Earlier quoted context omitted.
But all non-computable numbers are irrational.
No, there are also non-computable numbers that are imaginary, complex, or transfinite.
All rational numbers are real. Therefore all non-real numbers are irrational.
Inventor Claims to Have Solved Floating Point Error Problem
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Re: Inventor Claims to Have Solved Floating Point Error Problem
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The easiest way to understand it is to look at the question from a philosophical framework where it makes no sense to claim that there are "more" irrationals than rationals. And then untangle why it came to a different answer. In Constructivism, all statements have 3 possible values, not 2. They are true, false, and not proven. All possible objects must have a construction. So instead of talking about a vague "Cauchy…
Excellent explanation, thanks! But I feel compelled to point out that only a small minority of working mathematicians are constructivists.
This is true.
However there is no logical argument that can disprove the constructivist view, in which there aren't "more" irrationals than rationals. And therefore the logical argument that there is must have some hidden implicit assumptions.
Re: Inventor Claims to Have Solved Floating Point Error Problem
#203Re: Inventor Claims to Have Solved Floating Point Error Problem
#204Re: Inventor Claims to Have Solved Floating Point Error Problem
#205Earlier quoted context omitted.
Uh, no, "irrational" is defined as a subset of real numbers.
Non-rational, then. This is about as consequential as debating whether 1 is a prime number.
This thread was about "all the numbers that it's theoretically impossible to compute". If you think the difference between "irrational" and "non-rational" in this context is irrelevant, you have a weak grasp of number theory. Yes, all are non-computable, but in different ways.