The most beautiful construction in all of Mathematics is the standard Von Neumann construction in set theory which defines the natural numbers in terms of sets. One starts with the empty set 0:

Then one defines the natural number n to be (the cardinality of) the set S^{n}(0), where S(a) is the successor function defined by S(a)=a U {a} for every set a and the superscript denotes repeated composition. Explicitly:

- 0=S
^{0}(0)=0 - 1=S
^{1}(0)={0} - 2=S
^{2}(0)={0,{0}} - 3=S
^{3}(0)={0,{0},{0,{0}}} - 4=S
^{4}(0)={0,{0},{0,{0}},{0,{0},{0,{0}}}} - 5=S
^{5}(0)={0,{0},{0,{0}},{0,{0},{0,{0}}},{0,{0},{0,{0}},{0,{0},{0,{0}}}}} - 6=S
^{6}(0)={0,{0},{0,{0}},{0,{0},{0,{0}}},{0,{0},{0,{0}},{0,{0},{0,{0}}}},{0,{0},{0,{0}},{0,{0},{0,{0}}},{0,{0},{0,{0}},{0,{0},{0,{0}}}}}} - ...

The above construction is elegant and powerful:

- It uses only three literal symbols: "0", "{" and "}", therefore it is simple and thus elegant.
- By using the simple recursion S(a)=a U {a} it constructs an infinity of numbers, therefore it is powerful.

Consider the set of all letter words W of finite length. If i,j in N, w,w_{i} in W, and w[i] denotes letter i of word w, we define the maps C:W->W (the concatenation operator), SS:W x N^{2} -> W (the substring operator), and L:W->N (the word length operator) in the obvious way:

- C(w
_{1},...,w_{n})=w_{1}w_{2}(consecutive) iff n=2,

C(C(w_{1},...,w_{n-1}),w_{n}), otherwise. - SS(w,i,j)=w[i..j]=C(w[i],...,w[j]), iff i≤j,

'{' otherwise. - L(w)=length of word w.

The map VN(n)=S^{n}(0) of the Von Neumann construction now induces a natural map VS:N->W, from the natural numbers to the set of words W of finite length, defined by:

- VS(n)='0', iff n=0,

C(VS(n-1)[1..L(VS(n-1))-1],VS(n-1),'}'), otherwise.

VS(n) is essentially the string representation of the Von Neumann set VN(n) minus the commas.

We now define a letter replacement transformation R:W->W as follows:

- R(w)='0', iff w='0',

'1', iff w='{',

'2', iff w='}',

C(R(w[1..L(w)-1]),R(w[L(w)])), otherwise.

R basically takes the word w and replaces all occurances of '0', '{' and '}', with '0','1' and '2' respectively.

We next define the map V:W->N as follows:

- V(w)=0, iff w='0',

1, iff w='1',

2, iff w='2',

∑_{i=1..L(w)}V(w[i])*10^{i-1}, otherwise.

V basically gives the base 10 value of the number represented by the word w. Finally we look at the sequence defined by:

- a(n)=V(R(VS(n))).

Here are the terms a(0) to a(7):

- a(0)=0
- a(1)=102
- a(2)=101022
- a(3)=101021010222
- a(4)=101021010221010210102222
- a(5)=101021010221010210102221010210102210102101022222
- a(6)=101021010221010210102221010210102210102101022221010210102210102101022210102101022101021010222222
- a(7)=101021010221010210102221010210102210102101022221010210102210102101022210102101022101021010222221

010210102210102101022210102101022101021010222210102101022101021010222101021010221010210102222222 - ...

a(n) is an interesting sequence. For all n, 2|a(n) and 3|a(n), => 6|a(n). Call D(n) the number of digits of the number n. If n>0, then a(n) always contains D(a(n))/3 0's, 1's and 2's. The word representation of a(n) never contains the patterns {"00","11","12","20"}. It always contains the pattern "2...2" (n 2's) once and always ends with that pattern. a(n)/2 contains only the digits {0,1,5}. a(n)/3 contains only the digits {0,3,4,6,7}. a(n)/6 contains only the digits {0,1,3,5,6,7,8}.

Let's investigate it a little. It is easy to see that D(a(n)) is given (why?) by the recursion:

- D(a(n))=1, iff n=0,

D(a(n-1))+2, iff n=1,

2*D(a(n-1)), otherwise.

Solving the recurrence with Maple (or by hand if you are savvy), we get:

> rsolve({K(n)=2*K(n-1),K(1)=3},K(n));

K(n)=3/2*2^{n}, =>

D(a(n))=3/2*2^{n}.

