[rfergie]

One construction of the natural numbers that I've seen is

1. There exists an element 0 that is a natural number

2. For every natural number there is a "sucessor" that is also a natural number. (i.e. if n is a natural number then n+1 is a natural number)

This construction means there can't be an upper bound N because then step 2 couldn't be applied to N.

Maybe there are other constructions that could workaround this? I'm guessing not because you'd still struggle to define the usual rules of addition for all numbers in a bounded set

[naasking]

> This construction means there can't be an upper bound N because then step 2 couldn't be applied to N.

Bendegem discusses this problem at length in his paper [1]. As programming-heavy site, I assume we're all aware that computers have finite resources. The universe too has finite resources so no matter how big a computer you build, it too will be finite. Therefore the infinity that is so pervasive in math is unphysical in a very real sense. So what would math look like and how would theorems change if this finiteness were formalized? That's what various flavours of finitism aim to achieve.

So to get back to your question as to the nature of the naturals, it seems evident that yes, at some point, you literally can fail construct the natural number N+1 if you are given N, because you will run out of particles in the universe. What implications this will have for various theorems will be interesting for sure, but it isn't clear yet because finitism isn't given much funding.

Edit: however, it's clear that some very unintuitive results follow from the infinities embedded in mathematics, and that a finitist approach resolves some of them. For instance, the argument that "0.9999... = 1" is true in classical mathematics while this equality is arguably not true under strict finitism because "0.999..." does not exist, because infinite objects do not exist, and so it will never equal 1.

[1] See the section on continuous counting, https://www.researchgate.net/publication/288354797_A_Defense...

[Ferrotin]

Calling “0.999… = 1” very unintuitive is a very strange thing to say, because that makes perfect sense to most children. I’d like to see a result that truly is unintuitive, like what we get with the axiom of choice.

[naasking]

> Calling “0.999… = 1” very unintuitive is a very strange thing to say, because that makes perfect sense to most children.

Gotta call bullshit on that. First I don't think you have any robust empirical data on that question.

Second what's convincing to children that don't have enough knowledge of math to have formed any intuitions about it is not a compelling argument.

[Ferrotin]

I remember being a kid and having this discussion in elementary school. Kids have enough intuition to know that operations with fractions should get the same result as with decimal numbers. Or that 1-0.999… = 0.000…. Or that different lengths have a length in between them. All are legitimate and compelling arguments.

[amelius]

I think lots of students get lost with different orders of infinity (countable, uncountable, etc.), so I think there is definitely a point beyond which you can't push the intuition behind infinite objects.

[naasking]

> Or that different lengths have a length in between them. All are legitimate and compelling arguments.

Except that's wrong, not all lengths do have a length between them.

[rfergie]

Thank you for the link to the paper; very interesting!