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 t
Bruno Marchal wrote:
>
> 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
> 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
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 definiti
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
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
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.
>>
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
Hi Mirek,
Le 19-nov.-07, à 20:14, Mirek Dobsicek a écrit :
>
> Hi Bruno,
>
> thank you for posting the solutions. Of course, I solved it by myself
> and it was a fine relaxing time to do the paper work trying to be
> rigorous, however, your solutions gave me additional insights, nice.
>
> I am
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 "t
Hi Bruno,
thank you for posting the solutions. Of course, I solved it by myself
and it was a fine relaxing time to do the paper work trying to be
rigorous, however, your solutions gave me additional insights, nice.
I am on the board for the sequel.
Best,
Mirek
>
> I give the solution of the
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 theory" for the numbers.) The ordinary deduction r
Le 16-nov.-07, à 18:41, meekerdb (Brent Meeker) a écrit :
>
> 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 prop
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 prefe
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 natur
Bruno Marchal skrev:
Le 15-nov.-07, à 14:45, Torgny Tholerus a écrit :
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(
s(0))
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
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?
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 natural
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 n
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
potential infinite (accepted by
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 car
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
Hi David, Hi John,
OK, here is a first try. Let me know if this is too easy, too
difficult, or something in between. The path is not so long, so it is
useful to take time on the very beginning.
I end up with some exercice. I will give the solutions, but please try
to be aware if you can or can
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