Infinity of Naturals

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Double Think about Numerosity

According to standard mathematics, the Natural Numbers are given. Moreover they are given as a (completed) Infinite Set. This set is commonly denoted as: $$ \mathbb{N} = \left\{ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,\, \dots \right\} $$ Theorem. The set of all natural numbers is a completed infinity.
Proof. A set is infinite (i.e. a completed infinity) if there exists a bijection between that set and a proper subset of itself. Now consider the even naturals. They are a proper subset of the naturals and a bijection can be defined between the former and the latter. As follows:

  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 ...
  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...

However, according to the basic Axiom of this chapter, such a Completed Infinity does not exist. Which isn't necessarily going prevent anybody from still employing certain expressions, like for example $\,\forall n \in \mathbb{N}$ , as a well established "figure of speech".
For the sake of clarity, let's take a look at the mathematics laboratory par excellence, the digital computer. How is "the set of all naturals" represented in there? As numbers running from $1$ up to some maximum integer, usually of order $2^{32}$ or some such. That precise upper bound is not quite relevant. What is relevant is that there indeed does exist a limited upper bound. That limited set of naturals is well known in common mathematics too. It is called an initial segment of "the" standard Naturals, denoted as: $$ \mathbb{N}_n = \left\{ 1,2,3,4,5,6,7,8,9,\, \dots\, , n \right\} $$ Each such an initial segment contains all the naturals, starting from $1$, up to and including a largest natural $n$. With such an initial segment, it does not follow per se that $$ a \in \mathbb{N}_n \quad \mbox{and} \quad b \in \mathbb{N}_n \quad \Longrightarrow \quad (a+b) \in \mathbb{N}_n $$ Due to the fact that there is a finite upper bound on the initial segment, the sum $(a + b)$ can be beyond that bound. Which is clearly undesirable. Therefore sort of idealization is definitely needed, namely such that "beyond a bound" can no longer be the case. A tentative proposal could be to introduce The Naturals as the limit of an initial segment, where the largest natural $n$ approaches infinity. If you don't like this approach, please note that it will lead, in a few steps, to a result which is actually standard mathematics. So these intermediate steps could be accepted as sort of heuristic. Whatever, we shall simply assume, for the moment being, that The Naturals are defined as: $$ \mathbb{N} = \lim_{n\rightarrow\infty} \left\{ 1,2,3,4,5,\, \dots\, , n \right\} $$ Admittedly, the concept of a limit is somewhat stretched here. What we want to express is that The Naturals are quite resemblant to an initial segment, apart from an upper bound. So why not just say that the naturals are an initial segment without any upper bound. A difference with other approaches is that All of The Naturals therefore can only be known through the initial segments. Needless to say that Infinity will never be reached Actually - which is typical for limits anyway.
A significant issue in this context is Cardinality. When given a completed infinity of naturals, a bijection can set up between all naturals and all even naturals. This is no longer possible when initial segments are the only things that can be known about the naturals. A bijection can still be set up, but it does no longer cover "all" naturals in the segment. Instead, bijection reduces to a process that is known as counting. A count $E$ of all even naturals in an initial segment $\mathbb{N}_n = \left\{ 1,2,3,4,5,\, \dots\, , n \right\}$ where $n$ itself is even reveals that $E = n/2$:

  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 ...  n
  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |      |
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ... n/2 ...

And if $n$ is odd in $\mathbb{N}_n$ then $E = (n-1)/2$:

  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 ... (n-1)
  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |       |
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ... (n-1)/2 ...

Thus, for $n$ is even, we establish the limit: $$ \lim_{n\rightarrow\infty} \frac{\#\mbox{evens}}{\#\mbox{all}} = \lim_{n\rightarrow\infty} \frac{\#\left(\left\{ 1,2,3,4,\, \dots\, , n/2 \right\}\right)}{ \#\left(\left\{ 1,2,3,4,5,\, \dots\, , n \right\}\right)} = \lim_{n\rightarrow\infty} \frac{n/2}{n} = \frac{1}{2} $$ And, for $n$ is odd, we establish the limit: $$ \lim_{n\rightarrow\infty} \frac{\#\mbox{evens}}{\#\mbox{all}} = \lim_{n\rightarrow\infty} \frac{\#\left(\left\{ 1,2,3,\, \dots\, , (n-1)/2 \right\}\right)}{ \#\left(\left\{ 1,2,3,4,5,6,7,\, \dots\, , n \right\}\right)} = \lim_{n\rightarrow\infty} \frac{(n-1)/2}{n} = \frac{1}{2} $$ With the quotient of the two count functions available, note that it would have been only a technicality to define our tentative limit more rigorously. The inevitable conclusion is that the ratio $(\#\mbox{evens})/(\#\mbox{all})$ is equal to $1/2$. Therefore the "cardinality" of all even naturals divided by the "cardinality" of all naturals is not equal to $1$ but equal to $1/2$. Mind the scare quotes. Actually the technique as demonstrated is not quite unknown in common mathematics. Just look up Natural Density and you will find that it's just like it! So it turns out, in the first place, that no new mathematics is needed. More interesting facts are on that web page.
So here we are! This is what our proposal could have been: simply replace the standard notion of Cardinality by the other standard notion of Natural Density. However, experience learns that replacing things by other things is not as simple as that. A better strategy is to leave everything as it is; just consider Natural Densities as more relevant than Cardinalities, if the finitistic approach is to be preferred. Thus the gist of the above is not that common mathematics "doesn't know" about "finitistic cardinalities". It does. And the common theory of cardinalities is not so much wrong; I would rather call it redundant. Common mathematics may be just too much of the good.