Infinite counting numbers

In this discussion we explored the possibility of counting to and beyond infinity. Although it sounds like the realm of pseudoscience, our discussion actually touched on a fundamental part of set theory and mathematical analysis.

We began by wondering what would happen if a pez dispenser could keep dispensing pez forever. We took this as our representation of infinity because there are infinitely many pez. This quickly lead to the suggestion of a pez vending machine, which could keep dispensing pez dispensers forever. That is, the pez vending machine represented a compound infinity.

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Once motivated, here is how we began learning. Let’s denote a single unit by a * symbol. Thus the number five can be represented *****. The let’s denote ****…., that is, *’s forever, by the symbol ^. We then discovered:

Absorption law *^ = ^

To practice using the law, simplify the following:
*^**
**^^**
**^^*^**
^^^^^***
**^^**^^**^^
**^*^***^^^^***^*^*^*
What observations do you make?

Next we went to compound infinities, so let % denote ^’s forever.

Second absorption law: ^% = %

Once again, practice by simplifying the following.
^^%^%^
^^^%%^^%%
***^^^%%%
*^%*^%*^%*^%*
%%%^^^***
%%%^^^***%
What observations do you make? We noticed that the two absorption laws together imply a third one:

Simplification theorem: *% = %

After this, it is clear what you want to do next. Keep making up new symbols, writing new absorption laws, and finding new absorption theorems. Then write simplification exercises and solve them! There is a pattern that lets you quickly solve every one.

Extensions

Write all your infinity symbols, in increasing order, from left to right. Make up a new symbol to stand for that! What absorption laws are true for it?

Write the biggest infinite counting number that you can. Be creative with your symbol definitions!

Is addition of infinite counting numbers commutative? (In other words, is xy always equal to yx?)

Which infinite counting numbers can be written as a sum of two other counting numbers?

Can you define multiplication for infinite counting numbers? What laws would this satisfy? What about the distributivity law?

We have considered only infinities that go to the right. What if you made symbols for strings that are infinite both on the left and on the right?

Wrap-up

The concept that we are calling an “infinite counting number” is actually called ordinal. The ordinals get very, very large, far surpassing the limits of the imagination of even the most clever mathematicians. The ordinals we have described above are all very small—so small they are called countable. It is also known that uncountable ordinals exist, though objects so large cannot be easily visualized. The study of the higher infinite is one of the most attractive and mysterious subjects of modern mathematics research!

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