Possible to Multiply or Divide Infinities?

[ice109]

Huh? Almost all real numbers are irrationals and they are infinite decimals. How do you define real numbers?

And why can't you divide two infinite numbers? If you divide .6666... by .22222... would you not get .33333...? An irrational would take infinitely long to divide by another irrational, but in principle there is no reason it can't be done. Is there?

repeating decimals aren't irrational and there's a difference between a number who's magnitude is infinite and a number who's representation is infinite.

 

[tiny-tim]

So if your intuition that it makes sense to evaluate a distribution at a point... then your intuition is (probably) wrong.

Hi Hurkyl!

My intuition is evaluated as a distribution.

It is worthless almost everywhere, but becomes of value when I get to the point.

(On its own, it is meaningless, but the more convolved it gets … )

 

[Hurkyl]

and what is *? the cartesian product?

More or less. Cardinal arithmetic is defined by

$$|A| + |B| = |A \amalg B|$$
$$|A| \cdot |B| = |A \times B|$$
$$|A|^{|B|} = \left|A^B\right|$$

($\amalg$ is disjoint union)

 

[HallsofIvy]

Huh? Almost all real numbers are irrationals and they are infinite decimals. How do you define real numbers?

And why can't you divide two infinite numbers? If you divide .6666... by .22222... would you not get .33333...? An irrational would take infinitely long to divide by another irrational, but in principle there is no reason it can't be done. Is there?

The numbers you give, 0.666..., 0.222..., and 0.333... have an infinite number of decimal places. The number of decimal places is an artifact of the base 10 numeration system and not a property of the numbers themselves. They are not "infinite" themselves. There are no "infinite" real numbers whether rational or irrational.

And I know several ways of defining "real numbers"- Dedekind cuts, equivalence classes of Cauchy sequences, equivalence classes of increasing, bounded sequences, etc. In none of those are there "infinite" real numbers.

 

[Dragonfall]

The original question was about division by cardinals. I suggested using the image as division. That is, take a / b as the set {n: b * n = a} for cardinals a and b. So with that definition, $\aleph_0 / \aleph_0 = \mathbb{Z}^+\cup\aleph_0$, for example, with $\mathbb{Z}^+=\{1,2,3,\ldots\}$. An alternate definition in ZFC would take the least of these, so $\aleph_0 / \aleph_0 = 1$ in that case. Neither could 'handle' $\aleph_0 / \mathfrak{c}$; the first would give the empty set and the latter would be undefined.

Your former definition says, if I'm not mistaken, a/b = the number of ways that we can write a as b*n for some n. I don't think that this corresponds to what we mean by "division". The latter way makes more sense, in that $$\aleph_0 / \aleph_0 = 1$$. Seeing as cardinals are a sort of generalization of integers, we don't need to handle things like $\aleph_0 / \mathfrak{c}$, in the same sense that integer division does not need to handle 1/2.

 

[Dragonfall]

Also, in your former definition, 0/anything = class of all cardinals. While in the latter 0/anything = 0.