On 16 Mar 2015, at 23:33, LizR wrote:
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?
That gives example of representations. To use this as a definition of
numbers you will need to be circular: you will have to say something
like: 0 is a number, s(0) is a number, .... s(s(s(.... (0)))...) is a
number ... if "s" is repeated a *finite number of time*. There are
other ways, but they use implicitly the notion of finite number, or
more complex notions, like the notion of arbitrary set.
It can be proved that we cannot axiomatize the notion of finite things
in pure first-order logical theory. Second order logic somehow accepts
the intuition of finite, and build the rest from that. But then the
proofs are no more checkable, and we are no more in the formal frame.
It is math, no more logic.
Bruno
PS discover the post of the 16 mars today (!)
On 17 March 2015 at 05:39, Bruno Marchal <marc...@ulb.ac.be> 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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