On Sat, Sep 23, 2006 at 03:26:21PM +0200, Bruno Marchal wrote:
>
> Please allows me at this stage to be the most precise as possible. From
> a logical point of view, your theory of Nothing is equivalent to
> Q1 + Q2 + Q3. It is a very weaker subtheory of RA. It is not sigma1
> complete, you don't get the the UTM, nor all partial recursive
> functions FI or all r.e. set Wi. Actually you cannot recover addition
> and multiplication.
I'm not sure this is right, although I don't know what Q1, Q2 and Q3
are.
The Nothing itself does not have any properties in itself to speak
of. Rather it is the PROJECTION postulate that means we can treat it
as the set of all strings, from which any conscious viewpoint must
correspond to a subset of strings.
I should note that the PROJECTION postulate is implicit in your UDA
when you come to speak of the 1-3 distinction. I don't think it can be
derived explicitly from the three "legs" of COMP.
> But it is neither "nothing". It is the natural numbers without addition
> and multiplication, the countable order, + non standard models.
I disagree - it is more like the real numbers without order, addition
and multiplication group structures, but perhaps with the standard
topology, since I want to derive a measure.
But don't forget - this rich ontology is entirely due to the
PROJECTION postulate, not inherent to the Nothing.
> Or you have an implicit second order axiom in mind perhaps, but then
> you need to express it; and then you have a much richer ontology than
> the one expressed through RA.
>
Theres no implicit axioms in my mind, but it is always possible I have
unconsciously assumed something...
>
>
>
> >
> > One simply cannot observe this zero information object, one can only
> > observe somethings, descriptions in my terminology. Anything in
> > Sigma_1 is such a something.
>
> Sigma_1 is far richer. There are many sigma_1 true arithmetical
> sentences (provable by RA, PA, ZF, ...) not provable in your system.
>
Please send their proofs to me. In doing so you disprove your statement.
Anything provable by a finite set of axioms is necessarily a finite string of
symbols, and can be found as a subset of my Nothing.
>
> > Anything you can possibly to convey to me about
> > any mathematical object must also be extractable.
>
> Again, strictly speaking this is not true. (Unless your implicit axioms
> obviously ...)
>
How do you intend to convey the information to me, if not by some
finite string of symbols?
>
> > However, there are
> > possibly mathematical things not within the zero information objects,
> > but they are inherently noncommunicable (shades of you G*\G perhaps?).
>
> You are very well below. You cannot even prove the existence of a prime
> number in your theory.
>
>
> >
> > I think all that I say is that external reality is Nothing.
>
> No. Even your very weak theory as infinite models, and models of all
> cardinality. But it has no finite models, still less the empty model
> (which logicians avoid).
>
Why is the empty model "Nothing"? I don't think it is. Just as I don't
think the empty set is "Nothing". However, the empty string happens to
be identical to Nothing.
But it does have finite things, which curiously correspond to infinite
subsets (via duality).
--
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Mathematics
UNSW SYDNEY 2052 R.Standish.domain.name.hidden
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Received on Sun Sep 24 2006 - 04:13:18 PDT