951) had something to do with it?

Possibly. But where do you get the date 1951? Henkin's disseration is from
1947, and the paper on completeness in JSL from 1949 and the one on extension
to type theory from 1950.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

User: "Herman Jurjus"
Title: Re: Infinitesimal Arithmetic	29 May 2007 12:59:07 PM

Aatu Koskensilta wrote:

On 2007-05-29, in sci.logic, Herman Jurjus wrote:

Perhaps the appearance of the first Henkin-style completeness proof
(1951) had something to do with it?

Possibly. But where do you get the date 1951? Henkin's disseration is from
1947, and the paper on completeness in JSL from 1949 and the one on extension
to type theory from 1950.

You appear to be right. No idea where i got that 1951 from (has been
hanging around in my memory for years, so it must come from somewhere).
--
Cheers,
Herman Jurjus
.

User: "MoeBlee"
Title: Re: Infinitesimal Arithmetic	25 May 2007 12:06:18 PM

On May 25, 7:41 am, Tony Orlow <t...@lightlink.com> wrote:

MoeBlee wrote:

On May 24, 6:53 pm, Tony Orlow <t...@lightlink.com> wrote:

I
think that Robinson, while taking things from about the opposite
direction, essentially satisfies my intuitions, and that
transfinitological omegaulation really doesn't.

That's ridiculous for you to say as I've explained to you so many
times already. Robinson's mathematics is in classical mathematical
logic and set theory incuding choice principles. He even gives the
ORDER TYPE of the set of non-standard naturals, which is a TRANSFINITE
order type.

Yes, he appeases standard set theory,

He doesn't just "appease" set theory. He works IN set theory. His
results are BASED on set theory and mathematical logic. The GOAL of
his work in non-standard analysis was to develop it in set theory and
mathematical logic.

but his approach doesn't seem to
have much to do with it,

What in the world are you talking about?! You have the book right in
front of you. Open it up and read it why don't you? The development of
the non-standard systems has everything to do with set theory and
mathematical logic.

and if it were perfectly consistent with ZFC,
then there would be a smallest infinity in NSA, which there is not.

You are being WILLFULLY IGNORANT again.
I have explained this point about smallest infinity to you over a
couple of dozen times already.
Robinson is using the word 'infinity' in a different sense from that
of cardinality when he refers to the non-standard infinite numbers. It
is clear by context when he is doing that. He's not contradicting
infinity in its cardinality sense, but rather he's allowing the
informality of using an English word to have different senses in
different contexts of his mathematical discussion.
Robinson uses infinity cardinality all over the place. I JUST
mentioned that he even gives the infinite order type (using infinite
ordinal arithmetic) of the set of non-standard naturals. Rather than
recognize that, you just go right past to repeat the same canard you
always spout. Again, I've told you this over two dozen times, when
Robinson talks about infinite non-standard numbers he's using a
DIFFERENT sense of 'infinite', which is clear to anyone with half a
brain to actually read his development of the non-standard systems.
So, again for about the hundredth time:
There is 'infinite' in the sense of cardinality (or actually in the
sense of 'not finite' or in an equivalent (with choice) sense of
Dedekind infinite). Then there is a also DIFFERENT sense of
'infinite', which is not that of cardinality but rather as to certain
kinds of orderings. It is NOT a contradiction that there is a least
infinite cardinal but not a least infinite non-standard number since
'least infinite' as far as an ordering of non-standard numbers is not
as to infinite CARDINALITY.
TWO ***DIFFERENT*** SENSE OF 'INFINITE'. NOW do you finally get it?

Can
the two theories be entirely consistent, the one asserting the existence
of a number which the other excludes?

Set theory does NOT exclude the existence of non-standard numbers. The
existence of non-standard numbers is proven IN set theory. And with
Robinson's non-standard analysis, there is not a SEPARATE theory from
set theory, but rather his results are done IN set theory and with
mathematical logic that is itself formalizable in set theory. Then,
another approach, IST, is an EXTENSION of set theory.

Saying that your intuitions are captured by Robinson's results but not
by transfinite set theory is like saying, "Darwin's explanation of
the history of the species really fits my own way of thinking, not
like that theory of evolution, which is just bunk."

Uh, no. One may accept the general theory of evolution, using Darwin's
original thoughts as a springboard, while rejecting some more specific
theories regarding the actual mechanisms at work. Personally, it seems
to me that in that naturally scientific area, almost all proposed
mechanisms postulated for evolutionary development probably apply to a
greater or lesser degree in every instance. But, that's a quite
different area of inquiry than infinite sets.

