Double Think about Numerosity

Question

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. The "numerosity" of the evens is equal to the "numerosity" of the naturals. There are "as many" evens as there are naturals. 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 ...

Galileo's Paradox. The cardinality of the squares equals the cardinality of the naturals. There are "as many" squares as there are naturals. Proof. There exists a bijection between the naturals and the squares. The numerosity of the squares equals the numerosity of the naturals:

1 2 3  4  5  6  7  8  9  10  11  12  13  ..
| | |  |  |  |  |  |  |   |   |   |   |
1 4 9 16 25 36 49 64 81 100 121 144 169  ..

The cardinality of the powers of $7$ equals the cardinality of the naturals. There are "as many" powers of $7$ as there are naturals. Proof. There exists a bijection between the naturals and these powers:

1  2   3    4     5      6      7       8        9        10  ..
|  |   |    |     |      |      |       |        |         | 
7 49 343 2401 16807 117649 823543 5764801 40353607 282475249  ..

In general. Let there be defined a function $\;A(n) : \mathbb{N} \rightarrow \mathbb{N}$ . Assume that $\,A(n)\,$ is a sequence which is monotonically increasing with $\;n\;$ . Then we have the following bijection:

   1     2     3     4     5     6    7     8     9   .. 
   |     |     |     |     |     |    |     |     |
 A(1)  A(2)  A(3)  A(4)  A(5)  A(6) A(7)  A(8)  A(9)  ..

It can be concluded that the cardinality of the set $\;\left\{n\in\mathbb{N}:A(n)\right\}\;$ is $\;\aleph_0\;$, which is the cardinality of the naturals. Examples are:

• $A(n) = 2\cdot n\;$ (the even numbers)
• $A(n) = n^2\;$ (Galileo's paradox)
• $A(n) = 7^n\;$ (powers of seven)

But let's have a second thought about all this [ e.g with $A(n) = 7^n$ ]:

  1   2 3 4 5 6   7  8 9 10 .. 48  49  50 .. 343 .. 2401 ..
  0   0 0 0 0 0   1  1 1  1     1   2   2      3       4   : D(n)
A(0)            A(1)              A(2)       A(3)    A(4)

Let $D(n)$ be the number of $A(m)$ values (count) less than or equal to $n$, where $(m,n)$ are natural numbers. Then we have the following theorem, tentatively called the Inverse Function Rule: $$ \lim_{n\rightarrow \infty} \frac{D(n)}{ A^{-1}(n) } = 1 $$ Proof. $A(n)$ is monotonically increasing with $n$ , therefore the inverse sequence $A^{-1}(n)$ exists in the first place. And it is monotonically increasing as well. Furthermore, we see that $\,D(n)\,$ is $\,m\,$ for $\,A(m) \le n < A(m+1)$ .
Consequently: $\;A(D(n)) \le n < A(D(n)+1)\;$ , hence $\;D(n) \le A^{-1}(n) < D(n) + 1\;$ , therefore $\;A^{-1}(n) - 1 < D(n) \le A^{-1}(n)\;$ .
Divide by $\;A^{-1}(n)\;$ to get: $\;1 - 1/A^{-1}(n) < D(n) / A^{-1}(n) \le 1\;$ . For $\,n\rightarrow \infty\,$ now the theorem follows, because $\;A^{-1}(n) \rightarrow \infty\;$ . Written as an asymptotic equality: $\,D(n) \approx A^{-1}(n) $ .

Examples. Revealing the Double Think.

