Comment by dilippkumar
4 days ago
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.
> 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/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.
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.
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).
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.
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
>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.
Diagonal argument doesn’t work in a constructive ground. It’s not a matter of whether the conclusion is valid, but if we have blind faith in the premises and are fine about speaking of something we can’t build.
They are things that humans will never be able to construct, no matter how far their control over the universe surrounding them might go. To start with, humans can create the universe, — whether it’s infinite or not.
We can construct a function G, which, given a function f : N -> N -> {0,1} returns a function G(f) = h : N -> {0,1} defined by h(i):=not(f(i)(i))
This h=G(f) has the property that, for all i, there exists a j such that f(i)(j)≠h(j) . In particular, j=i will work for this.
It seems to me that this is all constructive.
The only out that I see is to not consider the class of “functions from N to {0,1}” to be something that exists (as a set, or type, or whatever).
Like, you can fairly reasonably hold the position that there is no powerset of the natural numbers, but you can’t reasonably hold the position that it exists and that there is a surjection from the natural numbers to it. (Likewise with any other set N. This isn’t specific to the natural numbers.)
We have a constructive refutation of that claim, in the sense that we have a construction of a function which, given such a surjection (as in, a function along with a promise that the provided function is such a surjection), produces a contradiction.
> 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.
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
1 reply →