On the other hand, for any natural number n we have: D(n)=[log_{10}(n)]+1, where [x]=floor(x), =>

[log_{10}(a(n))]+1=3/2*2^{n}, =>

log_{10}(a(n))+1~3/2*2^{n}, =>

log_{10}(a(n))~3/2*2^{n}-1, =>

a(n)~10^{3/2*2n-1}.

That's a fast growing sequence! Can we find a nice recursion for a(n)? Look carefully at the function VS, above, and try to convince yourself that the following works:

- a(n)=0, iff n=0,

102, iff n=1,

[a(n-1)/10]*10^{D(a(n-1))+1}+a(n-1)*10+2, otherwise.

Because there are always just 3 letters in the final word w ("0", "1", and "2"), one is tempted to define the last function V, alternatively as:

- V'(w)=0, iff w='0',

1, iff w='1',

2, iff w='2',

∑_{i=1..L(w)}V'(w[i])*3^{i-1}, otherwise.

which gives the base 3 value of the number represented by the word w, and then look at b(n) defined similarly:

- b(n)=V'(R(VS(n))).

Here are the terms b(0) to b(7):

- b(0)=0
- b(1)=11
- b(2)=278
- b(3)=202040
- b(4)=107371882880
- b(5)=30324991112647875920960
- b(6)=2418916677393691552133618294938847890243982720
- b(7)=15390805926573768302084260548391044837412074858741153833336080616936611834819590898337488640
- ...

It's more or less obvious that b(n) grows slower than a(n). How MUCH slower?

If we call D_{b}(n) the number of digits of the number n in base b, then D_{3}(n)=[log_{3}(n)]+1. Note however that D_{3}(b(n))=3/2*2^{n}. (why?). Therefore:

[log_{3}(b(n))]+1=3/2*2^{n}, =>

[log_{3}(b(n))]=3/2*2^{n}-1, =>

log_{3}(b(n))~3/2*2^{n}-1, =>

b(n)~3^{3/2*2n-1}.

Therefore b(n) grows much slower than a(n).

a(n) and b(n) made it into Neil Sloane's Encyclopedia of Integer Sequences, as sequences A129751 and A129754.

The natural thing to ask now, is whether 0.a(n) and 0.a(n)_{3} converge. Let's define those sequences formally. Let D(n) be as above. Then

- c(n)=a(n)/10
^{D(a(n))} - d(n)=c(n)
_{3}

Here are the first terms of c(n):

- c(1)=0.102
- c(2)=0.101022
- c(3)=0.101021010222
- c(4)=0.101021010221010210102222
- c(5)=0.101021010221010210102221010210102210102101022222
- ...

and of d(n):

- d(1)=0.407407407407...
- d(2)=0.38134430727023319615912208504801097393689986282579...
- d(3)=0.38017390453502834745531488914103352959218426880877...
- d(4)=0.38017228728208203343618615695064718198148619083135...
- d(5)=0.38017228727903885619326881478737833870768398645584...
- ...

Can we say anything about c(n) and d(n)? Yes!

c(n) converges! For all n we have c(n+1)<c(n) (can you see why?). Furthermore, for all n>0, c(n) is bounded: 0<c(n)<0.102 (why?), hence converges to some number 0<A<0.102 (why?). What about d(n)? It obviously converges to A in base 3 (why?) Can anything be said about A? Is it rational? Irrational? There is a quick and short proof that it's not only transcendental, but also *normal* in base 3. Can you see it?

Note that the construction above can generate many similar sequences, depending on the image of the letter set {"0","{","}"} under R(w). To be exact, there are 10 letters ("0".."9") to choose from, so we can immediately construct (10 choose 3)=120 sequences. To generate additional sequences, we can reorder the elements inside S^{n}(0) in decreasing cardinality or in some other way. Since there are n! ways to arrange the elements inside each such set, this process can generate an infinity of additional sequences.

__The Von Neumann Construction From a Metaphysical Perspective__

Let's take a little break from math now and examine the Von Neumann construct from a philosophical/metaphysical perspective. The Von Neumann construction is amazing in the sense that it requires only 3 elements: {"0","{","}"}. 0 is ultimately connected with emptiness and nothingness, a very basic characteristic of the universe. The two other symbols, "{" and "}" are ultimately connected with the properties of consciousness, separation and class. In a sense the first step of creation is consciousness "recognizing" the vast emptiness of space. As soon as this happens, the emptiness "separates" into "something" (that which has perceived the emptiness) and the emptiness itself. As soon as we have two things, a boundary is created: The 3rd element. The sequence thus is:

- Void: 0=0
- God: "I recognize that Void exists" ({0}=1).
- God: "Therefore, all that exists (for now) is Void and Me" ({0,{0}}=2)
^{[4]}.

