Branch of math that deals directly with infinity?

[greysky]

Is there a branch of mathematics that deals directly with infinity? For
example, calculus deals with quantities as they approach infinity, it
fails when a quantity is not merely approaching infinity, but already at
infinity... is there any branch of maths that does this?

[Ross A. Finlayson]

It might not be so available directly to the senses,
but mathematical infinity has been studied a long time.

Some have that real analysis is all about the study
of infinity. This is where the "continuum" of the
"continuum mechanics" (also known as infinitesimal
analysis, the integral calculus, real analysis) is
related to the discrete in terms of the infinite.

Usual regular foundations of modern mathematics
discuss the trans-finite, or trans-finite cardinals
vis-a-vis trans-finite ordinals.

It's "standard" that systems of real analysis have
the "finite yet unbounded" vis-a-vis, the "true
infinite", or a la the "potential" and "actual"
of the infinite. Then, various ideas of the
"non-standard" include the infinite and infinitesimal
as about and around the finite and discrete.

These include "points at infinity" and other usual
considerations deducing the properties of the infinite
(or the existence of the infinite) from the unboundedness
and dispersion of the finite.

So, yeah, there are lots of branches of mathematics
where infinity (or infinitesimals, or a continuum)
play a central or remote part.

[Bill]

I have found the most-thoughtful treatments of "infinity", something
beyond a mere ordinal, to be found in projective geometry. This was
suggested by "points at infinity" by another poster.

[William Elliot]

Look up transfinite numbers. There are two kinds, ordinal and
cardinal. In addition to those, there's the oo of calculus and the
point at infinity of projective geometry (which also has a line at
infinity).. In complex analysis, oo is the point at infinity of the
complex plain. This oo point stuff has been generalized to be the
point of a one point compactification.

[whei...@corunduminium.com]

There is an area of mathematics that deals with not only what it means to be infinite but also different sizes of infinity (for example, in a very understandable way one can show the number of reals is "greater than" the number of rationals).

I am guessing you are an undergraduate level math person, so please forgive me if I am wrong. If you know some topology, then you may be familiar with the "point at infinity" inserted to create "one point compactification" of manifolds such as the flat Earth morphed into a sphere.

Look up the theory of cardinal or ordinal numbers to get an idea. The infinite hotel problem gives a basic treatment in an understandable way - see if you can find it. This stuff will either make perfect sense or drive you nuts.

Unfortunately we can't do experiments to verify the conclusions, but that is what separates math from many of the other sciences.

Even in calculus, you don't approach infinity in the sense that you get close to it. The idea of such limits is that things settle down if you get far enough from the origin; for example [sin x]/x approaches 0 as x approaches infinity.

I hope these remarks are helpful. At one time I had the same question, and we are not alone!

[FredJeffries]

On Friday, March 3, 2017 at 7:39:48 PM UTC-8, greysky wrote:

Short answer: No, because there is no one single concept of "infinity" in mathematics.

Others have already given you good pointers, but you might find interesting this classic post by Professor Zdislav V. Kovarik wherein he gives "a long list of 'infinities' (with no claim to exhaustiveness)":

https://groups.google.com/forum/#!original/sci.math/wM6drf_B0Po/QO6yQP2avs8J

[John Gabriel]

That branch of math is called IDIOCY.

http://thenewcalculus.weebly.com

[burs...@gmail.com]

By IDIOCY you mean you are expert in this branch, with
your fake new calculus brid brain John Gabriel birdbrains?

[Ross A. Finlayson]

Ideas like the supertask involve where there's an index
of an event as of the naturals or ordinals, that then
actually "is" infinity when the event is reached. There
are examples including simple motion, where Zeno's Arrow
only gets half-way to its target, then half-again, and
so on, ad infinitum, that it reaches its target, and not
"only" "as close you'd like" as of the unbounded as converges
to a limit.

"Scalar" infinities and infinitesimals basically extend or
refine Newton's notions of fluents and fluxions, eg Sergeyev's.

Ehrlich has an extended survey of infinities in mathematics.

http://www.ohio.edu/people/ehrlich/

The "long line" of du Bois-Reymond reflects another reckoning
of properties of infinity, or continuity.

That there's a mathematical theory has that all these infinities
in their correctness or mututal or co-consistency have a place in
it, or "Hilbert's Infinite Living Museum".

Then, a theory would also include a continuum and continua, that
structurally all the models exist. Continuity these days in the
modern mathematics is often Dedekind's (or the equivalent, "field"
continuity), other examples include Brouwer's, then there's mine,
including a "line continuity", "field continuity", and a "signal
continuity", as formalisms as reflect the strong property of
continuity. In this manner then the points of geometry (which
are also all part of a theory of mathematics) are equipped with
properties as of their being individua of continua.

[Chris M. Thomasson]

Fractals are one possibly answer. Humm...

How many complex roots does the following equation have:

z^infinity + c

z and c are complex numbers...

[Me]

On Saturday, March 4, 2017 at 4:39:48 AM UTC+1, greysky wrote:

> Is there a branch of mathematics that deals directly with infinity?

Yes, it's called /set theory/.

> For example, calculus deals with quantities as they approach infinity, it
> fails when a quantity is not merely approaching infinity,

Right.

> but already at infinity...

So to say.

> is there any branch of maths that does this?

Yes, set theory. Actually, CANTOR the inventor of set theory called it "transfinite set theory".

"So, the essence of set theory is the study of infinite sets, and therefore it can be defined as the mathematical theory of the actual—as opposed to potential—infinite."

https://plato.stanford.edu/entries/set-theory/

[burs...@gmail.com]

Yep, he seems to have been a forerunner:

"Yet essentially in the middle of this period of
general exorcism of the infinite and infinitesimal,
Du Bois Reymond came up with a rigorous notion
leading to infinities and infinitesimals."

On the infinitary pantachie of Du Bois Reymond
Vladimir Kanovei, Vassily Lyubetsky - 2011

[burs...@gmail.com]

Gödel showed also some interest in pantachie, when
corresponding with Ulam, but then Gödel had a blind
spot about Ulams note in the Scottish Book, asking
about an inifinite (sic!) game:

Modifications of Mazur's Game:
1) Ulam:
There is a given set of real numbers E. Players
A and B give in turn the digits 0 or 1. A wins if
the number formed by these digits in a given
order (in binary system) belongs to E. For which E
does exist a method of win for player A (player B)?

http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka1pol.pdf
(PDF page 31)

[Ross A. Finlayson]

Some might read from that, and a knowledge that the
linear continuum of real numbers is complete, that
where there are infinitesimals among them, that they
are of them, the real numbers.

This notion of line continuity is a simple enough
constant monotone infinitesimal increasing course-
of-passage between zero and one as quite intuitive
(and formally sound as establishing another model
of the real numbers than modern mathematics' abstract
algebras equivalency classes of sequences that are
Cauchy as describe the real complete ordered field).



Theoretical mathematical physics and its mathematics
illustrate other relevant placeholders or frameworks
as about the point, local, global, and total.

Cantor, who's famous for set theory, reminds also of
the Absolute, the most infinitely grand as of where
mathematics ends and philosophy remains. This is
also usually "The Continuum" vis-a-vis a continuum,
as segment or ray or line or other connected region.


Understanding why there is and isn't a difference
between Nothing and Everything is much the same notion
as the difference between the One and the Infinite.