# Re: The Seventh Step 2 (Numbers and Sets: facultary!)

```
Hi John,```
```
I comment you two last posts. It all begins by a slight
misunderstanding to end with deep agreement (except perhaps on Plato,
and the simplicity of the totality "seen from outside", if that makes
sense).

>
> Still not clear; Axiom 1 says I is 'a' number, - OK.
> Axiom 2 sais "x" which I understand is general for "any" number.  So
> xI is not different from II. The example: (say) I is 2, x=3, xI=32
> and your 'II' is not 'a' number, but two numbers (definitely NOT by
> axiom 2, rather by axiom 1)
> I did not see 'axiom 3' acording to which III or IIII would be 'a'
> number (any). They would be e.g.
> 593 or 4983 (or even 666 and 2222).
> I am still missing the definitions of the digits. Not enough
> intuition?
>

The goal was to define the following object I II III IIII ..., and
illustrating the difficulty to define the "...". No digits appear here.

The axiom 2 says that if x is a number, then the result of the
concatenation of x with I is a number too.
But x, for being used to find or build a new number, needs to be
the start.

By axiom 1, I is a number. By axiom 2 if x is number, like I, then the
result of concatenation of x with I, i.e. II, is again a number.
By axiom 2 again, we have that III is a number. Indeed it is the
concatenation of II, which we have already as number at the preceding
step, with I.
And thus III is a number.
And then IIII, by axiom 2, again.
And the IIIII, by axiom 2 again.
etc.

The problem now is that the "etc" is fuzzy. In particular should we
consider the infinite expression IIII... as a number. It follows by a
*number* of application of the axiom 2. To avoid it we could stipulate
that the axiom 2 should be applied a finite number of times. But
"finite number" is what we tried to define with the axioms 1 and 2 to
begin with. The point is that we cannot define "..."
"..." is already a little god that the mathematician have to invoke
(under the form of an axiom in a set theory, for example; but this
will fundamentally described the dynamic of the symbols.

>
> *
> My question to the von Neumann (an older school- mate from my
> highschool) notation:
> "The generation of the universe of numbers proceed in stages,
> beginning with an empty universe..."
> WHO generates?
>
You. Or some machine, easy to build, or program. Using the dynamic of
the machine to implement the "..." in some local way.

> and HOW?
>
That is what I (try) to explain.

> especially an EMPTY one (whatever an empty universe may be? does
> 'it' have borders? (nonexitent ones, because it is empty), or
> volume? measures? space? that all in emptiness?{}
>

No. It is just the set having no element. It is mathematical set. In
modern math, all the notion of border, volume, measure, space are
defined or represented in term of sets. More modern approach use
function as primitive, or categories. But set are still used
implicitly by most mathematicians as a foundation, if only in the
background.

>
> ((your didactic epic is impressive).
> Why do you have a zero within the 1? To make it '2',
>
That is the cute feature of this set representation of the numbers,
and indeed of all ordinals. Each number or ordinal *is* the set of all
those numbers (ordinals) which are little than itself.

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

>
>  {{ },{{ }}} if 2 equals {0, 1} still empty sets? Or now you
> substitute for the inside unexplained digits?
> Your #3 is indeed {0,1,2} (took some reconsidering) and I really
> appreciate that you did not go all the way of the 7 days of
> Creation). Thanks. However I plead guilty not to have gone through
> all the 'omegas'.
> I guess that (*) means multiplication and (^) power. These have no
> explained meaning in the preceeding sets.
> Epsilon is noted generally as something tiny, e.g. Paul Erdos called
> little children 'little epsilons', (it became an accepted entry in
> Webster's dictionary).
>
> Where did 'ordinals' come from?

Georg Cantor. The discoverer of the fact that an infinite can hide
other infinities.

http://en.wikipedia.org/wiki/Georg_Cantor

>
> Still no connection with 'a' universe. Where did THIS enter? besides
> in my 'narrative' (not theory) there are innumerable and quite
> different 'universes' as being 'fulgurated' (appear and vanish) from

I am not sure I can figure out what you are talking about, here. When
I talk on the "universe" of numbers, I am using the term "universe" in
the most vague sense as possible. It could mean {0, 1, 2, 3 ... } as
it could mean all possible numbers, and even everything, according to
the context. Sorry if I have not been clear.

