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> For example: how many real numbers are there? The continuum hypothesis proposes a conservative answer, but since this is independent of the usual axioms of se

> For example: how many real numbers are there? The continuum hypothesis proposes a conservative answer, but since this is independent of the usual axioms of set theory, the question remains open: there could be vastly more real numbers than most people think. If don't agree with that. There are exactly as many real numbers as there are inifnite sequences consisting of zeroes and on (0,0,1,0,1,...). That is the same as the set of subsets of the natural numbers. That is well-defined. The continuum hypothesis asks if there are infinities smaller than this one, but bigger than the infinity of the natural numbers.

nyrikki 3y ago
The constructable numbers, which is what you are describing are aleph naught, or countable. The Cantor set is a way to think about this.
- red_trumpet 3y ago
  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.
Kranar 3y ago
No the size of the real numbers is not well defined. To be well defined means that there is an unambiguous interpretation of a proposition. The fact that the continuum hypothesis is independent of the usual axioms of set theory means that the size of the real numbers is not well defined (within those usual axioms), it's ambiguous whether the size of the real numbers is Aleph_1, Aleph_2, etc.... Just pointing out that the size of the real numbers is equal to the number of possible combinations of an infinite sequence of 0s and 1s doesn't give the size any well defined interpretation, it's just an alternative formulation of the size. Yes there are many ways to formulate the size of the real numbers, but unless you can take one of those formulations and pin it down to a specific and unambiguous value that can be operated on, then the size of the reals remains ambiguous. It's actually really open ended how large the size of the reals are. All that's known is that it's greater than Aleph_0, and that it's not equal to any limit cardinal such as Aleph_omega. https://en.wikipedia.org/wiki/Well-defined_expression https://en.wikipedia.org/wiki/Well-defined_expression
- red_trumpet 3y ago
  Do you have a background working with cardinals? I don't, so maybe I'm missing something here. But why would you consider the reals bigger, if there exists an Aleph_1 which is in between the reals and the naturals? After all, the reals are still the same set?
  - Kranar 3y ago
    Can you rephrase your question? Why would someone consider the reals to be bigger than what, Aleph_1? I'd say the majority of mathematicians consider the reals to be bigger than Aleph_1, including both Kurt Godel (believed it to be equal to Aleph_2) and Paul Cohen. Paul Cohen gave a talk where he said he believes the continuum is something unimaginably large, in fact if we take C to be the size of the continuum, then Cohen believes C = Aleph_C. There are a variety of motivations for these reasons, but they mostly boil down to believing that the continuum is something that is unreachable and non-constructible.
    - 3y ago

nyrikki 3y ago

The constructable numbers, which is what you are describing are aleph naught, or countable. The Cantor set is a way to think about this.

red_trumpet 3y ago
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.

Kranar 3y ago

No the size of the real numbers is not well defined. To be well defined means that there is an unambiguous interpretation of a proposition. The fact that the continuum hypothesis is independent of the usual axioms of set theory means that the size of the real numbers is not well defined (within those usual axioms), it's ambiguous whether the size of the real numbers is Aleph_1, Aleph_2, etc.... Just pointing out that the size of the real numbers is equal to the number of possible combinations of an infinite sequence of 0s and 1s doesn't give the size any well defined interpretation, it's just an alternative formulation of the size. Yes there are many ways to formulate the size of the real numbers, but unless you can take one of those formulations and pin it down to a specific and unambiguous value that can be operated on, then the size of the reals remains ambiguous. It's actually really open ended how large the size of the reals are. All that's known is that it's greater than Aleph_0, and that it's not equal to any limit cardinal such as Aleph_omega. https://en.wikipedia.org/wiki/Well-defined_expression https://en.wikipedia.org/wiki/Well-defined_expression

red_trumpet 3y ago
Do you have a background working with cardinals? I don't, so maybe I'm missing something here. But why would you consider the reals bigger, if there exists an Aleph_1 which is in between the reals and the naturals? After all, the reals are still the same set?
- Kranar 3y ago
  Can you rephrase your question? Why would someone consider the reals to be bigger than what, Aleph_1? I'd say the majority of mathematicians consider the reals to be bigger than Aleph_1, including both Kurt Godel (believed it to be equal to Aleph_2) and Paul Cohen. Paul Cohen gave a talk where he said he believes the continuum is something unimaginably large, in fact if we take C to be the size of the continuum, then Cohen believes C = Aleph_C. There are a variety of motivations for these reasons, but they mostly boil down to believing that the continuum is something that is unreachable and non-constructible.
  - 3y ago