4 ms·

> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. Certainly. To be clear, I don't mean to argue against any

> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ. Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. https://en.wikipedia.org/wiki/Peano_axioms#Formulation https://en.wikipedia.org/wiki/Peano_axioms#Formulation

JadeNB 10y ago
> Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; but, on the other hand, I suppose that one could argue that an argument using "only the most fundamental essence of the natural numbers" is one that, as it were, is allowed only to use a constant naming the least element, without pre-supposing any idea of what that name 'means' in any external axiomatic or intuitive sense.
- Bromskloss 10y ago
  I think I follow, and I agree. How I view the natural numbers on this fundamental level – and I would say it is the picture that the Peano axioms nail down – is as consisting of a first element and then an infinite chain of elements after it (equivalently: the natural numbers are a chain that extends infinitely in one direction but not in the other). As you say, it does not matter, then, if that first element is called "0", "1", or "17" (until we bestow additional structure on the numbers by introducing addition and multiplication operators).

JadeNB 10y ago

> Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; but, on the other hand, I suppose that one could argue that an argument using "only the most fundamental essence of the natural numbers" is one that, as it were, is allowed only to use a constant naming the least element, without pre-supposing any idea of what that name 'means' in any external axiomatic or intuitive sense.

Bromskloss 10y ago
I think I follow, and I agree. How I view the natural numbers on this fundamental level – and I would say it is the picture that the Peano axioms nail down – is as consisting of a first element and then an infinite chain of elements after it (equivalently: the natural numbers are a chain that extends infinitely in one direction but not in the other). As you say, it does not matter, then, if that first element is called "0", "1", or "17" (until we bestow additional structure on the numbers by introducing addition and multiplication operators).