With any analogy you can find some points of dissimilarity. You're
missing the main point: Robinson's non-standard analysis is NOT even
separable from set theory. His non-standard analysis DEPENDS on and is
PART of set theory. Sure, you can investigate other foundations for
non-standard analysis (though you wouldn't do that reasonably either,
since it would mean actually studying and understanding the
mathematics, which is something you don't want to do as you prefer
instead just to spout your judgements based on the merest vague
impressions you get from reading various phrases and formulas even as
you are blithe to the specific mathematical contexts and
developments); but when you specifically talk about Robinson non-
standard analysis, then that is non-standard analysis that is done in
set theory and mathematical logic.

Or, is it? Take a human, any human. Every human has a mother, who is a
human. Mathematically, then, there is an infinite set of mothers, right?

M=mankind.
Y=you.

EYeM
AxeM E yeM: y=mother(x)
~ExeM: Y=mother(x)

Does this not define an infinite regular sequence? Is there really an
infinite set of human mothers throughout history? Just an aside....

Just irrelevent nonsense.

Really, what is wrong with you, Tony? You have the Robinson book right
in front of you. It's STEEPED in transfinite set theory.

It uses regular expressions as a starting point, extending their
applicability from the finite naturals and reals to values that
transcend the finite. It doesn't start with set theory per se, but
offers a little genuflection in that direction, not doubt for political
expediency.

You're being an utter jerk. You even said that you don't understand
the mathematical logic that is the first chapter and the basis for the
rest of the book, yet you spout nonsense that Robinson uses set theory
for "political expediency"!
The VERY FIRST SENTENCE of the book desribes his intention to develop
non-standard analysis from the basis of classical mathematics (or
however he phrased it exactly). And from there on, the book uses set
theory of the infinite, and even with choice principles applied, as
the BASIS for non-standard analysis.

We needn't argue about that.

There's nothing to argue about. You simply ignore the very basis of
the book as the author himself announces it in the very first sentence
of the book and as is manifested in page after page after page of set
theoretic proofs.

Soon, I'll be able to get back
to studying it in more detail.

First you need to learn the basic set theory and mathematical logic
that that level of a book requires.

In any case, this was a comment to
Lester, suggesting that he at least take a cursory look at Robinson and
Conway, to see if it appeals to him.

And by taking a cursory look he'd be taking no less a look than you
have even as you spout your ignorant conclusions about it.

See it. Behold it. Embrace it, as temporarily as need be. But do take a
look. Robinson had the manners to make it, but NSA is quite an
achievement. Not the way I'd do it. But, then, I'm a jnani. :)

Not the way you would do it? You are such a complete blowhard. You
have no idea what is involved in ANY way of doing it. You might as
well say, "I really like that jet airplanes get us coast to coast in
just a few hours, though they're not designed the way I would design
them. Oh, and isn't medical technology something? Ultrasound for
example, what a great concept. Though, of course, it's not designed
the way I would design it." Sheesh.

Whatever, MoeBlee. You can get as indignant and insulting

YOU are insulting when you spout your ignorance on subjects you don't
know anything about.

as you need to
make yourself feel better.

Yes, it does make me feel a little better that even though you are
free to spout ignorance on a subject you don't know anything about, I
am free in return to call you on that.

I'll try not to take it too personally.

Too bad if you don't take it personally. It certainly applies
personally.
MoeBlee
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	25 May 2007 06:04:14 PM

On Thu, 24 May 2007 21:53:42 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Is an infinitesimal different than nothing?

And what the hell might "nothing" be?

What you seem to be saying
is that an infinitesimal addition to a finite does not produce a
different finite,

I don't seem to be saying it; I am saying it.

like a finite addition to an infinite, at least an
uncountable, does not produce a different infinite.

Yeah, Tony, look when I go to the trouble of defining "infinity" as
the "number of infinitesimals" I have a very limited tolerance for
jargon on the part of people who don't know how to define infinity.

However, it may
still produce a different value, as a combination of finites, infinites,
infinitesimals..

"Value"? What the hell is a "value"? I've stated the applicable rules
for combining finites and infinitesimals on the one hand and finites
and infinites on the other. If you want to argue the issue I suggest
you draw up a different set of rules and let your imagination run.
~v~~
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	25 May 2007 06:13:27 PM

On Thu, 24 May 2007 21:53:42 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Sorry, Lester, but I have to agree that r+1=r <-> r=0. Those are
equivalent statements. I agree that there is room for infinitesimals,
but I don't agree with that axiom.

What axiom? It's simply a statement of relevant predicates, Tony. And
I don't especially care whether you agree with it or not because all
you've done so far is run up a series of trial balloons for this and
that without being able to say how any of it is supposed to happen.

It was suggested that for the set of infinitesimals I,
ieI <-> A neN i<1/n

where:
"ieI" means "I is an infinitesimal"
"<->" means "is the same as"
"A neN" means "for all natural numbers"
"i<1/n" means "that infinitesimal is less than the reciprocal of any
natural number"

Whatever, Tony.

