Re: Bijections (was OM = SIGMA1)

2008-02-04 Thread Bruno Marchal
Le 30-janv.-08, à 13:43, Mirek Dobsicek wrote (in different posts): 2\ Bruno, you recently wrote that you do not agree with Wolfram's Principle of Computational Equivalence. As I understand to that principle, Wolfram says that universe is a big cellular automata. What is the evidence that

RE: Bijections (was OM = SIGMA1)

2007-11-29 Thread Jesse Mazer
Date: Tue, 20 Nov 2007 19:01:38 +0100 From: [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: Re: Bijections (was OM = SIGMA1) Bruno Marchal skrev: But infinite ordinals can be different, and still have the same cardinality. I have given

Re: Bijections (was OM = SIGMA1)

2007-11-21 Thread Bruno Marchal
Le 20-nov.-07, à 17:59, meekerdb a écrit : Bruno Marchal wrote: . But infinite ordinals can be different, and still have the same cardinality. I have given examples: You can put an infinity of linear well founded order on the set N = {0, 1, 2, 3, ...}. What is the definition of linear

Re: Bijections (was OM = SIGMA1)

2007-11-20 Thread Bruno Marchal
Le 19-nov.-07, à 17:00, Torgny Tholerus a écrit : Torgny Tholerus skrev: If you define the set of all natural numbers N, then you can pull out the biggest number m from that set.  But this number m has a different type than the ordinary numbers.  (You see that I have some sort of type

Re: Bijections (was OM = SIGMA1)

2007-11-20 Thread Torgny Tholerus
Bruno Marchal skrev: To sum up; finite ordinal and finite cardinal coincide. Concerning infinite number there are much ordinals than cardinals. In between two different infinite cardinal, there will be an infinity of ordinal. We have already seen that omega, omega+1, ... omega+omega,

Re: Bijections (was OM = SIGMA1)

2007-11-20 Thread Bruno Marchal
Le 20-nov.-07, à 12:14, Torgny Tholerus a écrit : Bruno Marchal skrev: To sum up; finite ordinal and finite cardinal coincide. Concerning infinite number there are much ordinals than cardinals. In between two different infinite cardinal, there will be an infinity of ordinal. We have

Re: Bijections (was OM = SIGMA1)

2007-11-20 Thread meekerdb
Bruno Marchal wrote: . But infinite ordinals can be different, and still have the same cardinality. I have given examples: You can put an infinity of linear well founded order on the set N = {0, 1, 2, 3, ...}. What is the definition of linear well founded order? I'm familiar with well

Re: Bijections (was OM = SIGMA1)

2007-11-20 Thread Torgny Tholerus
Bruno Marchal skrev: But infinite ordinals can be different, and still have the same cardinality. I have given examples: You can put an infinity of linear well founded order on the set N = {0, 1, 2, 3, ...}. The usual order give the ordinal omega = {0, 1, 2, 3, ...}. Now omega+1 is the

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread Torgny Tholerus
Quentin Anciaux skrev: Hi, Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit : What do you mean by "each" in the sentence "for each natural number"?  How do you define ALL natural numbers? There is a natural number 0. Every

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread Quentin Anciaux
Le Friday 16 November 2007 09:33:38 Torgny Tholerus, vous avez écrit : Quentin Anciaux skrev: Hi, Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit : What do you mean by each in the sentence for each natural number?  How do you define ALL natural numbers? There

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread Bruno Marchal
Le 15-nov.-07, à 14:45, Torgny Tholerus a écrit : Bruno Marchal skrev:Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit : What do you mean by ...? Are you asking this as a student who does not understand the math, or as a philospher who, like an ultrafinist, does not believe in the

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread meekerdb
Bruno Marchal wrote: ... If not, let us just say that your ultrafinitist hypothesis is too strong to make it coherent with the computationalist hypo. It means that you have a theory which is just different from what I propose. And then I will ask you to be ultra-patient, for I prefer to

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread Bruno Marchal
Le 16-nov.-07, à 09:33, Torgny Tholerus a écrit : There is a natural number 0. Every natural number a has a natural number successor, denoted by S(a). What do you mean by Every here?  Can you give a *non-circular* definition of this word?  Such that: By every natural number I mean

Re: Bijections (was OM = SIGMA1)

2007-11-16 Thread Torgny Tholerus
Bruno Marchal skrev: Le 15-nov.-07, 14:45, Torgny Tholerus a crit : But m+1 is not a number. This means that you believe there is a finite sequence of "s" of the type A = s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(

Re: Bijections (was OM = SIGMA1)

2007-11-15 Thread Bruno Marchal
Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit : Bruno Marchal skrev: 0) Bijections Definition: A and B have same cardinality (size, number of elements) when there is a bijection from A to B. Now, at first sight, we could think that all *infinite* sets have the same cardinality,

Re: Bijections (was OM = SIGMA1)

2007-11-15 Thread Torgny Tholerus
Bruno Marchal skrev: Le 14-nov.-07, 17:23, Torgny Tholerus a crit : What do you mean by "..."? Are you asking this as a student who does not understand the math, or as a philospher who, like an ultrafinist, does not believe in the potential infinite (accepted by

Re: Bijections (was OM = SIGMA1)

2007-11-15 Thread Quentin Anciaux
Hi, Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit : Bruno Marchal skrev: Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit : What do you mean by each x here? I mean for each natural number. What do you mean by each in the sentence for each natural number?  How

Re: Bijections (was OM = SIGMA1)

2007-11-14 Thread Torgny Tholerus
Bruno Marchal skrev: 0) Bijections Definition: A and B have same cardinality (size, number of elements) when there is a bijection from A to B. Now, at first sight, we could think that all *infinite* sets have the same cardinality, indeed the cardinality of the infinite set N. By N, I