My apologies obviously you did mean finite.
This is very interesting although probably too much for my brain at the
moment.
What is all the stuff about S(S(0)) and {}, {{}}, etc? Doesn't that define
finite numbers?
On 17 March 2015 at 05:39, Bruno Marchal <[email protected]> wrote:
>
> On 15 Mar 2015, at 21:29, meekerdb wrote:
>
> On 3/15/2015 10:03 AM, Bruno Marchal wrote:
>
> We cannot define the notion of finite number
>
>
> This will make it very difficult to interpret the output of your computer.
>
>
> I guess you are joking.
>
> In case you are serious, you really should study a good book on logic.
>
> Machines can handle many things that they cannot define.
>
> To make my statement more precise, it means that we cannot build a theory
> having all natural numbers and only the natural numbers as model, by using
> first order logic. In fact no theory of any finite things can be formalized
> in first order logic. There is no first order axiomatization of finite
> group theory, of finite field, etc. There are good theories, even first
> order theories, but they have infinite models.
>
> We can formalized finiteness in ... second order logic. But this is a
> treachery because this use the notion of finiteness (in explicit or
> implicit way).
>
> That is the root of the failure of logicism. Not only we have to assume
> the natural numbers and they additive and multiplicative structure, (if we
> want use them), but we can't interpret them categorically or univocally. It
> is a strange world where it can be consistent for a machine to be
> inconsistent.
>
> What I really meant was: we cannot define the notion of number without
> using the notion of finite number.
> You might try, as a game to define natural number without using the
> notion, like if explaining them to someone who does not grasp them at all
> (if you can imagine that).
>
> You might say I is a number, and: if x is a number, then Ix is a number.
> The difficulty is in avoiding the person believe that IIIII... become a
> number, with a variety of meaning for IIII...
>
> Bruno
>
>
>
>
>
> Brent
>
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