tl;dr : Maybe the reals are far more mysterious. All we ever have handle over are the countable sets. [0] Looking back, the core of my confusion was this. I h

tl;dr : Maybe the reals are far more mysterious. All we ever have handle over are the countable sets. [0] Looking back, the core of my confusion was this. I had thought: I can visualize what ℵ0 means; that’s just the infinity of integers. I can also visualize what C=2^ℵ0 means; that’s the infinity of points on a line. Those, therefore, are the two bedrocks of clarity in this discussion. By contrast, I can’t visualize a set of intermediate cardinality between ℵ0 and C. The intermediate infinity, being weird and ghostlike, is the one that shouldn’t exist unless we deliberately “force” it to. Turns out I had things backwards. For starters, I can’t visualize the uncountable infinity of real numbers. I might think I’m visualizing the real line—it’s solid, it’s black, it’s got little points everywhere—but how can I be sure that I’m not merely visualizing the ℵ0 rationals, or (say) the computable or definable reals, which include all the ones that arise in ordinary math? The continuum C is not at all the bedrock of clarity that I’d thought it was. Unlike its junior partner ℵ0, the continuum is adjustable, changeable—and we will change it when we build different models of ZFC. What’s (relatively) more “fixed” in this game is something that I, like many non-experts, had always given short shrift to: Cantor’s sequence of Alephs ℵ0, ℵ1, ℵ2, etc. [0] https://scottaaronson.blog/?p=4974 https://scottaaronson.blog/?p=4974