Re: Torkel Franzen on truth

On Dec 12, 7:57 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Newberry says...







On Dec 12, 6:34 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
No, I believe that this follows; either
1) We do not know if PA is consistent (I mean we know zilch, we do not
even have a good reason to presume that it is consistent.)
2) The human mind can perform non-computable functions
3) There exists a formalization of arithmetic that can prove its own
consistency. (That would explain our intuition that PA is consistent.)

No, those *DON'T* follow. For one thing, Godel's theorem is
about proof, while the claim "We know that PA is consistent"
or "We don't know that PA is consistent" is about *knowledge*.
What are you assuming about the relationship between knowledge
and proof?

I mean that knowledge is a superset of proofs.

Proofs from what theory? That doesn't make any sense. Every
statement can be proved in *some* theory.

hence the human mind surpasses any machine (if you exclude 1 and 3.)

Don't say "hence". It doesn't follow. Once again, *PLEASE* put
it in the form of a syllogism. Every statement should (in principle,
at least) follow from previously establish statements or assumptions,
plus the rules of logic. If your terminology changes from statement
to statement (for example, changing from "proof" to "knowledge") you
need additional bridging statements showing what you are assuming
about the connection between these terms.

I did put my argument in syllogism.
a) Humans are certain that PA is consistent

Okay, we grant that as an assumption.

b) No formal proof of PA has any cogency (TF explicitly admitted this)

I don't grant that at all. Why not say what I just said: There
is no proof of the consistency of PA except for those use principles
that go beyond PA.

I thought we already sttles this one.

No, we did not.

You admitted this a few times on this thread. Torkel Franzen says "the
soundness proof of PA is not intended to allay any doubts at all." p.
110

How do you know that ZFC + an axiom of infinity are consistent?


I am amazed that people claim that when a system proves its own
consistency the proof does not have any cogency but when you prove
it is in a stronger system it does.

Whether a proof is "cogent" is a *psychological* fact. Are
we convinced by it? There are convincing arguments for the
consistency of PA. They convince *me*, anyway.

So you would not be convinced by a system that proves its own
consistency but you would be convinced if were proven by principles
that go beyond it?

c) Machines are capable only of formal proofs

That's a ridiculous thing to say. That's completely
false. I can program a machine to play games, process
images, process sound, solve differential equations, etc.
That's not proving formal proofs.

Come on! Obviously I meant that machines cannot arrive at arithmetic
result by any other means than formal proofs.

What reason is there to say that? That's ridiculous. That's
completely false. Almost *no* computer programs work by
making formal proofs. Yet there are computer programs that
arrive at mathematical conclusions.

Examples?

What you are saying is nonsense.

So let's try the proof this way. Let us assume we are programmed as
ZFC with a strong axiom of infinity. So we can prove the consistency
of ZFC to ourselves. But we also know that we do not know if ZFC with
a strong axiom of infinity is consistent. If it is not than the proof
of ZFC's consistency is invalid. So we are convinced about ZFC's
consistency because ZFC with a strong axiom of infinity is our
horizont. We are fooling ourselves. If do not want to admit that we
are fools we have to drop the assumption that we are equivalent to ZFC
with a strong axiom of infinity.

Let PA = T_0, ZFC = T_1, ZFC + axiom of infinity = T_2. There probably
exists a theory T3 in which the consistency with a strong axiom of
infinity can be proved. The argument in the first paragraph applies to
any theory T_n for n > 1. That is are not equivalent to any theory
T_n.

How else can we possibly prove a consistency of a given theory T? A
theory could prove its own consistency. That is our option 3). I do
not know if a consistency of any theory T can be proven in a weaker
system but if you want we can add option 4.)

What else? A heuristic learning program cannot do it because
mathematics is not an empirical science. A game playing program cannot
do it because it doesn not even attempt to answer any arithmetic
questions. That a system is consistent can be proven only in some
theory. So in order to produce a consistency proof the computer has to
programmed as a theory.

So we have
A) an inreasing hierarchy of theories,and we have proven in the first
paragraph that either we are NOT equivalent to any of those or that we
do not know if PA is consistent.
B) theories proving their own consistencies
C) maybe a decreasing hierarchy of theories.

.

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