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But you can do arithmetic with uncomputable numbers. The probability of a the universal machine not halting on a random input is 1 - Ω. There, I just subtracted
by tromp 5y ago
But you can do arithmetic with uncomputable numbers. The probability of a the universal machine not halting on a random input is 1 - Ω.
There, I just subtracted an uncomputable number.
As you note, you cannot "know its digits", which is just rephrasing the "uncomputable" aspect of these numbers.
- orlp 5y agoI disagree that you have done arithmetic here. You have constructed an expression that describes arithmetic. Would you be satisfied if you entered 3 + 8 into a calculator it would simply give you back 3 + 8? Would you say this calculator has performed arithmetic? No. You described to it the arithmetic it must perform, and it failed to do so. In mathematics we almost exclusively work with the descriptions (expressions), using the rules for simplifying and manipulating those descriptions (algebra). But the foundation underlying the algebra is (currently) the reals, which contains uncomputable objects. Ultimately we want every description of a number to map to an actual number. Why are we allowing descriptions that are not computable, and then call the result a number anyway?
- auggierose 5y agoHow about instead of number you say "computable number" and call it a day? There, solved it for you. Oh no, I didn't, because the concept of computability is also not absolutely clear. Helpful reading: "The Nature of Mathematics", in "The Fabric of Reality" by David Deutsch.