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Not true. If you think about binary representation of numbers it is clear that there are as many infinite binary sequences as there are real numbers between 0 a

Not true. If you think about binary representation of numbers it is clear that there are as many infinite binary sequences as there are real numbers between 0 and 1. So that is uncountably infinite. Alternatively, apply the usual diagonal argument directly to the set of sequences to see that there are uncountably many of them. This has not much to do with constructible numbers, which are those (complex) numbers which can be constructed using compass and straightedge constructions starting from 0 and 1.

nyrikki 3y ago
The definable numbers, no matter if you use the constructible, algebraic, or computable numbers are countable infinities, or aleph naught. Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity. I realize that those concepts are difficult to build intuitions around, but what I said was correct. definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers. You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals. The reals being uncomputable 'almost everywhere' is well proven at this point. Cantor diagonalization being the typical way of showing it. 'Almost everywhere' as used above has a formal definition: A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero. The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...
- red_trumpet 3y ago
  Sure. What I took offense at was the "which is what you are describing" part. I described uncountable infinities.
  - nyrikki 3y ago
    Sorry, the way it was described I didn't take as the power set but the constructable sets.

nyrikki 3y ago

The definable numbers, no matter if you use the constructible, algebraic, or computable numbers are countable infinities, or aleph naught. Any segment of the real line, no matter how small you make it is aleph_1 or an uncountable infinity. I realize that those concepts are difficult to build intuitions around, but what I said was correct. definable,algebraic, and computable numbers have such a tiny percentage of the real line that added together their infinity of representations are still countable and equal in size to the natural numbers. You could map them all 1 to 1 to natural numbers and they would be the same size as the natural numbers, which is tiny compared to the reals. The reals being uncomputable 'almost everywhere' is well proven at this point. Cantor diagonalization being the typical way of showing it. 'Almost everywhere' as used above has a formal definition: A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero. The reals are uncomputable 'almost everywhere', the reals are not constructable 'almost everywhere' etc...

red_trumpet 3y ago
Sure. What I took offense at was the "which is what you are describing" part. I described uncountable infinities.
- nyrikki 3y ago
  Sorry, the way it was described I didn't take as the power set but the constructable sets.