Re: Infinities, cardinality, diagonalisation

2006-07-16 Thread Tom Caylor
Thanks. I guess I agree with the your quale and betting descriptions. Finiteness is an existence statement, there exists an end. If we are talking about an actual particular group of things, we need an observer to say where the end is, thus declaring it to be finite. But it is another thing

Re: Infinities, cardinality, diagonalisation

2006-07-15 Thread Bruno Marchal
Le 14-juil.-06, à 18:52, Tom Caylor a écrit : Here is where I believe the crux is: ... means you can continue to add the I as many times as you want. Actually, this is equivalent to: ... means you can continue to add the I as many times as you want and you can. It's just a little

Re: Infinities, cardinality, diagonalisation

2006-07-14 Thread Bruno Marchal
Hi Quentin, Tom and List, Of course, N is the set of finite positive integers: N = {0, 1, 2, 3, ...}. An infinite set A is countable or enumerable if there is a computable bijection between A and N. Forgetting temporarily the number zero, all finite number can be put in the shapes: | ||

Re: Infinities, cardinality, diagonalisation (errata)

2006-07-14 Thread Bruno Marchal
Le 14-juil.-06, à 14:34, Bruno Marchal a écrit : Hi Quentin, Tom and List, Of course, N is the set of finite positive integers: N = {0, 1, 2, 3, ...}. An infinite set A is countable or enumerable if there is a computable bijection between A and N. Please suppress the computable in

Re: Infinities, cardinality, diagonalisation

2006-07-13 Thread Quentin Anciaux
Hi, thank you for your answer.But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but then you enter the problem I've shown, so N shouldn't contains infinite length

Re: Infinities, cardinality, diagonalisation

2006-07-13 Thread Jesse Mazer
Quentin Anciaux wrote: Hi, thank you for your answer. But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but then you enter the problem I've shown, so N shouldn't

Re: Infinities, cardinality, diagonalisation

2006-07-13 Thread Tom Caylor
N is defined as the positive integers, {0, 1, 2, 3, ...}, i.e. the *countable* integers. (I am used to starting with 1 in number theory.) N does not include infinity, neither the countable infinity aleph_0 nor any other higher infinity. Infinite length integers fall into this category of

Re: Infinities, cardinality, diagonalisation

2006-07-13 Thread Tom Caylor
Technically, I should say that countable means that the set can be put into a one-to-one correspondence with *a subset of* N, to include finite sets. Tom Tom Caylor wrote: N is defined as the positive integers, {0, 1, 2, 3, ...}, i.e. the *countable* integers. (I am used to starting with 1

Infinities, cardinality, diagonalisation

2006-07-12 Thread Quentin Anciaux
Hi list, I have a question I've been thinking about for a while... It may seems stupid, but I need to understand where I'm wrong. So here it is... Does the set N contains infinite number ? I ask this because Cantor prove with the diagonalisation argument that the set R is uncountable and

Re: Infinities, cardinality, diagonalisation

2006-07-12 Thread Tom Caylor
I think my easy answer is to say that infinite numbers are not in N. I like to think of it with a decimal point in front, to form a number between 0 and 1. Yes you have the rational numbers which eventually have a repeating pattern (or stop). But you also have in among them the irrational