On 19 Nov 2014, at 16:44, Richard Ruquist wrote:
On Wed, Nov 19, 2014 at 5:12 AM, Bruno Marchal <[email protected]>
wrote:
On 19 Nov 2014, at 05:18, meekerdb wrote:
On 11/18/2014 4:57 PM, LizR wrote:
On 19 November 2014 06:45, meekerdb <[email protected]> wrote:
On 11/18/2014 5:00 AM, Bruno Marchal wrote:
On 17 Nov 2014, at 21:13, meekerdb wrote:
On 11/17/2014 2:55 AM, Bruno Marchal wrote:
The bible explains better (if we assume it is correct)
And if it isn't correct it doesn't explain anything. Which is
why science seeks to test correctness prior to explanatory power.
Ideally, or FAPP, perhaps.
But fundamentally, science cannot test correctness, not even
define it properly.
Are you saying that a theory cannot be tested an found incorrect??
I would think the obvious way to parse what Bruno has said here is
"science cannot show that something is correct".
Is that right, Bruno?
Yes.
Of course empirical tests are better at showing a theory is wrong
than showing it's right, which is Popper's observation.
Indeed.
I'm curious as to how you define correctness properly?
I can't do it for myself, nor can any machine do it for herself. But
a "sufficiently strong" machine can do it for a lesser strong
machine. You can define arithmetical truth and PA's correctness in
the set theory ZF for example. In that case "correctness" is defined
in the manner of Tarski: p is correct if it is the case that p is
satisfied by this or that mathematical structure, (for RA and PA,
you can use the usual (N,+, *) structure, and with computationalism,
that arithmetical truth (not definable in arithmetic) is enough).
This sounds like a description of which mathematical theories
suggest the existence of higher more-correct selves.
Not more correct, but knowing much more things. ZF knows that PA is
consistent, and ZF knows much more than PA about arithmetic, although
of course we still don't know if ZF knows the truth or the falsity of
Riemann hypothesis, but few doubt that ZF has any doubt about it.
Note that ZFC (ZF + the axiom of choice) does not know any more than
ZF. The axiom of choice has no consequences for arithmetic. (That is
not entirely easy to prove, but is a good exercise if you know Gödel's
constructible sets).
By Gödel's theorem, arithmetical truth is no exhaustible, so all
machines are superseded by other machines, and in fact this remains
true for the machine invoking Oracles (divine being which are supposed
to know the answer of Pi_1, Sigma_2, ... questions (by divine I just
mean here non computable, yet well definite in the standard model of
arithmetic (true or false).
ZF + kappa knows much more thing than ZF. In fact ZF + kappa believes
that ZF is consistent. And ZF+kappa believes vastly much more than ZF
about arithmetic, but is still under the jug of incompleteness, and
the hypostases apply to PA, ZF, ZFC, ZF+kappa, etc. (Assuming ZF+kappa
is consistent).
You can't be "more correct", as you are correct, or not. But the
spectrum of what you can believe in arithmetic can be very different.
The whole of the computable, Turing universality, is equivalent with
Sigma_1 complete. RA is already sigma_1 complete, and is quite humble
in her arithmetical knowledge. From PA and the extension, you have the
Löbianity (PA is not only sigma_1 complete, but PA knows that it/he/
she is sigma_1 complete, and it knows the plausible reason why it has
to be humble with respect to the arithmetical truth, on which it can
only point, without explicit definition).
Sigma_1 completeness, the ability to prove all true sentences having
the shape ExP(x), with P recursive/decidable, is universal with
respect to computability, but is very humble with respect of
provability, and there is no "universal provability" notion: all
provability predicate or machine can be extended (even mechanically)
to a more powerful machine, where powerfulness is measure in term of
classes of arithmetical propositions. There is just no complete
theory, definable by a machine, for the arithmetical reality. Gödel's
and Tarski's theorems makes the arithmetical truth quite
transcendental for the machines.
Bruno
Richard
Bruno
Brent
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