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.

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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/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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