On 10 Jun 2009, at 01:50, Jesse Mazer wrote:
> Isn't this based on the idea that there should be an objective truth
> about every well-formed proposition about the natural numbers even
> if the Peano axioms cannot decide the truth about all propositions?
> I think that the statements that cannot be proved are disproved
> would all be ones of the type "for all numbers with property X, Y is
> true" or "there exists a number (or some finite group of numbers)
> with property X" (i.e. propositions using either the 'for all' or
> 'there exists' universal quantifiers in logic, with variables
> representing specific numbers or groups of numbers). So to believe
> these statements are objectively true basically means there would be
> a unique way to "extend" our judgment of the truth-values of
> propositions from the judgments already given by the Peano axioms,
> in such a way that if we could flip through all the infinite
> propositions judged true by the Peano axioms, we would *not* find an
> example of a proposition like "for this specific number N with
> property X, Y is false" (which would disprove the 'for all'
> proposition above), and likewise we would not find that for every
> possible number (or group of numbers) N, the Peano axioms proved a
> proposition like "number N does not have property X" (which would
> disprove the 'there exists' proposition above).
>
> We can't actual flip through an infinite number of propositions in a
> finite time of course, but if we had a "hypercomputer" that could do
> so (which is equivalent to the notion of a hypercomputer that can
> decide in finite time if any given Turing program halts or not),
Such an hypercomputer is just what Turing called an "oracle". And the
haslting oracle is very low in the hierarchy of possible oracles.
And Turing results is that even a transfinite ladder of more and more
powerful oracles that you can add on Peano Arithmetic, will not give
you a complete theory. Hypercomputing by constructive extension of PA,
with more and more powerful oracles does not help to overcome
incompleteness, unless you add non constructive ordinal extension of
"hypercomputation".
This is the obeject of the study of the degrees of unsolvability,
originated by Emil Post.
Arithmetical truth is big. No notion of hypercomputing can really
help. Yet, the notion of arithmetical truth is well understood by
everybody, and is easily definable (as opposed to effectively
decidable or computable) in usual set theory. That is why logician
have no problem with the notion of standrd model of Peano Arithmetic,
for example.
> then I think we'd have a well-defined notion of how to program it to
> decide the truth of every "for all" or "there exists" proposition in
> a way that's compatible with the propositions already proved by the
> Peano axioms.
Hypercomputation will not help. Unless you go to the higher non
constructive transfinite. But of course, in that case you are using a
theory much more powerful than peano Arithmetic and its extension by
constructive ordinal. You have to already believe in the notion of
truth on numbers to do that.
> If I'm right about that, it would lead naturally to the idea of
> something like a "unique consistent extension" of the Peano axioms
> (not a real technical term, I just made up this phrase, but unless
> there's an error in my reasoning I imagine mathematicians have some
> analogous notion...maybe Bruno knows?)
Just go to set theory. Arithmetical truth, or standard model of PA,
can play that role. It is not effective (constructive) but it is well
defined. Mathematicians used such notions everyday. If you belive in
the excluded middle principle on closed arithmetical sentences, you
are using implictly such notions.
> which assigns truth values to all the well-formed propositions that
> are undecidable by the Peano axioms themselves.
You can do that in set theory. Of course, this is not an effective way
to do it, but we know, by Godel, that completeness can never be given
in any effective way. Set theory can define the standard model (your
"unique extension") of PA, but it is not a constructive object. In set
theory, few object are constructive.
> So this would be a natural way of understanding the idea of truths
> "about the natural numbers" that are not decidable by the Peano
> axioms.
After Godel, truth, even on numbers can be well defined, in richer
theory, but have to be non effective, non mechanical. It is not a
reason to doubt about the truth of the arithmetical propositions. On
the contrary it shows that such truth kicks back and refiute all
effective definition we could belive in about that whole truth.
>
> Of course even if the notion of a "unique consistent extension" of
> certain types of axiomatic systems is well-defined, it would only
> make sense for axiomatic systems that are consistent in the first
> place. I guess in judging the question of the consistency of the
> Peano axioms, we must rely on some sort of ill-defined notion of our
> "understanding" of how the axioms should represent true statements
> about things like counting discrete objects.
We have to rely on our intuition of numbers and sets. It is ill-
defined, but today we know there is nothing better, and there there
will never anything better. That is why it is difficult to refute an
ultra-intuitionist, or why it is difficult to refute a zombie. Notion
like "natural numbers" or "consciousness" just cannot be defined so as
to be comprehensible by someone who does not already grasp those
notions. That is the main reason of the failure of logicism.
> For example, we understand that the order you count a group of
> discrete objects doesn't affect the total number, which is a
> convincing argument for believing that A + B = B + A regardless of
> what numbers you choose for A and B.
A + B = B + A, for all A and B, is already not provable in the
Robinson arithmetic, but you can prove in Peano Arithmetic. You need
the schema of axioms of induction.
> Likewise, we understand that multiplying A * B can be thought of in
> terms of a square array of discrete objects with the horizontal side
> having A objects and the vertical side having B objects, and we can
> see that just by rotating this you get a square array with B on the
> horizontal side and A on the vertical side, so if we believe that
> just mentally rotating an array of discrete objects won't change the
> number in the array that's a good argument for believing A * B = B *
> A. So thinking along these lines, as long as we don't believe that
> true statements about counting collections of discrete objects could
> ever lead to logical contradictions, we should believe the same for
> the Peano axioms.
I would say it is a complete mystery why we believe in those axioms,
but I have never meet someone who does not believe in it. Even Torgny
is forced to believe in it so as to be able to assert that he does not
believe in it. A real ultrafinitist cannot assert that he is
ultrafinitist. A real ulrafinitist should just ask, what do you mean
by natural numbers. And the only answer we can give him is "sorry pal,
but this is stidied in primary school, and if you have not understand
it (as opposed to some building of a sophisticated philosophical
argument against them), there is nothing we can do for you. Of course
Torgny know very well what we are talking about, like he knows very
well what consciousness is.
You can see the UDA+AUDA as a reduction of the mind-body appearance to
the mystery of numbers. The beauty of numbers, is that we can explain
today in detail why, IF someone believe in numbers, then by work alone
he can understand why the numbers are impossible to define.
But it is a fact that they are very easy to grasp.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Jun 10 2009 - 18:03:26 PDT