In other words, for a number to be infinitesimal, it must be smaller
than the inverse of any standard natural. Where you have hypernaturals,
i.e. infinite values, defined, then you can define their inverses as
infinitesimals. Your dr is derived from the infinite subdivision of a
definite interval.

Yeah look, Tony, until you can explain how circular rotation actually
occurs I'm really not interested in entertaining more of your guesses.
~v~~
.

User: "Jonathan Hoyle"
Title: Re: Infinitesimal Arithmetic	25 May 2007 11:14:33 PM

Yeah look, Tony, until you can explain how circular rotation actually
occurs I'm really not interested in entertaining more of your guesses.

I wasn't aware that the occurrence of circular rotation required any
further explanation. What in particular about circular rotation
disturbs you so that doesn't seem to bother the rest of the world?
Jonathan Hoyle
Eastman Kodak
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	26 May 2007 11:18:01 AM

Lester Zick wrote:

On Thu, 24 May 2007 21:53:42 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Sorry, Lester, but I have to agree that r+1=r <-> r=0. Those are
equivalent statements. I agree that there is room for infinitesimals,
but I don't agree with that axiom.

What axiom? It's simply a statement of relevant predicates, Tony.

That's kinda what an axiom IS, Lester.
And

I don't especially care whether you agree with it or not because all
you've done so far is run up a series of trial balloons for this and
that without being able to say how any of it is supposed to happen.

Huh?

Whatever, Tony.

Yeah look, Tony, until you can explain how circular rotation actually
occurs I'm really not interested in entertaining more of your guesses.

~v~~

Circular rotation of WHAT? You want to define the circular path of a
mathematical point? Make its vertical motion proportional to its
horizontal location with respect to the center, and its horizontal
motion proportional to the negative of the vertical position. Voila! You
draw a circle. Or take any finitely moving point, and apply a finite
tansverse acceleration to it, if you prefer. If your acceleration varies
infinitesimally, though, you're going to have something infinitesimally
different from a circle.
01oo
.

User: "G. Frege nomail@invalid"
Title: Re: Infinitesimal Arithmetic	25 May 2007 06:43:58 PM

On Fri, 25 May 2007 16:13:27 -0700, Lester Zick
<dontbother@nowhere.net> wrote:
Slightly corrected Tony's presentation.

Sorry, Lester, but I have to agree that r+i=r <-> i=0. Those are
equivalent statements.

Right. Since Mr. Brainless takes "r+i=r" to be a definition if
"infinitesimal", this means that the only infinitesimal in his
"theory" (or whatever) is 0.

It was suggested that for the set of infinitesimals I,

i e I <-> An e N : 0 < |i| < 1/n.

where

"i e I" means "I is an infinitesimal"
"<->" means "if and only if"
"An e N" means "for all natural numbers"
"0 < |i| < 1/n" means "|i| is less than the reciprocal of any
natural number"

Whatever, Tony.

Yes, whatever.

In other words, for a number to be infinitesimal, it must be smaller
than the inverse of any standard natural. [And hence smaller than any
finite number.]

Right.
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	26 May 2007 11:19:49 AM

G. Frege wrote:

On Fri, 25 May 2007 16:13:27 -0700, Lester Zick
<dontbother@nowhere.net> wrote:

Slightly corrected Tony's presentation.

Sorry, Lester, but I have to agree that r+i=r <-> i=0. Those are
equivalent statements.

Right. Since Mr. Brainless takes "r+i=r" to be a definition if
"infinitesimal", this means that the only infinitesimal in his
"theory" (or whatever) is 0.

Whatever, Tony.

Yes, whatever.

I liked your definition. Did I spell it out clearly enough, do you think?

In other words, for a number to be infinitesimal, it must be smaller
than the inverse of any standard natural. [And hence smaller than any
finite number.]

Right.

I won't get my hopes up. ;)
Tony
.

User: "G. Frege nomail@invalid"
Title: Re: Infinitesimal Arithmetic	26 May 2007 11:32:37 AM

On Sat, 26 May 2007 12:19:49 -0400, Tony Orlow <tony@lightlink.com>
wrote:

It was suggested that for the set of infinitesimals I,

i e I <-> An e N : 0 < |i| < 1/n

where

"i e I" means "i is an infinitesimal"
"<->" means "if and only if"
"An e N" means "for all natural numbers n"
"0 < |i| < 1/n" means "|i| is less than the reciprocal of n
[but bigger than 0]"

Whatever, Tony.

Yes, whatever.

I liked your definition. Did I spell it out clearly enough, do you think?

Sure. Though you might avoid the reverence to the set of
infinitesimals I and just define (the predicate):
infinitesimal(x) :<-> An e N : 0 < |i| < 1/n.

In other words, for a number to be infinitesimal, it must be smaller
than the inverse of any standard natural [and hence smaller than any
finite number].

Right.

