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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