Thinking about ordinals
I always found ordinal numbers pretty intuitive, but cardinals somewhat less so. It might be because I'd thought about a related concept at a pretty young age.
Things exist. That's 0. We can think about things, that's S0 = 1. We can think about thinking about things, which is SS0 = 2. We can think about thinking about thinking about things, which is SSS0 = 3. The axiom of infinity only comes in because we are capable of noticing this pattern, and thinking about, as an entity, this pattern of thinking about thinking about [...] things to any arbitrary level. This is \omega. :O Then we can think about thinking about this process, etc. I wonder if there's any connection to the term "comprehension" in set theory. Of course, the analogy doesn't go too far, but it provides some intuition, at least for me.
Things exist. That's 0. We can think about things, that's S0 = 1. We can think about thinking about things, which is SS0 = 2. We can think about thinking about thinking about things, which is SSS0 = 3. The axiom of infinity only comes in because we are capable of noticing this pattern, and thinking about, as an entity, this pattern of thinking about thinking about [...] things to any arbitrary level. This is \omega. :O Then we can think about thinking about this process, etc. I wonder if there's any connection to the term "comprehension" in set theory. Of course, the analogy doesn't go too far, but it provides some intuition, at least for me.