I won't get my hopes up. ;)

Right!
F.
P.S.
It's a nice exercise to use the proposed definition (from above) to
show that there are no infinitesimals in the set of real numbers R.
--
E-mail: info<at>simple-line<dot>de
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 09:55:05 AM

G. Frege wrote:

On Sat, 26 May 2007 12:19:49 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Whatever, Tony.

Yes, whatever.

I liked your definition. Did I spell it out clearly enough, do you think?

Sure. Though you might avoid the reverence to the set of
infinitesimals I and just define (the predicate):

infinitesimal(x) :<-> An e N : 0 < |i| < 1/n.

Do you think one approach is better than the other? I mean, is there
ultimately any difference between saying "x is an infinitesimal" and "x
is a member of the set of infinitesimals"? This seems to come down to
the Leibnizian concept that the identity of an object is equivalent to
the set of all predicates which apply to it. Perhaps there is a current
convention I should be aware of.

In other words, for a number to be infinitesimal, it must be smaller
than the inverse of any standard natural [and hence smaller than any
finite number].

Right.

I won't get my hopes up. ;)

Right!

F.

P.S.
It's a nice exercise to use the proposed definition (from above) to
show that there are no infinitesimals in the set of real numbers R.

Sure, there are no infinitesimals in R, and that follows from the above,
but R* can include not only infinitesimals, but specific infinite
values. In fact, the definition of a unit infinity leads directly to its
reciprocal. I think that's the better direction for the derivation.
That's the kind of model I'm comfortable with, as opposed to simply
"diverges" or "uncountable". I'd like to see more precise measure for
infinities. Of course, I'm perhaps in a minority, but at least it's not
a singleton set. ;)
Tony
.

User: "G. Frege nomail@invalid"
Title: Re: Infinitesimal Arithmetic	27 May 2007 10:19:36 AM

On Sun, 27 May 2007 10:55:05 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Whatever, Tony.

Yes, whatever.

I liked your definition. Did I spell it out clearly enough, do you think?

Sure. Though you might avoid the reverence to the set of
infinitesimals I and just define (the predicate):

infinitesimal(x) :<-> An e N : 0 < |i| < 1/n.

Do you think one approach is better than the other? I mean, is there
ultimately any difference between saying "x is an infinitesimal" and
"x is a member of the set of infinitesimals"?

Sure there is. Just look! (I'm serious.) The definition I mentioned
is "more elementary". (Actually, in this case "infinitesimal(x)" is
just an _abbreviation_ for the expression "An e N: 0 < |i| < 1/n".)
Another point (which might or might not be relevant here, but which
MIGHT be relevant in certain cases), what if the "collection" of
all infinitesimals were NOT a set, but a proper class?! --- In this
case your proposed definition would fail! (Note that I'm NOT
claiming that this actually is the case here.)
For example, consider the following definition of /set/
set(x) :<-> Ey(x e y) v x = 0
in a framework which is some variant of Z, which allows for
urelements (objects which are neither sets nor classes, etc.)
In this case there were no set V of all objects. Hence the
following "definition"
x e V <-> Ey(x e y) v x = 0 ,
where V is meant to be the set of all objects (of our theory),
would fail.

This seems to come down to the Leibnizian concept that the identity
of an object is equivalent to the set of all predicates which apply
to it.

This may or may not be the case.
On the other hand, you may consider my considerations to be
"inspired" by "Occam's razor" ("entities should not be multiplied
beyond necessity").

It's a nice exercise to use the proposed definition (from above) to
show that there are no infinitesimals in the set of real numbers R.

Sure, there are no infinitesimals in R, and that follows from the above,
but R* can include not only infinitesimals, but specific infinite
values.

Yes.

[...] I think that's the better direction for the derivation.

This may very well be the case.
F.
--
E-mail: info<at>simple-line<dot>de
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 11:51:34 AM

G. Frege wrote:

On Sun, 27 May 2007 10:55:05 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Whatever, Tony.

Yes, whatever.

I liked your definition. Did I spell it out clearly enough, do you think?

Sure. Though you might avoid the reverence to the set of
infinitesimals I and just define (the predicate):

infinitesimal(x) :<-> An e N : 0 < |i| < 1/n.

Do you think one approach is better than the other? I mean, is there
ultimately any difference between saying "x is an infinitesimal" and
"x is a member of the set of infinitesimals"?

Is it really more elementary, though, to define a new predicate, versus
a new set?

Another point (which might or might not be relevant here, but which
MIGHT be relevant in certain cases), what if the "collection" of
all infinitesimals were NOT a set, but a proper class?! --- In this
case your proposed definition would fail! (Note that I'm NOT
claiming that this actually is the case here.)

Hmmm... Unless the elements of the "set" were themselves considered
sets, leading to some kind of possible nested self-inclusion, this
wouldn't be a concern, would it?

