3 ms·

Nice write-up, though I wish you had expanded a bit more on this: "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product

by lisper 2y ago

Nice write-up, though I wish you had expanded a bit more on this: "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..." Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference. BTW, you might find this interesting: https://blog.rongarret.info/2023/01/an-intuitive-counterexample-to-axiom-of.html https://blog.rongarret.info/2023/01/an-intuitive-counterexam...

ColinWright 2y ago
> Nice write-up, Thank you. > ... I wish you had expanded a bit more on this: >> "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..." > Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference. I'll see if I can hunt out an accessible reference. Problem is, I think it uses Cohen's forcing, and that's pretty tough going in any kind of detail. I might add a note ... thanks for the suggestion. > BTW, you might find this interesting: https://blog.rongarret.info/2023/01/an-intuitive-counterexample-to-axiom-of.html https://blog.rongarret.info/2023/01/an-intuitive-counterexam... I've bookmarked that for when I have a coffee, biscuit, and 30 minutes. Thank you.
- lisper 2y ago
  > I think it uses Cohen's forcing You don't have to give a full proof, just an intuitive explanation of why the "obvious" arguments fail. For example, we can prove the non-emptiness of product sets for finite sets. Why can that not simply be extended in the "obvious" way to infinite sets? How can making the sets "bigger" suddenly make the product empty?
  - ColinWright 2y ago
    Hmm. I'll have to think more about that. The problem I have is that "intuitive arguments" differ between people. What one person thinks is intuitively obviously true, another can think is intuitively obviously false. I'll have to think about what I would be trying to say. It's a good point ... it's now explicitly on my list. Thank you.

by lisper 2y ago

ColinWright 2y ago

> Nice write-up, Thank you. > ... I wish you had expanded a bit more on this: >> "We can show that using the axioms of Zermelo-Fraenkel Set Theory we cannot prove the product of an infinite collection of non-empty sets to be non-empty. That seems daft..." > Indeed, and it's exactly the sort of thing that you should not simply proclaim to be true with no explanation or reference. I'll see if I can hunt out an accessible reference. Problem is, I think it uses Cohen's forcing, and that's pretty tough going in any kind of detail. I might add a note ... thanks for the suggestion. > BTW, you might find this interesting: https://blog.rongarret.info/2023/01/an-intuitive-counterexample-to-axiom-of.html https://blog.rongarret.info/2023/01/an-intuitive-counterexam... I've bookmarked that for when I have a coffee, biscuit, and 30 minutes. Thank you.

lisper 2y ago
> I think it uses Cohen's forcing You don't have to give a full proof, just an intuitive explanation of why the "obvious" arguments fail. For example, we can prove the non-emptiness of product sets for finite sets. Why can that not simply be extended in the "obvious" way to infinite sets? How can making the sets "bigger" suddenly make the product empty?
- ColinWright 2y ago
  Hmm. I'll have to think more about that. The problem I have is that "intuitive arguments" differ between people. What one person thinks is intuitively obviously true, another can think is intuitively obviously false. I'll have to think about what I would be trying to say. It's a good point ... it's now explicitly on my list. Thank you.