Note however that the (cardinality of the) set {0,{0}}=2 is not included in God's universe (2 not in {0,1}), therefore it implies a separate "existence" from that of 0 and 1. This existence/consciousness, the consciousness that corresponds to perceiving 2 (as a whole), or if you will, the consciousness that *is* 2, is The Firstborn^{[5]}^{[6]}.

Then, The Firstborn similarly claims: "All that exists (for now) is Void, God and Me" ({0,{0},{0,{0}}}=3). But again, the cardinality of the latter is not included in the previous universe! From that point on, the same thing happens with every new cardinality, so the moment God finishes step 3 above, we have an explosion or chain reaction of existences, which generates the first infinite hierarchy in existence: The first Big Bang: The Creation of The Natural Numbers! Tremendous!^{[1]}^{[2]}^{[3]}^{[7]}^{[8]}^{[9]}

The natural numbers in turn realize that their existence implies the existence of their negatives. The integers are created. The integers in their turn realize that their existence implies the existence of the rationals and so on and so forth. The irrationals are created. Then the reals. Then the complexes, the quaternions, the octonions, sedenions, vector spaces, fields, algebras, etc. The existence of Mathematics is established and its kingdom is set above all other kingdoms, except that of music (which is a direct creation of the rational numbers, therefore precede the reals).

The two most famous transcendental constants of mathematics for example are created directly from the Naturals as shown in the Yin-Yang symbol.

From that point on, it's a one way street: God now has a powerful tool at his disposal: Mathematics, with which He can generate pretty much all of reality and characterize existence down to any desirable detail by using Applied Mathematics to model it. The second Big-Bang occurs and the universe follows... The rest is history.

The whole construction is thus ultimately related to the three numbers {0,1,2}, and hence to the base 3 and the number 3. The numbers {0,1,2} therefore (or if you prefer the numbers {1,2,3}) form an *archetype* upon which all of reality stands. It is not a coincidence therefore that the number 3 is connected with various interesting facts:

- 3 is the minimum number of elements which make consciousness possible: perceiver, object perceived and perception of object.
- 3 is the number of the faces of God in most modern Trinitarian dogmas with Jesus explicitly associated with the number 3.
- 3 is the first digit of π=3.14159..., which is connected to the circle in geometry, which is again connected to metaphysics, religion and God.
- 3 is the number of spatial dimensions in our space.
- 3 is the minimum number of humans needed to form a small family, which is the smallest social enterprize that operates somewhat efficiently in society.
- 3 is the number of the firstborn, created from the union of the parents 1 and 2, as 1+2=3.
- 3 is the least number of points in geometry needed to define a plane, which has dimension 2.

It seems then that the Trinitarian dogma of 2 is a remnant (or evolute) of 1, so things are simpler than what the theologians make them to be. If we call V=Void, G=God, F=Firstborn, then the sequence above is described pictorially as:

The sequential birth of everything from V: G=S(V), F=S(G), 3=S(F), 4=S(3),...,...

- Pythagorean School Motto: "All is Number".
- Pythagoreans: "Void exists, and it enters the heaven from the unlimited breath of it, so to speak, breathes in void. The void distinguishes the natures of things, since it is the thing that separates and distinguishes the successive terms in a series. This happens in the first case of numbers; for the void distinguishes their nature.".
- Diogenes Laertius: "...from the monad evolved the dyad; from it numbers; from numbers, points; then lines, two-dimensional entities, three-dimensional entities, bodies, culminating in the four elements earth, water, fire and air, from which the rest of our world is built up...".
- John 1:1-3: "1: In the beginning was the Word, and the Word was with God, and the Word was God. 2: He was with God in the beginning. 3: Through him all things were made; without him nothing was made that has been made.".
- Colossians 1:15-16: "15: He is the image of the invisible God, the firstborn over all creation. 16: For by him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things were created by him and for him.".
- Hebrews 1:5-6: "5: For to which of the angels did God ever say, "You are my Son; today I have become your Father? Or again, "I will be his Father, and he will be my Son"? 6: And again, when God brings his firstborn into the world, he says, "Let all God's angels worship him."".
- Tao Te Ching: (tr. Mair 1990:9): "The Way (nothingness) gave birth to unity (God), Unity gave birth to duality (Firstborn), Duality gave birth to trinity, Trinity gave birth to the myriad creatures.".
- Tao Te Ching, Chapter 40: "All of creation is born from substance. Substance is born of nothingness...".
- Tao Te Ching, Chapter 42: "Tao begets one; one begets two; two begets three; three begets all things.".