For example, consider the following definition of /set/

set(x) :<-> Ey(x e y) v x = 0

Should this be "set(x) :<-> Ey(y e x) v x = 0"?

in a framework which is some variant of Z, which allows for
urelements (objects which are neither sets nor classes, etc.)

In this case there were no set V of all objects. Hence the
following "definition"

x e V <-> Ey(x e y) v x = 0 ,

where V is meant to be the set of all objects (of our theory),
would fail.

You refer to Russell's paradox. I can see that, but I am not sure that
applies to the set of infinitesimals on the real line, if they are
considered urelements in *R. I am sure you know more about that than I.

This seems to come down to the Leibnizian concept that the identity
of an object is equivalent to the set of all predicates which apply
to it.

This may or may not be the case.

On the other hand, you may consider my considerations to be
"inspired" by "Occam's razor" ("entities should not be multiplied
beyond necessity").

You mean we should not define more sets than are required? Does that
also apply to predicates? Somehow, they seem rather equivalent to me.

It's a nice exercise to use the proposed definition (from above) to
show that there are no infinitesimals in the set of real numbers R.

Sure, there are no infinitesimals in R, and that follows from the above,
but R* can include not only infinitesimals, but specific infinite
values.

Yes.

[...] I think that's the better direction for the derivation.

This may very well be the case.

F.

Have a good one.
T.
.

User: "G. Frege nomail@invalid"
Title: Re: Infinitesimal Arithmetic	27 May 2007 12:07:42 PM

On Sun, 27 May 2007 12:51:34 -0400, Tony Orlow <tony@lightlink.com>
wrote:

[...] The definition I mentioned is "more elementary". (Actually, in
this case "infinitesimal(x)" is just an _abbreviation_ for the ex-
pression "An e N: 0 < |i| < 1/n".)

Is it really more elementary, though, to define a new predicate, versus
a new set?

You didn't listen... ;-)
--> Actually, in this case "infinitesimal(x)" is just an
_abbreviation_ for the expression "An e N: 0 < |i| < 1/n" <--
This is VERY elementary.
Remember, I said: "...you may consider my considerations to be
'inspired' by 'Occam's razor' ("entities should not be multiplied
beyond necessity").

Hmmm... Unless the elements of the "set" were themselves considered
sets, leading to some kind of possible nested self-inclusion, this
wouldn't be a concern, would it?

I don't know, at least this is not the (potential) problem I was
referring to here.

For example, consider the following definition of /set/

set(x) :<-> Ey(x e y) v x = 0

Should this be "set(x) :<-> Ey(y e x) v x = 0"?

Right. Thanx.
Another correction:

In this case there were no set V of all sets. Hence the
following "definition" ~~~~

x e V <-> Ey(y e x) v x = 0 ,
~ ~
where V is meant to be the set of all sets (of our theory),
would fail. ~~~~

You refer to Russell's paradox.

No, not really. I'm referring to the fact that in ZFC there's no
universal set. (Though there are other set theories which would
allow for such a set.)

I can see that, but I am not sure that applies to the set of
infinitesimals on the real line [...]

Right. Hence I said (above): "Note that I'm NOT claiming that this
actually is the case here." --- But it MIGHT be the case. (Hence
your definition relies an a _proof_ that I actually is a set. While
mine doesn't.)

This seems to come down to the Leibnizian concept that the identity
of an object is equivalent to the set of all predicates which apply
to it.

This may or may not be the case.

On the other hand, you may consider my considerations to be
"inspired" by "Occam's razor" ("entities should not be multiplied
beyond necessity"). ~~~~~~~~

You mean we should not define more sets than are required?

Entities in general.

Does that also apply to predicates?

Sure.

Somehow, they seem rather equivalent to me.

They AREN'T. Now _that's_ where Russell's paradox comes in. :-)
F.
--
E-mail: info<at>simple-line<dot>de
.

User: "G. Frege nomail@invalid"
Title: Re: Infinitesimal Arithmetic	27 May 2007 10:24:10 AM

On Sun, 27 May 2007 17:19:36 +0200, G. Frege <nomail@invalid>
wrote:
Ooops... Typo! (Should read "sets" not "objects".)

For example, consider the following definition of /set/

set(x) :<-> Ey(x e y) v x = 0

in a framework which is some variant of Z, which allows for
urelements (objects which are neither sets nor classes, etc.)

In this case there were no set V of all sets. Hence the
following "definition" ~~~~

x e V <-> Ey(x e y) v x = 0 ,

where V is meant to be the set of all sets (of our theory),
would fail. ~~~~

Sorry, for that.
F.
--
E-mail: info<at>simple-line<dot>de
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 11:53:11 AM

G. Frege wrote:

On Sun, 27 May 2007 17:19:36 +0200, G. Frege <nomail@invalid>
wrote:

Ooops... Typo! (Should read "sets" not "objects".)

