On 05 Nov 2012, at 13:45, Roger Clough wrote:

Hi Bruno Marchal


Is sigma_6 truth truth with only a 6 sigma possibility of error ?

let P(x) be a decidable number property. Like being prime.

Note that if P(x) is decidable, then ~P(x) is decidable too. P(x), and ~P(x) are said sigma_0

Then, thanks to a theorem of Mostowski, you have a natural ladder of degrees of insolubility:

sigma_1 the proposition with shape ExP(x) = "it exists a number x such that it is the case that P(x)"

You can see that if a sigma_1 proposition is true, then, if you have enough time you can know it in principle. Just test P(x) on 0, then on 1, then on 2, etc. If "ExP(x)" is true, you will find that x eventially with that method.

Pi_1 the negation of of sigma_1 proposition. That is ~ExP(x), which is equivalent with Ax~P(x). Do you see that. If is is false that a number exists with the property P, it means that all numbers have the property ~P.

Now Pi_1 are a priori more complex to prove that the sigma_1, as you have to very that 0 has not p, and then 1 has not p, ad infinitum, in case the proposition is true. Note that you can still refute such a proposition in case it false, as you have just to verify the "ExP(x)" to refute it.

Sigma_2   =    Ay Ex P(x, y)      Pi_2   Ey Ax P(x, y)
etc.

So a sigma_6 proposition would be AxEyAzErAtEkAnEmP(x,y,z,r,t,n,m)

Very complex proposition. For example with an oracle for the halting problem, you can decide all sigma_1 truth or falsity, but you can't decide a sigma_2 proposition.

Note this: sigma_1 completeness (the ability to decide the true, (bt not necessarily the false) sigma_1 sentences) is equivalent with Turing universality.

There is no direct relationship with error correction.

Bruno






Roger Clough, rclo...@verizon.net
11/5/2012
"Forever is a long time, especially near the end." -Woody Allen


----- Receiving the following content -----
From: Bruno Marchal
Receiver: everything-list
Time: 2012-11-04, 08:56:01
Subject: Re: The two types of truth


On 03 Nov 2012, at 12:45, Roger Clough wrote:

Hi Bruno Marchal and Stephen,

http://www.angelfire.com/md2/timewarp/leibniz.html

"Leibniz declares that there are two kinds of truth:
truths of reason [which are non-contradictory, are always either
true or false],

We can only hope that they are non contradictory.
And although true or false, they are aslo known or unknown, believed
of not believed, disbelieved or not disbelieved, etc.




and truths of fact [which are not always either true or false].

Why? They are contextual, but you can study the relation fact/context
in the higher structure level.



Truths of reason are a priori, while truths of fact are a posteriori.
Truths of reason are necessary, permanent truths. Truths of fact are
contingent, empirical truths.
Both kinds of truth must have a sufficient reason. Truths of reason
have their
sufficient reason in being opposed to the contradictoriness and
logical inconsistency
of propositions which deny them. Truths of fact have their
sufficient reason in
being more perfect than propositions which deny them."

Unfortunately, this is acceptable below Sigma_1 truth, but doubtable
above, so even in the lower complexity part of arithmetic, things are
not that simple.

Bruno





Roger Clough, rclo...@verizon.net
11/3/2012
"Forever is a long time, especially near the end." -Woody Allen


----- Receiving the following content -----
From: Bruno Marchal
Receiver: everything-list
Time: 2012-11-03, 07:13:24
Subject: Re: Numbers in the Platonic Realm


On 02 Nov 2012, at 23:12, Stephen P. King wrote:

On 11/2/2012 1:23 PM, Bruno Marchal wrote:
I can understand these symbols because there is at least a way
to physically implement them.

Those notion have nothing to do with "physical implementation".

So your thinking about them is not a physical act?

Too much ambiguous. Even staying in comp I can answer "yes" and
"no".
Yes, because my human thinking is locally supported by physical
events.
No, because the whole couple mind/physical events is supported by
platonic arithmetical truth.
Dear Bruno,

Where is the evidence of the existence of a Platonic realm?

It is part of the assumption. We postulate arithmetic. I try to avoid
the use of "platonic" there, as I used the term in Plato sense. In
that sense Platonia = the greek No?, and it is derived from
arithmetic and comp.

All you need is the belief that 43 is prime independently of "43 is
prime".



The mere self-consistency of an idea is proof of existence

Already in arithmetic we have the consistence of the existence of a
prrof of the false, this certainly does not mean that there exist a
proof of the false. So self-consistency is doubtfully identifiable
with truth, and still less with existence.



but the idea must be understood by a multiplicity of entities with
the capacity to distinguish truth from falsehood to have any
coherence as an idea!

Not at all. 43 is prime might be true, even in absence of universe and
observer.



We cannot just assume that the mere existence of some undefined acts
to determine the properties of the undefined. Truth and falsity are
possible properties, they are not ontological aspects of existence.

Truth is no more a property than existence. It makes no sense.

Bruno


http://iridia.ulb.ac.be/~marchal/



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