# Re: Infinities, cardinality, diagonalisation

```
Bruno Marchal wrote:
> 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.```
```
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 redundant to say it that way.
Now A and B *know*, as well as anyone can even know, what finite means.
All they have to do is perform some experimentation to get the idea
that, after a while of adding "I" they eventually get tired and/or
loose interest, so they have to *stop*.  What's so difficult about
understanding what stopping is?  Even the word "finite" has "fin" in
it, i.e. "end".  The notion is defined by invariance.  Something
similar (invariant) is happening (adding "I" at one step is considered
the same action as adding "I" in another step) and then the invariance
disappears, i.e. the adding of the "I" is no longer happening.

> 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
>
>
>

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