On May 9, 12:58 pm, Lester Zick <dontbot...@nowhere.net> wrote:

On 8 May 2007 18:50:45 -0700, "Ross A. Finlayson"



<r...@tiki-lounge.com> wrote:

On May 8, 12:17 pm, Uncle Al <Uncle...@hate.spam.net> wrote:

Lester Zick wrote:

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.

Hey Zick, perserverative boring spammer, tell us about algebras
defined by

not(not(p or q) or not(p or not(q)) = p

--
Uncle Alhttp://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)http://www.mazepath.com/uncleal/lajos.htm#a2

By infinitesimal, do you mean delta x as delta x goes to zero, dx, the
differential?

Technically I'm more interested in the definite infinitesimal integral
between 0 and dt of v=dr/dt which is the radially directed finite v
resulting from the definite infinitesimal integral of centrifugal a.
The magnitude of dr is infinitesimal so r remains finitely constant.

Then, if you sum those over the unit interval, you get 1, so, if "i"
was the differential and "I" the infinity of differentials on the unit
interval, then you might think something along the lines as the
inventors of the calculus (infinitesimal analysis) did, that Ii = 1 =
iI. (Perhaps that "I" is better thought of as the single unit scalar
infinity or as Sergeyev calls it (1), and the integral is not a sum
but a multiple, with then Lebesgue sums of the products.) Lester, do
you think Ii = 1? Then, it is not a transfinite infinity that you
consider.

Well this an interesting question, Ross, I've actually pondered. I'm
inclined to the opinion that i=r/I and I=r/i. However I'm disinclined
to make a federal case out of it at this point because the calculus
makes clear the relation between finite acceleration and velocity so
the only novel aspect of what I'm trying to emphasize is the concept
of infinitesimal definite integration which clearly shows the presence
of finite centripetal velocity and infinitesimal dr in the presence of
centripetal acceleration.

Then, in consideration of a "real number" r (read: definite real) in
sum with an infinitesimal (read: unit scalar infinitesimal), and that
the sum is not definitely different than the original real number
(definitely the same, indefinitely the next), there's a notion that
there would be I many to those to increment r by one, and some portion
of I many to increment less than one, defined in terms of again
definite real numbers on the unit interval.

I'm not sure I'm construing your remarks here correctly, Ross. This is
more than just a mathematical curiosity. It's a real problem in the
mathematical interpretation of angular mechanics and time as an
independent variable in infinitesimal contexts.

~v~~

Consider a non-self-intersecting space-filling curve, that is a curve
(in smooth infinitesimal analysis, where the circle is an infinitely-
sided regular polygon, perhaps a collection of line segments sharing
endpoints) that has as a point on the curve each point in an area,
say, the unit square or unit circle.

Those space-filling curves are generally accepted to exist, in certain
proscribed forms, as fractals, but consider something along the lines
of an analog of a line from zero to one on the one-dimensional unit
interval, that starts at zero and somehow spirals outwards thus that
it is continuously incident to itself yet fills area and forms
variously the open or closed unit disc or two-dimensional coordinate
plane for any two linearly independent coordinate axes, or polar/
circular coordinates of two variables. (The derivative is
infinitesimal, recoverable in bookkeeping.)

(Consider an entire plane or infinitesimal patch of area: is it a
disc or square or any regular polygon in between?)

That novitiates in pre-calculus recognize and may ably use delta-
epsilonics does not erase the thousands of years pre-, post-, (and
what's during, contra-?, intermedio-?) the infinitesimal analysis'
"rigorization". That is to say, the attempts to bring conceivable
notions of the real numbers that lie outside what is today the
standard are legion, because there is a perceived acknowledgment in
the academic corpi that there is more to the real numbers than their
standard treatment.

In delta-epsilonics, it is for no finite difference that true
analytical results corresponding to the obvious geometric results hold
true: it is not for any finite difference that they hold true, and
were the differences zero then there would not be non-trivial results
about the non-trivial. So, the differential is less than any definite
finite quantity, and non-zero. Then, perhaps you might consider that
delta-epsilonics provides what may be seen as a subset of Tony's
"inverse function rule", in a way.

A lot of people spend a lot of time considering the foundations of
mathematics and the perceived paradoxoi/en/es in the philosophical
mathematical logic construed to underpin it, because it's fascinating
to those with an introspective yet holistic mindset. For millenia it
is the case that some of what are perceived as the deepest thoughts
are about these things. So, when there gets to be consideration that
there couldn't be an empty set or there can't be infinity, where by
the same token by the very deep embedding of those concepts as
definitive of all in between they can't not, then, reconciliation
leads to duality, observable in many systems, and, all.

Generally where I use the word universe it means the domain of
discourse, the _collection_, in set theory a _set_, in physics the
physical universe, which includes all things. So, the null axiom
theory is the null axiom set theory, the null axiom number theory, the
null axiom geometric theory, and the null axiom physical theory.

Compare God's "it was dark, then there was light" to Lee's "from the
void springs everything." Posit: nothing, or not, ad infinitum,
itself the continuum and Ouroboros, Yggdrasil's map is a theosophical
diagram.

In theory, truth should equate to utterability (or trivially lack
thereof, with the NAT reinforcing itself), in a consistent theory:
provability.

There's only one theory with no axioms.

Ross