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Ok, let's take another one. Consider me to be doing fuzzy experimental math philosophy. I am not a mathematician and never want to be one so the universe won't

Ok, let's take another one. Consider me to be doing fuzzy experimental math philosophy. I am not a mathematician and never want to be one so the universe won't explode. I'm going to reject the Continuum Hypothesis, ok? (Math tells me I can take it or leave it, so I'm going to leave it.) Now I take a subset of the reals, including Q, with lower cardinality but uncountable still. So now "almost all" of reals aren't these guys? And "almost all" of these guys are transcendental or uncomputable? And I can keep injecting infinities in there like this, ok... Math doesn't tell me I can or can't do this!! So I think I can choose my intuition, and math isn't even going to tell me otherwise.

RiderOfGiraffes 16y ago
> I'm going to reject the Continuum Hypothesis, ok? OK. > Now I take a subset of the reals, including Q, > with lower cardinality but uncountable still. Call it K, so K contains Q, and |Q| < |K| < c=|R|. OK. > So now "almost all" of reals aren't these guys? Yes. Almost all the K aren't Q, and almost all the R aren't K. > And I can keep injecting infinities in there like > this, ok... Yes. > Math doesn't tell me I can or can't do this!! Correct. > So I think I can choose my intuition, and math > isn't even going to tell me otherwise. Non-sequitur. The purpose of intuition is to guide you in developing proofs or solving problems. Feel free to use your own intuition and reject that of others. If you're not going to be proving theorems or solving problems, there's nothing wrong with having an intuition at odds with everyone else's. However, as a data point, the intuition that's developed over the last two centuries or so, the one you reject, seems to work in proving theorems and solving problems. The concepts of infinity contain much that seems contradictory. Intuitions based on your finite world simply don't work. The best way to develop new ones is to work with these things, or to accept the visualisations that others provide.

RiderOfGiraffes 16y ago

> I'm going to reject the Continuum Hypothesis, ok? OK. > Now I take a subset of the reals, including Q, > with lower cardinality but uncountable still. Call it K, so K contains Q, and |Q| < |K| < c=|R|. OK. > So now "almost all" of reals aren't these guys? Yes. Almost all the K aren't Q, and almost all the R aren't K. > And I can keep injecting infinities in there like > this, ok... Yes. > Math doesn't tell me I can or can't do this!! Correct. > So I think I can choose my intuition, and math > isn't even going to tell me otherwise. Non-sequitur. The purpose of intuition is to guide you in developing proofs or solving problems. Feel free to use your own intuition and reject that of others. If you're not going to be proving theorems or solving problems, there's nothing wrong with having an intuition at odds with everyone else's. However, as a data point, the intuition that's developed over the last two centuries or so, the one you reject, seems to work in proving theorems and solving problems. The concepts of infinity contain much that seems contradictory. Intuitions based on your finite world simply don't work. The best way to develop new ones is to work with these things, or to accept the visualisations that others provide.