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:
|
||
|||
||||
|||||
||||||
|||||||
||||||||
...
This raises already an infinitely difficult problem: how to define
those finite numbers to someone who does not already have some
intuition about them. The answer is: IMPOSSIBLE. This is part of the
failure of logicism, the doctrine that math can be reduced to logic.
technically this can be explained through mathematical logic either
invoking the incompleteness phenomenon, or some result in model theory
(for example Lowenheim-Skolem results).
But it is possible to experience somehow that impossibility by oneself
without technics by trying to define those finite sequence of strokes
without invoking the notion of finiteness.
Imagine that you have to explain the notion of positive integer, or
natural number greater than zero to some extraterrestrials A and B. A
is very stubborn, and B is already too clever (as you will see).
So, when you present the sequence:
| || ||| ...
A replies that he has understood. Numbers are the object "|",and the
object "||", and the object "|||", and the object "...”. So A conclude
there are four numbers. You try to correct A by saying that "..." does
not represent a number, but does represent some possibility of getting
other numbers by adding a stroke "|" at their end. From this A
concludes again that there is four numbers: |, ||, |||, and ||||. You
try to explain A that "..." really mean to can continue to add the "|";
so A concludes that there are five numbers now. So you will try to
explain to A that "..." means you can continue to add the "I" as many
times as you want. From this A will understand that the set of numbers
is indefinite: it is
{|, ||, |||, ||||, |||||} or {|, ||, |||, ||||, |||||, ïïïïïï} or some
huge one but similarly ... finite.
Apparently A just doesn't grasp the idea of the infinite.
B is more clever. After some time he seems to grasp the idea of "...",
and apparently he does understand the set {|, ||, |||, ||||, |||||,
...}. But then, like in Tom's post, having accepted the very idea of
infinity through the use of "...", B, in some exercise, can accept the
infinite object
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||...
itself as a number. How will you explain him that he has not the right
to take this as a finite number. He should add that the rule,
consisting in adding a stroke "|" at the end of a number (like
"|||||||"), can only be applied a finite number of times. OK, but the
problem was just that: how to define "a finite number" ....
The modern answer is that this is just impossible. The set of positive
integers N cannot be defined univocally in any finite way.
This can take the form of some theorem in mathematical logic. For
example: it is not possible to define the term "finite" in first order
classical logic. There is not first order logic theory having finite
model for each n, but no infinite models.
You can define "finite" in second order logic, but second order logic
are defined through the intuition of finiteness/non-finiteness, so this
does not solve the problem.
This can be used to show that comp will make the number some absolute
mystery.
Now, note that B, somehow, can consider the generalized number:
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||...
as a number. Obviously, this corresponds to our friend the *ordinal
omega*. From the axiom that you get a number by adding a stroke at its
end: you will get
omega+1, as
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||...|
omega+2, as
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||...||
omega+3
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||...|||
...
omega+omega
||||||||||||||||||||||||||||||...||||||||||||||||||||||||||||||...
...
omega+omega+omega
||||||||||||||||||||||||||||||...||||||||||||||||||||||||||||||...||||||
||||||||||||||||||||||||...
...
omega*omega
|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...
|||...|||...|||...|||...|||...|||... ...
... and this generate a part of the constructive countable ordinals.
And we stay in the domain of the countable structure, unless you decide
to go up to the least non countable ordinals and beyond. For doing this
properly you need some amount of (formal) set theory. In all case, what
"..." expressed is unavoidably ambiguous.
Bruno
Le 13-juil.-06, à 15:29, Quentin Anciaux a écrit :
> 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
> positive integer number. But if they aren't natural number then as the
> set seems uncountable they are in fact real number, but real number
> have a decimal point no ? so how N is restricted to only finite length
> number (the set is also infinite) without infinite length number ?
>
> Thanks,
> Quentin
>
> On 7/13/06, Tom Caylor <Daddycaylor.domain.name.hidden> wrote:
>> 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 numbers which are uncountable. (Hey this reminds me of
>> the fi among the Fi.)
>>
>> To ask what is the next number after an infinite number, like
>> 11111...11111... is similar asking what is the next real number after
>> 0.11111...11111...
>>
>> Tom
>>
>>
>>
>>
>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Jul 14 2006 - 08:35:55 PDT