On 06 Oct 2011, at 23:14, Craig Weinberg wrote:
On Oct 6, 12:04 pm, Bruno Marchal <[email protected]> wrote:
On 04 Oct 2011, at 21:59, benjayk wrote:
The point is that a definition doesn't say anything beyond it's
definition.
This is deeply false. Look at the Mandelbrot set, you can intuit that
is much more than its definition. That is the base of Gödel's
discovery: the arithmetical reality is FAR beyond ANY attempt to
define it.
Can't you also interpret that Gödel's discovery is that arithmetic
can
never be fully realized through definition?
The usual model (N, +, *), taught in school, and called "standard
model of arithmetic" by logician fully "realize" it, and is definition
independent.
This doesn't imply an
arithmetic reality to me at all, it implies 'incompleteness'; lacking
the possibility of concrete realism.
The word "concrete" has no absolute meaning. Comp is "many types---no
Token".
So, the number 17 is always prime because we defined numbers in the
way. If
I define some other number system of natural numbers where I just
declare
that number 17 shall not be prime, then it is not prime.
No. You are just deciding to talk about something else.
I think Ben is right. We can just say that 17 is also divisible by
number Θ (17 = 2 x fellini, which is 8.5),
8.5 is not a 0, s(0), s(s0)), .... You are just calling "natural
number" what we usually call rational number. You illustrate my point.
You talk about something else, and you should have disagree with the
axioms that I have already given.
and build our number system
around that. Like non-Euclidean arithmetic.
That already exists, even when agreeing with the axioms, of, say,
Peano Arithmetic. We can build model of arithmetic where we have the
truth of "provable(0=1)", despite the falsity of it in the standard
model, given that PA cannot prove the consistency of PA. This means
that we have non standard models of PA, and thus of arithmetic. But it
can be shown that in such model the 'natural number' are very weird
infinite objects, and they do not concern us directly. But "17 is
prime" is provable in PA and is thus true in ALL interpretations or
models of PA. Likewise, the Universal Dovetailer is the same object in
ALL models of PA.
All theorems of PA are true in all interpretations of PA (by Gödel's
completeness theorem).
Primeness isn't a reality,
it's an epiphenomenon of a particular motivation to recognize
particular patterns.
They have to exist to be able of being recognized by some entities, in
case they have the motivation. The lack of motivation of non human
animal for the planet Saturn did not prevent it of having rings before
humans discovered them.
Bruno
http://iridia.ulb.ac.be/~marchal/
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