Sorry, for that.

F.

No problem. I think that confused me for a minute, but then I saw what
you were saying, anyway. :)
Tony
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	25 May 2007 05:57:12 PM

On Thu, 24 May 2007 21:53:42 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Well, since hyper-real arithmetic is just that: "infinitesimal
arithmetic" (or rather arithmetic allowing for "infinitesimals"),
it's hard to see what you are after here.

What I'm after is infinitesimal arithmetic and what you're after is to
insist I mean hyper real arithmetic while I would suggest you start a
thread on hyper real arithmetic if that's what interests you instead
of infinitesimal arithmetic which is what interests me but not you.

Oh, Lester, honestly, now.

As opposed to the usual dishonesty, Tony?

If the only known well-defined infinitesimals
in mathematics are those defined by Robinson in NSA, or by Conway as
surreals,

"If this, if that". Why don't you grow up for a change and start
trying to think about "what is" and "what isn't" instead of these
sophomoric fantasies regarding "maybe this" and "maybe that"?

then you can't expect to have a fruitful conversation without
at least trying to learn whether one of those formulations satisfies the
intuitions that feel offended by whatever it is you're reacting to.

Why should I worry about surreals and hyperreals when I've already got
tangential v=dr/dt and transverse acceleration a=dv/dt to explain
circular rotation together with a bunch of modern mathematikers who
can't explain circular rotation but can hypothesize a bunch of points
on a real number line which doesn't even exist to approximate it.

I
think that Robinson, while taking things from about the opposite
direction, essentially satisfies my intuitions, and that
transfinitological omegaulation really doesn't. In any case, it would be
helpful if you could state as clearly as possible what your root
assumptions are, and then we can try to see where they lead.

You should know better than anyone that I've already clearly stated
the one assumption I make and have demonstrated the truth of that
assumption in terms of itself ad nauseum and have no aspiration to
explain the obvious to mathematikers who are too lazy or stupid to
think for themselves in terms of anything but their assumptions of
truth and their idiotic guesses about their assumptions of truth.
~v~~
.

User: "Jonathan Hoyle"
Title: Re: Infinitesimal Arithmetic	25 May 2007 11:25:47 PM

On May 25, 6:57 pm, Lester Zick <dontbot...@nowhere.net> wrote:

Why should I worry about surreals and hyperreals when I've already got
tangential v=dr/dt and transverse acceleration a=dv/dt to explain
circular rotation

That's the problem, Lester: you don't already have them. You only
think you do.
Of course, if you are willing to accept your argument and believe that
the rest of the mathematical, logical and physical communities are all
in error, why do you post this here? Even more to the point: why do
you think that people who actually understand these things disagree
with you? Are you not willing to acknowledge even the possibility
that you are in error and that people with years of training in these
areas might possibly know what they are talking about? If you are not
so willing to consider that, does it really surprise you that you
would be considered a crank by virtually all your readers here?
Please take some time to consider these questions before responding,
so that you don't give flaming knee-jerk response.
Regards,
Jonathan Hoyle
Eastman Kodak
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	26 May 2007 03:18:39 PM

On 25 May 2007 21:25:47 -0700, Jonathan Hoyle <jonhoyle@mac.com>
wrote:

On May 25, 6:57 pm, Lester Zick <dontbot...@nowhere.net> wrote:

Why should I worry about surreals and hyperreals when I've already got
tangential v=dr/dt and transverse acceleration a=dv/dt to explain
circular rotation

That's the problem, Lester: you don't already have them. You only
think you do.

Aha! At last the big picture begins to emerge. Could you guys all just
be a bunch of curved space-time wackos? Boy are you antediluvian
anachronisms. No wonder PD is so coy about explanations for his
opinions. He ought to be embarrassed too. Talk about fossil science.
Right back to thirteenth century Europe. Who else is in the cabal?
Obviously Randy and Stephen. But they're just minor characters.
Certainly Bob Kolker, you, PD, and presumably most of the rest as
well. That explains a lot. No point arguing with faith based physics.
Just wish I'd known to begin with. No sense arguing faith with truth.
Mea culpa, mea culpa, mea maxima culpa.

Of course, if you are willing to accept your argument and believe that
the rest of the mathematical, logical and physical communities are all
in error,

Well in the first place, pixilated Johnny, "accepting" my argument and
belief that the rest of these communities are all in error is not an
argument. I spent the bulk of my time on the usenet over the last two
years trying to get these communities to formulate arguments,
alas to no avail. They all just pursue their assumptions of truth and
guesswork apace.

why do you post this here?

Where better to cast my net than among the heathen?

Even more to the point: why do
you think that people who actually understand these things disagree
with you?

