But if infinity will be eliminate, numbers will be infinite or not?

If we will find the largest number we can  currently store in a computer, we can plus 1 and we will have Another largest number.

Proper mathematical meaning? It stands for nothing real, therefore it is an abstraction. A computer cannot hold it. It does not fit their idea.

I agree with you. I see infinity like a way to define our universe. As I said before if we find the biggest number in the world we can add one and we will have a new number, bigger than the previous number. So we can go on we will never find a final result, but we can't prove that infinity exists because we will never reach infinity, in each field of science. We can only theorize his existence, But I think that infinity is a fundamental concept for understand mathematics.

hi,

There have been several discussion threads around the subject of infinity, sum to infinity and whether 0.99999 recurring = 1.

I think these all miss the point.

There are many branches of mathematics; no one knows them all; they are not all consistent with each other.

What makes a branch of mathematics is a set of consistent axioms and the theorems that can be proved from these.  Let me give an example first of an inconsistent set.

axiom 1.  a + a = b
axiom 2.  a + a = c
axiom 3.     b ≠ c

Straight away you can see these are inconsistent.  Any two are OK.  But add the third and it all goes horribly wrong.

Anyone may make up a new branch of mathematics by stating the axioms.

Others may criticise if they can find an inconsistency or of they find a flaw with a proof but that's the extent of their options.  You don't have to use the theory; you can ignore it.

Now there is a perfectly consistent set of axioms for allowing the concept of infinity into number theory.  With these added, lots of new, and useful things can be done.  (calculus for instance)  Or you can leave infinity out and limit your maths to what can be done without it.  It's a free choice.

You can, if you want, limit yourself to {counting numbers} with + and x.  Plenty of useful things can be done with just these and some people get by for their whole lives with just that.  Or you can add new axioms as you find you need them.   Lots of maths courses leave out complex number theory and you just have to say that some quadratics have no real solution. 

So no one has the power to un-invent infinity; as long as one person (**) wants to keep it, that's enough.

** That'll be me. 


Bob

bob bundy wrote:

axiom 1.  a + a = b
axiom 2.  a + a = c
axiom 3.     b ≠ c

I don't wish to be a stick in the mud, but that set of axioms is not incosistent, because you haven't defined + to be either commutative or transitive.

What substitution?

Hi Bob;

I think you see the problem. You defined a system consisting of only 3 axioms and people are already questioning its validity. Imagine mathematics with its thousands of theorems and axioms.

Stefy wrote:

I am questioning its invalidity.

So you think I've made a valid set of axioms. hhhhmmmmm!

Bob

Hmm, the issue is, I don't think it's impossible to...officially take it out.  There will always be those who use it, and it would likely be remembered throughout history, maybe even added again, but that doesn't mean it can't be officially taken out either.  It is why I argue for its case; I definitely see potential in it if it doesn't already have anything that's really necessary.  But arguably to that point, I'm sure you could argue quite a lot more isn't absolutely necessary, yet without it, math would in turn be much harder.  Infinity is a good idea.  It may provide confusion, especially with all the branches of math using it, some in arguably contradictory ways, but without it, I feel it would do more damage then not.

I've worked with infinity a lot so I'll chime in.  I also take issue with some of the conventional wisdom surrounding ∞ so the following is non-standard.

There are different uses of the term infinity in math.  Perhaps the most common is in reference to transcendental numbers.  Pi for example is an infinite decimal expression, but as a distance it's not ∞.  It exists somewhere between 3 and 4.  Pi is a finite distance from 0 (even if we can't quite pin down the exact distance).

When I refer to ∞ I mean the farthest possible distance from 0.

0 is the lower limit, ∞ is the upper limit. (think radial or scalar dimension)

0 * 0 = 0 (can't get any smaller)
∞ * ∞ = ∞ (can't get any larger)

Like 0, ∞ is non-polar.  +∞ and -∞ are the same thing.  If there is a way for +∞ and -∞ to be distinct, there also must be a way for +0 and -0 to be distinct (which there might well be but I'm not willing to get into that much detail just yet).

As measured from 0 to the tangent, ∞/∞ = 0 ± 1.  Details here, because words alone don't do it justice: [ http://www.perspectiveinfinity.com/root_grid.html#infdivinf ]
0 * ∞ offers a similar result but exists 90 degrees from ∞/∞ on the unit circle.  All points on the unit circle can be understood as distinct fractional representations of ±1.

In addition to 0 ± 1, ∞/∞ also = 0 ± every other natural number.  From an xy perspective when tangent is parallel with the x axis, ∞/∞ results in the entire y-axis.  Only by measuring to specific tangents can we limit the result, such as is the case with 0 ± 1.

0 is an origin.  When we count from 0, everything is relative to 0.  We don't preface every number with 0+ or 0-, even though that's exactly what we're doing.  ∞ can also be used as an origin.  Now instead of counting from 0 towards ∞ we can count from ∞ towards 0.  ∞ + 1 is a finite distance from ∞, but as far as 0 is concerned ∞ + 1 = ∞.  Different perspectives, different answers.

I refer to ∞ along with numbers relative to ∞ as "shadow numbers" since ∞ is not considered a "real number".

Also:
∞ + ∞ = ∞
∞ - ∞ = ∞
Remember, non-polar!  Like zero, infinity is timeless, non-dualistic.  Try splitting something timeless into finite parts and all we split is our perception of it.


Nothing is undefined.

I agree with anonimnystefy, infinity-infinity is undefined.