• $A(n) = 2.n \;\Longrightarrow\;\lim_{n\rightarrow\infty} D(n)/[\,n/2\,] = 1 \;\Longrightarrow\; D(n) \approx n/2$
  The numerosity of the evens is not equal but  half  the numerosity of the naturals
• $A(n) = n^2 \;\Longrightarrow\;\lim_{n\rightarrow\infty} D(n)/\sqrt{n} = 1\;\Longrightarrow\; D(n) \approx \sqrt{n}$
  The numerosity of the squares is the square root of the numerosity of the naturals
• $A(n) = 7^n \;\Longrightarrow\;\lim_{n\rightarrow\infty} D(n)/[\,\ln(n)/\ln(7)\,] = 1\;\Longrightarrow\; D(n) \approx \log_7(n)$
  The numerosity of the powers of 7 is the 7-logarithm of the numerosity of the naturals

Still another example at the Wikipedia page about Fibonacci numbers : For large $\,n$ , the Fibonacci numbers are approximately $F_n \approx \phi^n/\sqrt{5}$ . Giving a limit similar to the one for the powers of seven: $$ A(n) = F_n \quad\Longrightarrow\quad \lim_{n\rightarrow\infty} \frac{D(n)}{\ln(n.\sqrt{5}) / \ln( \phi )} = 1 \qquad \mbox{where} \quad \phi = \frac{1}{2}(1+\sqrt{5}) $$ Question. My problem is nomenclature in the first place. The term Inverse Function Rule seems to have been coined up somewhere on the internet but I could't find any application resemblant to the above - there also is a rumour that it goes all the way back to Carl Friedrich Gauss, but I coun't find any sensible reference confirming this.
Wouldn't even dare to ask about the more important thing, namely what is the "correct" way to no double think about numerosities: cardinalities or via the "inverse function rule" ?

Answer 1

There are some stabs being made at expanding the idea of asymptotic densities and measure to account for our basic intuition that the whole should be greater than the part, even for infinite sets. Vieri Benci, Mauro DiNasso and Marco Forti, among others, are doing work along these lines.

A rather advanced treatment is here http://arxiv.org/abs/1011.2089

Recently, this nice, readable article made its way into press at the Journal of Logic and Analysis: Elementary Numerosity and Measure http://arxiv.org/pdf/1212.6201.pdf

Answer 2

If I understood the question correctly, the OP is asking for one of these:

1. An established name for the "Inverse function rule" stated in the question
2. An established name for a concept of "numerosity" based on that rule
3. A resolution to the discrepancy between this concept and set theoretical cardinality.

1.

The expression $\lim_{n\rightarrow\infty}\tfrac{D(n)}{A^{-1}(n)}$ raises the question of how is $A^{-1}(n)$ defined for $n$ not in the image of $A$. The theorem seems to state that any extension of $A^{-1}$ to a monotone function on the reals is asymptotically equivalent to $D$. I am inclined to believe there's no established name for this result in the literature.

2.

I don't believe there's an established name for this either. While similar to natural density, the "numerosity" here is given not as a number, but as a function (or rather, an asymptotic equivalence class). There exist other examples of classifying objects by asymptotic behaviour, computational complexity being the main exponent.

3.

The answer would depend on what question is one trying to address by the concept.

Cardinality is the natural answer to the question "how many elements are there in a collection?", based on the idea that if you can put the elements of one collection in one-to-one correspondence with the elements of the other, the collections are the same size (the OP's own diagrams illustrate this quite well). This makes no allusion whatsoever to any properties of the elements, or their relation to one another—it only requires that enough elements exist on each side to match the other side. One can rearrange or relabel the elements of a set at will, and the cardinality will remain unchanged. Much of set theory is about the question of when and how can one construct such one-to-one correspondences, to the point where one can argue that the only really "interesting" property of a set is its cardinality (more precisely, the category of sets is equivalent to the category of cardinal numbers).

Both natural density and "numerosity", on the other hand, are different properties of the distribution of a subset of naturals within the whole, namely different ways to answer the question "how often does one encounter elements of this subset among the naturals?", likely to be of interest in probabilistic number theory. Both depend strongly on the order structure of the naturals—if we were to rearrange the naturals (and the elements of $A$ accordingly), both measures could vary wildly. Each measure corresponds to a different form of approximation of the relative frequency of finite prefixes of the naturals, one as a single number and the other as a curve. In the latter case, the fact that the distribution of a subset generated by a monotone function $A$ is approximated by the inverse function $A^{-1}$ seems akin to the relation between inverse functions and their derivatives.