Hi Kim, hi John, hi People,

Kim provided me with an excellent answer to my preceding post (out-of- 
line though). And John told me he was impatient to see "my definition"  
of the natural numbers (and some other numbers) in term of sets. So I  
make a try. Nothing is important here for the sequel, but it can help  
too.

This is in line with our future goal to figure out what a computation  
is, and what is the difference between a computation and a description  
of a computation. This plays a probably subtle role in the seventh  
step of UDA, and also in the eight step. So just another example of a  
well standard set theoretic representation of the natural numbers (and  
the transfinite ordinals which extends them) can be useful, if only as  
a reservoir of examples of structures later.

John has perhaps believed I was trying to define the numbers, (by  
which I always mean "natural numbers", that is 0, 1, 2, 3, ...), but I  
don't try to do that. I try just to help people with different view of  
those numbers with an emphasis on what they are, as opposed to how to  
represent them.

I have already mentioned the notation I, II, III, IIII, ...

We could capture this number's representation by axioms (and implicit  
rule), like

Axiom 1: I is a number
Axiom 2: if x is a number, then xI is a number.

So I is a number (by axiom 1), so II is a number (by axiom 2), so III  
is a number (by axiom 3), so IIII is a number (by axiom 3).

Is IIIIIIIIIIII.... a number? To avoid it we should need a rule saying  
that we can apply axiom 2 only a finite number of time. But "finite  
number" is what we were trying to define, so, well, we can't define  
them, and I will rely on your intuition.

* * *

So, let me give you a nice representation of the natural numbers in  
terms of sets. This material will not been used in the sequel, so take  
it easy. it is a glimpse of "beyond infinity". This is due mainly to  
to von Neumann. He showed that we can generate "the universe of  
numbers" (actually of ordinals) from "nothing", or from an empty  
universe, by using two powerful principles: the principle of set  
comprehension, and the principle of set reflexion. I have tested  
successfully this idea with young people.

The generation of the universe of numbers proceed in stages, beginning  
with an empty universe. At each state we try 1) to comprehend the  
whole universe, and 2) (it is the rule of the game) to put what we  
have comprehend in the universe. 1) and 2) are the comprehension rule  
and the reflexion rule.

Well, we still need a notation to describe the result of the  
comprehension. On a board a use circles or ellipses, but here I will  
use the more standard accolades. For example I comprehend John and  
Kim, means I conceive the set {John, Kim}.

Let us go: (please do it yourself alongside, with {} a circle, { { } }  
a circle with a little circle inside, it is easier to read, more cute,  
and you will see the growing fractal:


Day 0: I wake up and I observe the universe. But the universe is  
empty. Nothing. My comprehension of the universe at this stage is  
represented by the empty set { }. It is my model of the universe at  
that stage. And, well I will define or represent the number 0 by { }.  
It is my conception of the universe at the middle of the day 0. We  
have 0 = { }
But then I have to obey to the reflection rule, and I have to put { }  
in the universe, and then I go to bed.

Day 1: I wake up and I observe the universe. But the universe contains  
{ }. It contains 0. My comprehension of the universe at this stage is  
represented by the set containing the empty set {{ }}. And, well I  
will define or represent the number 1 by {{ }}.  It is my  
comprehension of the universe at the middle of the day 1. We have 1 =  
{{ }}
But then I have to obey to the reflection rule, and I have to put  
{{ }} in the universe, and then I go to bed.

Day 2: I wake up and I observe the universe. But the universe contains  
{ } and {{ }}. It contains 0, and 1. My comprehension of the universe  
at this stage is represented by the set containing {{ }, {{ }}}. And,  
well I will define or represent the number 2 by {{ }, {{ }}}.  It is  
my comprehension of the universe at the middle of the day 2. We have 2  
= {0, 1}
But then I have to obey to the reflection rule, and I have to put {{ }  
{{ }}} in the universe, and then I go to bed.

Day 3: I wake up and I observe the universe. But the universe contains  
{ } and {{ }} and {{ } {{ }}}. It contains 0, and 1, and 2. My  
comprehension of the universe at this stage is represented by the set  
{{ },  {{ }},  {{ } {{ }}}}. And, well I will define or represent the  
number 3 by {{ },  {{ }},  {{ } {{ }}}.  It is my comprehension of the  
universe at the middle of the day 3. We have 3 = {0, 1, 2}
But then I have to obey to the reflection rule, and I have to put  
{{ },  {{ }},  {{ } {{ }}}} in the universe, and then I go to bed.

