User blog comment:P進大好きbot/New Issue on Traditional Analyses/@comment-34422464-20190828003038/@comment-39541634-20190828104357

@Anybody who is interested in the topic of infinity and set theory

The really mind-boggling thing is that even the "small" ordinal ω is already infinite. I'm assuming you already know that ω is bigger than every finite number, but have you ever actually stopped to think of the implications?

It leads to the following paradox:

On the one hand, since ω is larger than any finite number, it encapsules - in some sense - every finite concept within itself. Every expressible idea can be written as a finite string of symbols, and is therefore within the grasp of ω.

On the other hand, ω is the smallest infinity possible. So every time we talk about different kinds of infinities (ordinals, cardinals, infinite sets and so on) we are refering to the "world" beyond ω.

Seems like a contradiction, doesn't it? Welcome to the strange world of set theory. I can't really give a layman-friendly explanation for how both of the above paragraphs can be true(*), but suffice to say that this bizzare situation is the very soul of set theory.

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(*) In an earlier draft of this comment, I tried. I wrote something along the lines of "set theory circumnavigates the problem by modelling infinite objects with finite concrete concepts"... which is kinda close to being correct, but also appallingly wrong from a techincal view point. I'm simply at a loss of how to translate these technical concepts into non-techincal English.