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
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