Dear Bruno, this is my reply to your "SeventhStep-2" post.
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?
*
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? and HOW? 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?{}
((your didactic epic is impressive).
Why do you have a zero within the 1? To make it '2',
{{ },{{ }}} 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?
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 the Plenitude. We
know only about THIS one.
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.
John M
On Fri, Mar 6, 2009 at 2:03 PM, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
> 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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Received on Fri Mar 06 2009 - 16:11:21 PST