How can we think about infinity?

Question

Consider the following abstraction.

We have a skyscraper with an infinite number of floors.

The first floor contains the first type of infinity $\aleph_0$. So I guess finite numbers can live in the basement (or nowhere if we just talk about transfinite cardinals).

A horizontal shift (0, 1, 2, ...) will be grouped together on one floor. A vertical shift ($\aleph_0, \aleph_1$, ...} will correspond to moving up a floor.

The building has no ceiling, but there's an infinite number of floors, so you can't escape it. The Absolute Infinity exists outside of this building, but no matter how far up the building you go, you will never reach it.

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But then we have cases like $\aleph_\omega$ and $\aleph_{\aleph_\omega}$ and so on. Is it better to think about these different types of infinities as existing in different dimensions?

So the countable infinity $\aleph_0$ exists in the first dimension, $\aleph_1$ in the second dimension, etc.

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Is one of these abstractions preferable in thinking about infinity?

Maybe infinity is so abstract that one can never truly understand it.

Answer 1

Since you've relegated the "finite numbers" to the basement, I should mention that there are an infinite number of finite numbers. Perhaps the basement is not the best place for them.

But this comment leads to the more salient point. There are no infinite versus finite numbers because infinity isn't a number.

But yes, your infinite building is making my head spin.

Answer 2

I think you might be better served by thinking directly about ordinals. The finite ordinals are in the basement; the countable ordinals on the next floor; and so on. There are (lots and lots of) ordinals so big that they don't inject into the reals. If we let $\alpha$ be one such ordinal, then the $\alpha$th floor will not fit anywhere in our upwards-pointing hotel if we're going to hope to embed it in real space. You're going to need some new kind of "dimension" for it.