It's not a huge problem in intellectual behavioral mechanics. The
behaviorists do it all the time. It's simply what you've been taught.
Your entire academic careers you've been educated, intimitdated, and
browbeaten into conformance with the academic-scholastic status quo.
If one argues a paradigm of truth in demonstrable, mechanical terms,
one can scarcely be surprised when all of those who have been trained
otherwise say it ain't so. My problem was I had no idea that modern
academic, scientific, and mathematical paradigms had no interest in
truth because historically western philosophies of science and math
have always been oriented towards truth in universal terms and have
always at least paid lip service to that ideal whether they honored it
in practice or not.

Are you not willing to acknowledge even the possibility
that you are in error

Of course. Why just the other day I posted an egregious error which I
promptly apologized for and withdrew. What you and others fail to
recognize in such contexts however are the proofs for what I contend
and the paradigm of mechanically reducible truth in exhaustive terms.
I've spent the last two and half years trying to ascertain whether I
had made some mistake in these proofs and if so how and by and large
all I've gotten are denials, denials, denials without explanations to
the point now where I consider the principle true and established. Why
even Tony can cite the basic principle at the drop of a hat even if he
has no idea what it means in practice.

and that people with years of training in these
areas might possibly know what they are talking about?

I don't doubt they do know exactly what they are talking about. They
just don't have any substantial idea what I'm talking about or whether
what they're talking about is actually true or just an educated guess.
Your own ignorance of the very idea of actual, literal truth is more
than indicative of that. How can a problematic paradigm possibly be
maintained without knowing it's true? So people with years of training
don't bother me at all when they're ignorant of what's true and how to
demonstrate that truth.

If you are not
so willing to consider that, does it really surprise you that you
would be considered a crank by virtually all your readers here?

Well actually they consider me a crackpot by virtue of the fact that I
don't agree with them not that what I say isn't true. Big deal. How
could one expect otherwise when a uniform paradigmatic basis for truth
and agreement is absent? All their wailing and gnashing of teeth
really amounts to is a lamentation to the effect I don't agree with
them and am presumptuous enough to hold truth as the only criterion of
science not contrary conventions and their approval.

Please take some time to consider these questions before responding,
so that you don't give flaming knee-jerk response.

Well you have to understand that this has been going on for the last
two and a half years and you've only just gotten here. So I've been
pretty thoroughly routinized to all the reactionary personalities and
knee jerk responses of which yours was not uncharacteristic. You just
assumed what you knew was sufficiently true as to be unassailable.
It's what distinguishes you as an empiric from me as a scientist that
your arguments rely on assumptions of truth whereas my arguments rely
on demonstrations of truth instead. My demonstrations may be incorrect
but at least they are critically reviewable demonstrations.
Now I can understand you don't like my style and don't appreciate
being tagged with all the historical acrimony associated with such
contentious subjects. But I've given up being polite to those whose
first reaction is contemptuous dismissal just because my paradigm of
science and mathematics is different from theirs and what they've been
taught which they can't demonstrate is necessarily true in universal
terms when I can and do demonstrate my paradigm true universally.
~v~~
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 10:25:21 AM

Lester Zick wrote:
<personal silliness snipped>
Of course. Why just the other day I posted an egregious error which I

promptly apologized for and withdrew. What you and others fail to
recognize in such contexts however are the proofs for what I contend
and the paradigm of mechanically reducible truth in exhaustive terms.
I've spent the last two and half years trying to ascertain whether I
had made some mistake in these proofs and if so how and by and large
all I've gotten are denials, denials, denials without explanations to
the point now where I consider the principle true and established. Why
even Tony can cite the basic principle at the drop of a hat even if he
has no idea what it means in practice.

Even Tony? Ahem! Lester, I tried to go through it step by step with you,
precisely to ascertain exactly what your proof meant, and whether the
flaw I detected was accurate, and it came down to asking you a yes-or-no
answer, which you refused to answer, repeatedly. I can recite your proof
"at the drop of a hat" because I have given it some thought. I don't see
that effort reciprocated. So, I rather have my doubts that you have any
actual interest in "whether [you] had made some mistake in these
proofs". You raise interesting topics at times, but you don't actually
seem to want to discuss them, but rather, as I've said, to simply "stir
the pot".

and that people with years of training in these
areas might possibly know what they are talking about?

You talk about science in your usual vague terms. Why don't you try
defining exactly what you think the scientific method is? That might be
a good exercise.

If you are not
so willing to consider that, does it really surprise you that you
would be considered a crank by virtually all your readers here?

I've tried to discuss the basic mechanics of truth with you, but you
don't seem interested. I guess that makes ME stupid and lazy, eh?

Please take some time to consider these questions before responding,
so that you don't give flaming knee-jerk response.

Except that you don't explain yourself well enough to be critically
reviewed, and requests for clarification, as simple as a yes or no, go
unanswered. So, that kinda disingenuous, dear Lester. Sorry to be so blunt.

