It's hard to know where to begin. In particular, you are trying to use your experience with finite numbers to decide what "must" or "ought" to be true with infinite sets. That way lies multiple inconsistencies, and simple proofs that 1=2, for example.

You can't "cancel out" things when you include infinities. One thing that we've learned is that "division" has to be an "undoing of multiplication" in order to get consistency. This is why we can't divide by zero, because once we've multiplied the number line by zero we've collapsed it to a single point, and there's no way to undo that.

If you're just dealing with the sizes of sets then the only way to compare sizes in general is to try to pair them up. This seems reasonable, and then you find that the contents of a bag with two copies of the natural numbers can exactly be paired up with the contents of a bag with 3 copies of the natural numbers. That seems to suggest that 2 * oo = 3 * oo. You now have a choice - accept that, and its consequences, or go with your intuition that these things "ought not be equal."

And you really do have a choice here. The thing is that over 130 years of mathematics has shown that the "equality of size means pairable" approach has no inconsistencies, and leads to useful results.

So when you say this:

If my definition of infinity is the set of all integers,
why isn't this true -

(2 * infinity) / (3 * infinity) = 2/3

Even if the definition is something else, it's the same set,
and it should get canceled out.

Well, firstly, we have a fair idea what it means to divide 2 by 3. We also derive the rules of arithmetic and algebra (things like "cancelling") from those meanings. but why should those processes carry over to the infinite case? You would need first to define what you mean by dividing infinite sets.

Now, you have the choice of trying to make that work, but there are problems. you want to define the LHS to be the RHS, but you'll need to tell me why sets that can be paired off exactly, with none left over on either side, should not be thought of as the same size.

> For Cantor, it simply must be accepted that the inner
> and outer circles have the same number of points — our
> intuition that the outer circle has more points, according
> to Cantor, is just wrong.

We know that the outer circle circumference is larger than
the inner circle circumference. We can even find the ratio
very easily.

Yes

How do you transition from that - to counting infinitely
small dots on those circles and saying they have the same
number of those dots? I don't see the connection from that
exercise to the mathematical rules.

I can pair them off, so that every point on the inside circle gets associated with one and only one point on the outside circle. Why should I not then declare these two collections to be the same size?

As I said, there is a choice, and this is the one that has proven to be more fruitful, and lead to no inconsistencies (yet).

Normal logic dictates that there should be more real
numbers between 1 and 10 than there are between 1 and 2.

Why? Consider this function:

f : x -> 1 + 9 * (x-1)

This function exactly pairs off the numbers in one interval with the numbers in the other. Given I can do that, why should I accept your intuition that these collections are of different sizes?

Because 1-10 contains all the numbers that are in 1-2,
and then some!

That's finite thinking. Yes, you can choose to try to develop an arithmetic that builds on that experience and that intuition, and you'll get an entirely different transfinite arithmetic, but you'll have to work very hard to avoid many

But that's not true!?

Define "True."

Mathematics is full of conventions, some of which seem natural, some seem natural after you've worked with them for long enough, and some seem inevitable. But your intuition can lead you astray, and saying that something must be true based purely on your direct experience of a few small numbers runs the risk of simply being wrong.

Added in edit: You can define oo to be a "number" and then work with it as a number, and you can define your rules, but then you need 1/oo, and that's an infintesimal. Then you get an entire copy of the reals at every point of the reals, and there are "numbers" that are bigger than 0, but smaller than every positive number. Personally, I found non-standard analysis an interesting diversion, but eventually uninteresting and unengaging.