Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous. You don't need any ellipses and there is no ambiguity.
Right, it's important that one may define ℕ without resorting to "...". Only once everybody agrees what this set should be and that it exists, we can write ℕ = {0, 1, 2, 3, ...} as shorthand or as a reminder, not as a definition.
In the beginning, it's acceptable to "just believe" that ℕ and ℝ exist, and later replace them with more rigorous definitions.
When one wants a completely rigorous definition of ℕ within ZFC, the most popular system of axioms for set theory, it gets elaborate: It's common to define ℕ as the smallest infinite von-Neumann ordinal. This happens to be the set of all finite von-Neumann ordinals, which we then interpret as natural numbers. But that requires an understanding of ordinals first, and why a definition as "the smallest ordinal that satisfies X" is sound.
I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?
Does a
train suffice instead of a basket?
But more seriously:
Naively, a set
X is a mathematical object such that, given any mathematical object
y, the statement
y ∊
X "makes sense", i.e., it is either true or false. Also, sets may not be "too big" such that one runs into
Russel's paradox or similar problems.
Thus, the set ℕ of all natural numbers exists; you can tell me with a straight face that 3, 5, and 329 belong to ℕ (because they are natural numbers), and you can tell me that ♥ and
M aren't natural numbers, thus don't belong to ℕ.
As long as one accepts the existence of infinite mathematical objects, one can feel reasonably safe that the existence of ℕ doesn't yet trigger any paradoxes or contradictions: Decades of work haven't found any. We can't prove that it's really paradox-free, but the fundamental obstacle is not infinite sets, it is that for any sufficently rich system of axioms, one
cannot prove from itself that it's free of contradictions. So this is really the best that we can get.
If one is not satisfied with naive definitions and wants something more rigorous and founded on classical logic, ZFC is the most popular formal system of set-theoretic axioms.
In ZFC,
everything that exists is a set. The axioms force that some sets exist, such as the empty set ∅, an infinite set (doesn't matter which), the two-element set for any two given elements, and some more things. Note that the existence of infinite sets is explicitly forced in ZFC, it's part of the design of this system of axioms.
Because everything is a set, but we still want to do mathematics similarly to how we're used to, we'll model our desired mathematical things using sets. E.g., we define the natural number 0 to be the empty set ∅, the natural number 1 is the 1-element set {∅}, the number 2 is {∅, {∅}}, an ordered pair of two things
x and
y is the set {
x, {
x,
y}}, and a function, a.k.a. a mapping, is a set of such ordered pairs.
It's similar to how everything in a computer (text, numbers, images, sound) is only a sequence of bytes under the hood. It's rarely necessary to consider the bytes: We don't talk about bytes much at all, we talk about numbers and texts and images, and it all makes sense to both of us. But
if the need arises, we know how to look under the abstraction and examine the raw bytes.
Infinite sets are exactly those sets
X that admit injections
X →
X that miss elements. (An injection is a function that never takes the same value at two different inputs.) For example, ℕ is infinite because the function
n ↦
n + 3 takes only different values for different inputs
n and misses (never takes as value) the first three natural numbers 0, 1, 2.
Finite sets are exactly those such that their size is expressible using a natural number.
It's not obvious that every set satisfies exactly one of these two, but it's provable in ZFC.
The nature of infinity as a philosophical or artistic idea will likely be fundamentally different to such a property of a set to admit certain functions to itself.
If one rejects the readily-existing infinite sets, one can still consider ℕ as building instructions to produce ever more mathematical objects, each itself finite, beginning with 0, and call them numbers. This is the basic idea of Finitism.
When one dives into formal logic, one will eventually separate two languages:
- In the finitistic metalanguage, we conduct proofs. Each proof can only have a finite number of steps and argue about a finite number of symbols. (It doesn't matter that, inside the theory, that symbol means something infinite. The proof treats it as one thing that has properties, e.g., being infinite, whatever that may mean in the theory.)
- In the domain-specific language that only makes sense when talking about objects of the theory, such as ZFC, we can use the term "infinite" even though that makes no sense in the metalanguage.
E.g., we can argue about the magic in the Harry Potter books, and our argument will use magic-related words that have meaning within that theory, even though nothing magic-related makes sense in our outside world.
It's useful to keep the metalanguage as weak as possible, to avoid introduction of inconsistencies. Reason is again: We can't prove that the metalanguage is consistent merely by using the metalanguage.
It's similar to how, in software, we don't want features unless there are good reasons to have them, as every feature has the potential to introduce bugs.
I have a Master's degree in Music
Hats off, then. It's one of the hardest subjects to even get admitted.
Regarding music, the "most infinite" thing that comes to my mind is the
Shepard glissando. But maybe you have even better examples.
-- Simon