>
> I gave a 'way' for their appearance and demise, no unexplained
> 'behavior' or 'activity' involved.

?

> One final adition: I have the feeling that 'your' universe is still
> an empty set what jibes well with the illusionary 'physical world'
> with matter imagined, that in the final analysis does not contain
> anything 'matterly' (why QM calls the partitioned 'energy' (also a
> meaningless name) as "particles"). All the rest is figment, base it
> on fictional numbers or else. And indeed, 'computation' requires a
> factor that computes.
> So far I still have my questions. Sorry.
>

I was just trying to give you another question: what do we mean by
"...".  And I was illustrating we just can't do it.

At the same time I was illustrating the notion of recursive
definition. I did already such a thing on the list when I define the
combinators (or their syntax).

I said

Axiom 1 K is a combinator
Axiom 2 S is a combinator
Axiom 3, if x is a combinator, and y is a combinator, then (x y) is a
combinator.

From this you can already build all combinators: they are K, S, (K
K), (K S), (S K), (S S), (K (K K)), (K (K S)), (K (S K)), (K (S S)),
((K K) K), ...

(And then you have already a full "programming language" or "universal
machine" with two little computation rules, which you can find here,
in those links to my combinatory posts to the everything list:

http://github.com/raganwald/homoiconic/blob/master/2008-11-12/combinator_chemistry.md

Let me comment your next post here:

John wrote:

> Bruno, - again the bartender...
> *
> Initial remark:
> I like Gunther's parenthetical condition of arithmetic consistency -
> which I find not assured in DIFFERENT universes.

Formally, the Löbian machine agrees, perhaps serendipitously, with
you. It is the godelian consistency of inconsistency. For each Lobian
machines, there are mathematical universes, making true the axiom of
arithmetic, or the specification of the machine, in which there exist
"proofs" that there are inconsistent. I put "proof" in quote, because
those "proof" are different from the usual one (based on our
incommunicable intuition of what we mean by "finite", again).

> As I said axioms (2+2=4) are
> in my opinion thought - conditions to make one's theory workable and
> so they are  conditioned by the circumstances.

"2+2 = 4" is true in all those (quite different) mathematical
structures. I don't see how "2+2 = 4" could depend on circumstances. I
guess you are interpreting "2+2 = 4" as something else. You are not
referring to the natural numbers.

> *
> What I try to add is the 'mind-body' problem. While I have no
> definition for "mind",

Assuming digital mechanism in cognitive science, there is a simple
definition of mind: the object study of computer science. You can then
define theology by the gap between computer science and computer's
computer science.

> we 'all' think to know what it means: a non-material mentality which
> encompasses the tool's (brain(function)) genetic built differences -
> i.e. enhanced  or reduced ease of connectivity-building in select
> topical domains - plus the sum of previous experience helping one's
> personal interpretation (and maybe more) including one's faith in a
> "soul" as well,

OK.

> while the 'body' is the formulation of a figment in the 'physical
> world' upon phenomena that are (mis/poorly) understood when received
> and both are parts of the complexity of us.

Nice. OK.

> I cannot figure a 'separation' of substantial parts of a complexity
> without destruction of the complexity in its entirety, so a
> "transport" can be only the entire complexity - or none.

I follow that intuition. Lobian machine too, of course proVably only
in a technical sense.

> Aristotle had it easy with his simple cognitive level of the
> 'physical world' so there was an easy possibility of thinking
> separately about the physical body and the rest of it not fitting
> into such.

It was an ingenious methodological assumption which has made possible
many discoveries. But it is a metaphysical trap which could lead to
the elimination of person in theory, if not in practice. OK.

>
> In brief: I se no 'mind-body' problem, only when we try the ancient
> (I may say: obsolete) ways of separating the 'physical world
> figment' from the total (complexity).

Plato didn't commit *that* error! nor Plotinus, nor any rationalist
mystics, from East, or West, or North, or South.
Also the total can be simple (it is the main agreed assumption of the
"everything" idea), and complexity appears from inside the totality.

> ((you promised an explanatory post to my askings - I am in a hurry
> to write down these remarks, because MAYBE after your explanations
> these would not make sense<G>))

Too late :)

Best regards,

Bruno

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

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