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I haven't studied math beyond what was needed for my engineering courses. However, I also am starting to believe that infinity doesn't exist. Or more specific

I haven't studied math beyond what was needed for my engineering courses. However, I also am starting to believe that infinity doesn't exist. Or more specifically, I want to argue that infinity is not a number, it is a process. When you say {1, 2, 3, ... } the "..." represents a process of extending the set without a halting condition. There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further. There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.

skulk 10mo ago
> There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further. Sure, but ordinal numbers exist and are useful. It's impossible to prove Goodstein's theorem without them. https://en.wikipedia.org/wiki/Ordinal_number https://en.wikipedia.org/wiki/Ordinal_number https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.
- dilippkumar 10mo ago
  Thanks for the pointer. I went down the rabbithole, and as far as I can tell, you have to axiomatically assume infinities are real in order to prove Goodstein’s theorem. I challenge the existence of ordinal numbers in the first place. I’m calling into question the axioms that conjure up these ordinal numbers out of (what I consider sketchy) logic. But it was a really fun rabbithole to get into, and I do appreciate the elegance of the Goodstein’s theorem proof. It was a little mind bending.
  - skulk 10mo ago
    yes, if you want ordinal numbers in ZFC you need to take the axiom of infinity. Other than that it's a pretty straightforward construction. If you reject the axiom of infinity you also essentially reject all of standard analysis (using limits to study reals often implicitly invokes the axiom of infinity).
anvuong 10mo ago
Whether you think infinity exists or not is up to you, but transfinite mathematics is very useful, it's used to prove theorems like Goodstein's sequence in a surprisingly elegant way. This sequence doesn't really have anything to do with infinity as first glance.
- alexnewman 10mo ago
  What did we build with this useful math. Who was fed? What businesses did it create?
soulofmischief 10mo ago
Actually, all numbers are functions in Peano arithmetic. :) For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on. There is no end of a number line. There are lines, and line segments. Only line segments are finite. > There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist. You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with. There are infinities with higher cardinality than others. Infinity relates to set theory, and if you try to simply imagine it as a "position" in a line of real numbers, you'll understandably have an inconsistent mental model. I highly recommend checking out Cantor's diagonal argument. Mathematicians didn't invent infinity as a curiosity; it solves real problems and implies real constraints. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
- drdeca 10mo ago
  > For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on. S is a function symbol. S(0) (in PA) is not a function. It is an expression involving one.
  - alexnewman 10mo ago
    Let’s keep it simple. What physics or engineering is easier? Let us ignore mathematics for its own sake . If you can’t show use there … id argue it’s our math aesthetics that are wrong
    - drdeca 10mo ago
      Why have you picked that comment of mine to reply with that reply? I was just correcting an error. Your reply seems irrelevant to my comment. I was just saying that “one more than 0” isn’t a function just because in Peano arithmetic, the successor function, along with the constant 0, is used to denote natural numbers.
  - 10mo ago

skulk 10mo ago

> There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further. Sure, but ordinal numbers exist and are useful. It's impossible to prove Goodstein's theorem without them. https://en.wikipedia.org/wiki/Ordinal_number https://en.wikipedia.org/wiki/Ordinal_number https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.

dilippkumar 10mo ago
Thanks for the pointer. I went down the rabbithole, and as far as I can tell, you have to axiomatically assume infinities are real in order to prove Goodstein’s theorem. I challenge the existence of ordinal numbers in the first place. I’m calling into question the axioms that conjure up these ordinal numbers out of (what I consider sketchy) logic. But it was a really fun rabbithole to get into, and I do appreciate the elegance of the Goodstein’s theorem proof. It was a little mind bending.
- skulk 10mo ago
  yes, if you want ordinal numbers in ZFC you need to take the axiom of infinity. Other than that it's a pretty straightforward construction. If you reject the axiom of infinity you also essentially reject all of standard analysis (using limits to study reals often implicitly invokes the axiom of infinity).

anvuong 10mo ago

Whether you think infinity exists or not is up to you, but transfinite mathematics is very useful, it's used to prove theorems like Goodstein's sequence in a surprisingly elegant way. This sequence doesn't really have anything to do with infinity as first glance.

alexnewman 10mo ago
What did we build with this useful math. Who was fed? What businesses did it create?

soulofmischief 10mo ago

Actually, all numbers are functions in Peano arithmetic. :) For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on. There is no end of a number line. There are lines, and line segments. Only line segments are finite. > There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist. You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with. There are infinities with higher cardinality than others. Infinity relates to set theory, and if you try to simply imagine it as a "position" in a line of real numbers, you'll understandably have an inconsistent mental model. I highly recommend checking out Cantor's diagonal argument. Mathematicians didn't invent infinity as a curiosity; it solves real problems and implies real constraints. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

drdeca 10mo ago
> For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on. S is a function symbol. S(0) (in PA) is not a function. It is an expression involving one.
- alexnewman 10mo ago
  Let’s keep it simple. What physics or engineering is easier? Let us ignore mathematics for its own sake . If you can’t show use there … id argue it’s our math aesthetics that are wrong
  - drdeca 10mo ago
    Why have you picked that comment of mine to reply with that reply? I was just correcting an error. Your reply seems irrelevant to my comment. I was just saying that “one more than 0” isn’t a function just because in Peano arithmetic, the successor function, along with the constant 0, is used to denote natural numbers.
- 10mo ago