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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 c

by 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.
  - red_trumpet 3y ago
    A set M is said to have the same size as the reals R if there is a bijection M -> R. M has size less or equal than R (|M| <= |R|) if there is an injection M -> R, and |R|<=|M| if there is an injection R -> M. That is my working definition of "size". The continuum hypothesis asks if |R| = Aleph_1, or not? If not, then Aleph_1 < |R|, meaning there is an infinite set M with an injection M -> R, but no bijection to R or the natural numbers N (|N|=Aleph_0). But the existence or non-existence of such a set M does not at all change the size of R in my mind. Why would it?
    - Kranar 3y ago
      I suppose it's not too clear to me what you mean then, but of course this is an abstract topic so that's to be understood. A size just by itself is meaningless, if I said I have Aleph_0 units of happiness well for all you know I could be depressed, even if I said I have 2^Aleph_0 units of happiness I could still be depressed. It's only if you can compare what those units of happiness are to things you are familiar with, like how many units of happiness is eating chocolate, or being able to rest after a hard and productive day at work, that you would be able to get a sense of how happy (or depressed) I am. So sure, just knowing the size of the real numbers on its own without comparing it to anything doesn't change its significance in and of itself because it's meaningless. But if we could begin to identify certain sets that were smaller than the size of the reals (and larger than the naturals), and we could identify properties of these sets that gave us a sense of their expressive power, their reasoning power, etc... then we might start to appreciate just how large the real numbers are. We know categorically that Aleph_1 is the size of the set of the first uncountable ordinal number w_1, so any ordinal less than w_1 is countable. If the size of the continuum is Aleph_1 then that would have rather unintuitive implications involving the types of functions that are possible to define and reason about along with their relationship to probabilities and randomly choosing real numbers within the interval [0, 1] depending on whether the size of the real numbers is Aleph_1 or bigger. Probably one of the more accessible arguments for why this is is linked below [1]. [1] https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry

by Kranar 3y ago

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.
- red_trumpet 3y ago
  A set M is said to have the same size as the reals R if there is a bijection M -> R. M has size less or equal than R (|M| <= |R|) if there is an injection M -> R, and |R|<=|M| if there is an injection R -> M. That is my working definition of "size". The continuum hypothesis asks if |R| = Aleph_1, or not? If not, then Aleph_1 < |R|, meaning there is an infinite set M with an injection M -> R, but no bijection to R or the natural numbers N (|N|=Aleph_0). But the existence or non-existence of such a set M does not at all change the size of R in my mind. Why would it?
  - Kranar 3y ago
    I suppose it's not too clear to me what you mean then, but of course this is an abstract topic so that's to be understood. A size just by itself is meaningless, if I said I have Aleph_0 units of happiness well for all you know I could be depressed, even if I said I have 2^Aleph_0 units of happiness I could still be depressed. It's only if you can compare what those units of happiness are to things you are familiar with, like how many units of happiness is eating chocolate, or being able to rest after a hard and productive day at work, that you would be able to get a sense of how happy (or depressed) I am. So sure, just knowing the size of the real numbers on its own without comparing it to anything doesn't change its significance in and of itself because it's meaningless. But if we could begin to identify certain sets that were smaller than the size of the reals (and larger than the naturals), and we could identify properties of these sets that gave us a sense of their expressive power, their reasoning power, etc... then we might start to appreciate just how large the real numbers are. We know categorically that Aleph_1 is the size of the set of the first uncountable ordinal number w_1, so any ordinal less than w_1 is countable. If the size of the continuum is Aleph_1 then that would have rather unintuitive implications involving the types of functions that are possible to define and reason about along with their relationship to probabilities and randomly choosing real numbers within the interval [0, 1] depending on whether the size of the real numbers is Aleph_1 or bigger. Probably one of the more accessible arguments for why this is is linked below [1]. [1] https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry