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>I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.

andrewla 1y ago
Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
- blueplanet200 1y ago
  Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
  - andrewla 1y ago
    I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
    - CyLith 1y ago
      I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
      - blueplanet200 1y ago
        All that BT tells me is that when I break up a set (sphere) into multiple sets with no defined measure (how the construction works) I shouldn't expect reassemlbing those sets should have the same original measure as the starting set.
    - dullcrisp 1y ago
      Won’t the reals we can construct by any computation be enumerable? What measure can they have if not zero?
      - andrewla 1y ago
        Yes, they have measure zero. So the question becomes whether "measure" is a useful concept at all. In my opinion, no, it is not. It's just another artifact of non-constructive and meaningless abstractions. Many modern courses in analysis skip measure theory except as a historical artifact because the gauge integral is more powerful than the Lebesgue integral and doesn't require leaving the bounds of sanity to get there.
  - axolotliom 1y ago
    > I don't see how you can be _skeptical_ of those ideas. Well you can be skeptical of anything and everything, and I would argue should be. Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_acceptance https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_... I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities. You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > Math is math, if you start with ZFC axioms This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities. https://news.ycombinator.com/item?id=44739315 https://news.ycombinator.com/item?id=44739315
    - SabrinaJewson 1y ago
      > Addressing your issue directly, the Axiom of Choice is actively debated: The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset. Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.
      - codebje 1y ago
        Perhaps this is an ignorant question, but wouldn't you need AC to select the s ⊆ A whose existence the contradiction depends on? A constructive proof, at least the ones I'm trying to build in my head, stumbles when needing to produce that s to use in the following arguments.
  - orangecat 1y ago
    My cranky position is that I'm very skeptical of the power set axiom as applied to infinite sets.

Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.

blueplanet200 1y ago
Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
- andrewla 1y ago
  I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
  - CyLith 1y ago
    I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
    - blueplanet200 1y ago
      All that BT tells me is that when I break up a set (sphere) into multiple sets with no defined measure (how the construction works) I shouldn't expect reassemlbing those sets should have the same original measure as the starting set.
  - dullcrisp 1y ago
    Won’t the reals we can construct by any computation be enumerable? What measure can they have if not zero?
    - andrewla 1y ago
      Yes, they have measure zero. So the question becomes whether "measure" is a useful concept at all. In my opinion, no, it is not. It's just another artifact of non-constructive and meaningless abstractions. Many modern courses in analysis skip measure theory except as a historical artifact because the gauge integral is more powerful than the Lebesgue integral and doesn't require leaving the bounds of sanity to get there.
- axolotliom 1y ago
  > I don't see how you can be _skeptical_ of those ideas. Well you can be skeptical of anything and everything, and I would argue should be. Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_acceptance https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_... I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities. You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > Math is math, if you start with ZFC axioms This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities. https://news.ycombinator.com/item?id=44739315 https://news.ycombinator.com/item?id=44739315
  - SabrinaJewson 1y ago
    > Addressing your issue directly, the Axiom of Choice is actively debated: The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset. Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.
    - codebje 1y ago
      Perhaps this is an ignorant question, but wouldn't you need AC to select the s ⊆ A whose existence the contradiction depends on? A constructive proof, at least the ones I'm trying to build in my head, stumbles when needing to produce that s to use in the following arguments.
- orangecat 1y ago
  My cranky position is that I'm very skeptical of the power set axiom as applied to infinite sets.