B Dividing by infinity, exactly, finally!

[jaketodd]

Then what would, say, $2 \cdot \infty$ or $5 \cdot \infty$ reduce to, if as you maintain, they would be different?

I think you're on a wild goose chase.

You'd have to ask the people who came up with the alternative number systems I've mentioned. I'm just concentrating on their ability to divide by infinity, mainly. I think what you're asking is kind of like asking what does 2x or 5x reduce to.

 

[jaketodd]

No, if you define infinity as your link in OP ($-\infty<a<+\infty,{}\forall a\in\mathbb{R}$), then $2\times\infty\leq\infty$ must be true by definition (if multiplication by infinity is even defined).

This one: https://en.wikipedia.org/wiki/Projectively_extended_real_line
It only adds one notion of infinity, defined, instead of two. It doesn't have +/- infinities, just one encompassing one.

 

[TeethWhitener]

This one: https://en.wikipedia.org/wiki/Projectively_extended_real_line
It only adds one notion of infinity, defined, instead of two. It doesn't have +/- infinities, just one encompassing one.

Ok so then either $2\times\infty$ is infinity or it is a finite real number. So @jbriggs444 ‘s point is still valid.

 

[FactChecker]

This one: https://en.wikipedia.org/wiki/Projectively_extended_real_line
It only adds one notion of infinity, defined, instead of two. It doesn't have +/- infinities, just one encompassing one.

Exactly. In one very important application, it is useful to think of $\infty$ as a single point. Clearly, there are other important applications where you would not want $-\infty$ and $+ \infty$ to be the same.
Infinity is complicated.

 

[jaketodd]

Ok so then either $2\times\infty$ is infinity or it is a finite real number. So @jbriggs444 ‘s point is still valid.

I think we need to study the alternative number systems more. I just know that infinity is defined in both I mention, instead of being undefined and just an idea. You say "finite real number." Both of the alternative number systems I mention are different than the real number system, as mentioned in those two Wikipedia articles. Related, but different.

 

[jaketodd]

Ok so then either $2\times\infty$ is infinity or it is a finite real number. So @jbriggs444 ‘s point is still valid.

Is 2x5 the same as 2x10? Infinity can be defined, so no.

 

[jbriggs444]

Is 2x5 the same as 2x10? Infinity can be defined, so no

But can it be defined with the properties that you wish?

That the non-standard reals contain at least one infinite element is well accepted. However, as I understand it, there is no definition available within the non-standard reals that can distinguish between "finite" and "infinite". That is, the set of infinite reals is an external set, not an internal set. So you are going to have a hard time constructing $\infty$ or usefully defining it.

Note that the non-standard reals are something completely different from the two point compactification of the real line (with exactly two infinities) or the one point compactification of the complex plane (with exactly one infinity) that were mentioned in the OP.

 

[FactChecker]

I think we need to study the alternative number systems more. I just know that infinity is defined in both I mention, instead of being undefined and just an idea. You say "finite real number." Both of the alternative number systems I mention are different than the real number system, as mentioned in those two Wikipedia articles. Related, but different.

You are correct. There are alternative number systems with definitions of $\infty$ that have been studied very thoroughly. They may not be as simple as you indicated in your original post, but they are very interesting in certain applications.

 

[TeethWhitener]

Is 2x5 the same as 2x10? Infinity can be defined, so no.

This is a non sequitur. $2\times5\neq2\times10$ regardless of whether infinity is defined.

Here’s the deal. You need to define what $2\times\infty$ means. If it’s a number in the extended real number system or the projective system, then it’s either equal to $\infty$, $-\infty$, or a real number. Otherwise you have to add $2\times\infty$ separately to the system with its own arithmetic rules. That’s fine. People have done that; they’re called transfinite ordinal numbers, and there’s a rich theory surrounding them (including an internally consistent arithmetic). The main piece of the puzzle is how an element like $2\times\infty$ fits into the order of the system. Is it greater, less than, or neither in relation to the other numbers? On the extended real line, all the numbers can be ordered from least to greatest, and every subset can as well. In the projective real line, there is no good way to order numbers, since infinity is simultaneously greater than and less than all the numbers. So in a real sense, we can’t say that $2<1$ is false in this system. This is exactly analogous to saying we can’t tell whether one point on a circle is greater or less than another point.

The point of all this is to say that every system is going to have its trade offs. As @pbuk pointed out, these systems don’t satisfy the field axioms. In particular, the infinity elements do not have a multiplicative or additive inverse.

 

[pbuk]

I think we need to study the alternative number systems more.

I think you need to study these systems more; they have already been studied extensively by mathematicians and their properties and limitations are well known as you should be able to see from the depth and consistency of the replies in this thread.

 

[DrClaude]

I think you need to study these systems more; they have already been studied extensively by mathematicians and their properties and limitations are well known as you should be able to see from the depth and consistency of the replies in this thread.

And on this note, time to close the thread. Thanks to all that have participated.