Day 4: I wake up and I observe the universe. But the universe contains  
{ } and {{ }} and {{ }, {{ }}} and {{ } , {{ }},  {{ }, {{ }}}}. It  
contains 0, and 1, and 2, and 3. My comprehension of the universe at  
this stage is represented by the set {{ },  {{ }},  {{ }, {{ }}},  
{{ } , {{ }},  {{ }, {{ }}}}}. And, well I will define or represent  
the number 4 by {{ },  {{ }},  {{ }, {{ }}}, {{ },  {{ }},  {{ },  
{{ }}}}}.  It is my comprehension of the universe at the middle of the  
day 4. We have 4 = {0, 1, 2, 3}
But then I have to obey to the reflection rule, and I have to put  
{{ },  {{ }},  {{ }, {{ }}}, {{ },  {{ }},  {{ }, {{ }}}}} in the  
universe, and then I go to bed.

Well, at this stage, or a bit later, some people tell me already "OK,  
we have understood, we got the idea". But "to understand" is the  
english for the latin "comprehendere" (comprendre, in french). It  
seems that now, your conception of the universe is

{   { },    {{ }},    {{ }, {{ }}},      {{ },  {{ }},  {{ },  
{{ }}}}     ... }

This is day omega. Omega is the first infinite number. It is an Other  
number (note). not a natural number. It is the unavoidable infinite  
number IIIIIII.....   omega = {0, 1, 2, 3, ...}. It is the well known  
set of all natural numbers.
OK, but if I "comprehend it" I have to put it in the universe by the  
reflexion rule. So at the middle of the day omega+1, my conception of  
the universe:

{   { },    {{ }},    {{ }, {{ }}},      {{ },  {{ }},  {{ },  
{{ }}}}     ...  {   { },    {{ }},    {{ }, {{ }}},      {{ },   
{{ }},  {{ }, {{ }}}}     ... }}

omega+1 is {0, 1, 2, 3, ... omega}

Ok, but if I comprehend it, I have to put it in the universe, so I get

{   { },    {{ }},    {{ }, {{ }}},      {{ },  {{ }},  {{ },  
{{ }}}}     ...  {   { },    {{ }},    {{ }, {{ }}},      {{ },   
{{ }},  {{ }, {{ }}}}     ... } {   { },    {{ }},    {{ },  
{{ }}},      {{ },  {{ }},  {{ }, {{ }}}}     ...  {   { },     
{{ }},    {{ }, {{ }}},      {{ },  {{ }},  {{ }, {{ }}}}     ... }}}

omega+2

get it? after some infinite time again the universe looks like

{0, 1, 2, ... omega, omega+1, omega+2, omega+3, omega+4, omega+5, ...}

This is omega+omega,
and thus this continues

omega+omega+1, omega+omega+2, omega+omega+3, omega+omega+4, omega+omega 
+5, ...

Which leads to omega+omega+omega
omega+omega+omega+1
omega+omega+omega+2
omega+omega+omega+3
...
which leads to
omega+omega+omega+omega
omega+omega+omega+omega+1
omega+omega+omega+omega+2
omega+omega+omega+omega+3
...
which leads to
omega+omega+omega+omega+omega
omega+omega+omega+omega+omega+1
...
which leads to
omega+omega+omega+omega+omega
...
omega+omega+omega+omega+omega+omega
...
omega+omega+omega+omega+omega+omega+omega
...
omega+omega+omega+omega+omega+omega+omega+omega
...
omega+omega+omega+omega+omega+omega+omega+omega+omega
...
omega+omega+omega+omega+omega+omega+omega+omega+omega+omega
...
...
which leads to

omega*omega
omega*omega+1
omega*omega+2
...
omega*omega+omega

and you can guess (making giant steps):

omega*omega*omega
... ...
omega*omega*omega*omega
... ...
omega*omega*omega*omega*omega
... ...
omega*omega*omega*omega*omega*omega
... ...
omega*omega*omega*omega*omega*omega*omega
... ...
omega^omega

and speeding
omega^omega^omega
...
omega^omega^omega^omega

leading to
omega^omega^omega^omega^...

which is named epsilon zero. It is a star in logic, by playing some  
role in proof theory. Epsilon zero is still a very little ordinals, as  
such "other" infinite, transfinite number are called.

We will need only omega. Computability theory need even much higher  
ordinal than epsilon zero, but don't worry now about that. But natural  
numbers are like that, they behave so weirdly that you have to  
introduce many kind of "other numbers" to help to figure out what  
they, the natural numbers, are capable of.

It is good to met the ordinals at least once.
Do you think is is possible to comprehend *all* ordinal numbers? To  
get a picture of the whole universe of number and ordinals? (subject  
of reflexion).

Thanks for your work Kim and Russell,

Best regards to John and the others for they kindness,

Bruno

PS I will take the opportunity of JOUAL, and our conversations, to try  
sum up UDA (and AUDA?) in a short paper, and I have already accepted  
to participate to a mini-colloquium (by psychologists which are kind  
with me) this month in Brussels, so I have to write two papers, and  
this to say that I am a bit busy this month, so:  it is hollyday Kim!   
No more math until april! Take all your time for swallowing those  
ordinals, and don't be afraid to ask question (perhaps online so other  
can learn something too). Next lesson, in April: I say a bit more on  
those other numbers.


http://iridia.ulb.ac.be/~marchal/




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