Now I can understand you don't like my style and don't appreciate
being tagged with all the historical acrimony associated with such
contentious subjects. But I've given up being polite to those whose
first reaction is contemptuous dismissal just because my paradigm of
science and mathematics is different from theirs and what they've been
taught which they can't demonstrate is necessarily true in universal
terms when I can and do demonstrate my paradigm true universally.

~v~~

You have yet to demonstrate that. A simple yes or no to a simple
question would be a big step for you, and clear things up for all of us.
01oo
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	27 May 2007 12:57:32 PM

On Sun, 27 May 2007 11:25:21 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Lester Zick wrote:
<personal silliness snipped>

See, Tony, this is exactly why you get "silliness" and "truth"
confused; you can't tell the difference. To you it's all alike.
~v~~
.

User: "Tony Orlow"
Title: Re: Infinitesimal Arithmetic	27 May 2007 06:12:10 PM

Lester Zick wrote:

On Sun, 27 May 2007 11:25:21 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Lester Zick wrote:
<personal silliness snipped>

See, Tony, this is exactly why you get "silliness" and "truth"
confused; you can't tell the difference. To you it's all alike.

~v~~

What I snipped were a bunch of personal aspersions and irrelevant
banter. You can look back, if you like, and see whether you think I
snipped anything particular that was germane to the discussion.
01oo
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	27 May 2007 01:59:21 PM

On Sun, 27 May 2007 11:25:21 -0400, Tony Orlow <tony@lightlink.com>
wrote:

You have yet to demonstrate that. A simple yes or no to a simple
question would be a big step for you, and clear things up for all of us.

A simple yes or no to your simplistic assumptions of truth wouldn't
clear anything up for anybody, Tony. And don't partronize me by trying
to tell me what a big step would be for me when you just do your own
assumptions of truth by giant leaps and bounds without any
demonstration or even attempts at demonstrations of truth.
~v~~
.

User: "Lester Zick"
Title: Re: Infinitesimal Arithmetic	27 May 2007 01:39:02 PM

On Sun, 27 May 2007 11:25:21 -0400, Tony Orlow <tony@lightlink.com>
wrote:

Even Tony? Ahem! Lester, I tried to go through it step by step with you,

And failed precisely because in the case of conjunctions you insisted
on assuming I had assumed an "or" relation between objects without
being able to show where the "or" relation you claimed I had assumed
was. In point of fact all I did was assume different objects, which I
needn't even assume, and point out the consequences of compounding and
applying various sequences of "not" in succession without further
assumptions, and investigating the results.

precisely to ascertain exactly what your proof meant, and whether the
flaw I detected was accurate, and it came down to asking you a yes-or-no
answer, which you refused to answer, repeatedly.

I routinely refuse to answer pejorative yes-no questions posed in
prejudicial terms of your own underlying assumptions as to what
constitutes truth when you can't demonstrate the truth of your own
assumptions. If you can't understand why the reason is because it is
the demonstration of truth versus the assumption of truth which is at
issue. So leave truth alone and move on to other assumptions of truth.

I can recite your proof

"at the drop of a hat" because I have given it some thought.

That I can believe.

I don't see
that effort reciprocated.

You don't see any effort on my part for analyzing your assumptions of
truth? Gee. I wonder why. Maybe it's because you can't demonstrate
your assumptions of truth whereas I can demonstrate mine and your
assumptions of truth don't interest me anymore than anyone elses do.

So, I rather have my doubts that you have any
actual interest in "whether [you] had made some mistake in these
proofs".

You are welcome to your doubts. You are also welcome to my doubts
regarding your ability to analyze the demonstration of truth without
the ability to assume your own assumptions of truth. If you want to
assume the demonstration of truth by assumption of truth go right
ahead. Just don't try to tell me it has anything to do with the true
demonstration of truth. Calling things "truth values" doesn't have
anything to do with demonstrations of truth and whether they're true.

You raise interesting topics at times, but you don't actually
seem to want to discuss them, but rather, as I've said, to simply "stir
the pot".

I certainly don't want to discuss them in terms of your assumptions of
truth which you can't demonstrate true when I can demonstrate the
truth of my assumptions and do so in rigorously reduced exhaustive
terms. You don't seem able to grasp the nature of the problem in terms
of technique. I make the single assumption "not" and ask what the
implications are of its compounding in various ways: the universal
truth of "not" because "not not" is self contradictory, the sequential
and parallel compounding of parametric "not( )" to produce boolean
conjunctions without the use of other conjunctions, and so on.
Then you come along and say "well let's just analyze the demonstration
in terms of truth tables, conjunctions, binary logic, and so forth" as
if your calling such things "true" and "truth" or "logic" had anything
with whether any of these things were actually demonstrably true or
just further systematized formalisms having nothing to do with actual
demonstrations of truth in fully reduced mechanical terms.
~v~~
.

User: "Tony Orlow"