Re: Godel's incompleteness and formal language

From: tausyn (descent_software_at_!hey-you!.com)
Date: 02/03/05 

Date: Thu, 03 Feb 2005 19:37:59 GMT

Acme Diagnostics wrote:

> Truth Detector <TD@evaluator.com> wrote:
> >Jim Spriggs <jim.sprigs@ANTISPAMbtinternet.com.invalid> wrote in
> >>
> >> Two important things: it's all formal systems that contain a certain
> >> amount of arithmetic.
> >
> >Yes, I forgot to include that. But it's not just arithmetic, but the
> >equivalent to something more than '+' and '*'. First order logic doesn't
> >use arithmetic, per se, but does have equivelents.
> >>
> >> And it's "... are true but cannot be _proved_."
> >>
> >I don't know how you could show something to be true without proving it.
>
> Not being a mathematician either, I asked the same question and got
> this answer from Chris Menzel:
>
> "Well, in fact, we don't ever need to be able to *tell* which of the
> unprovable statements are true (though often we can) to know *that*
> there are true statements that are unprovable (in a given system).
> Godel showed that any reasonable set of axioms for arithmetic will be
> incomplete. That means that, for any such set of axioms, there are
> statements A in the language of arithmetic such that both A itself *and*
> its negation not-A are unprovable. By basic logic, one or the other has
> to be true. So even if we don't know which it is, we know that there
> are true statements that are not provable from the axioms."
>
> That sounds pretty clear to me.
>
> On what is a "true sentence", quoting Jeffrey Ketland:
>
> "Tarski's Definition of Truth for Arithmetic:
> Tr is the smallest set X of arithmetic sentences such tha t,
> (a) for any atomic sentence t=u, t=u is in X iff the value
> of t is the same of (sic?) u
> (b) for any sentence A, ~A is in X iff A is not in X
> (c) for any sentences A, B, A&B is in X iff both A and B
> are in X
> (d) for any sentence "forall x A(x)", "forall x A(x)" is in X iff,
> for any number n, A(n) is in X.
>
> When we say that an arithmetic sentence is true, we mean that
> it is an element of Tr. "
>
> From this it seems that "true" refers to a definitional truth,
> e.g. "Tarski's Definition," as opposed to an empirical truth and
> as opposed to a real-world (e.g. cause and effect) logical
> "true." When a math system is devised, "true" statements
> are simply defined.
>
> I also asked about "true" v. "fact" and received no clear answer
> that I can recall. It appears that mathematical logicians use the two
> interchangeably which is confusing to people like myself who think
> of facts as empirically confirmed predictions. If my computer does
> a logic calculation and then displays the result, that would be a
> logical "true" that was also a "fact." But since a Turing Machine
> is entirely a theoretical concept, I don't see how it could produce
> a "fact" in this sense, but only a logical "true" which in theory to
> the best of my knowledge is again a definitional truth.
>
> Larry
> ``

i believe it was russell who differentiated truth and fact along the following
lines: truth and falseness apply to hypotheticals, only. once you know the
"truth" of a statement you have a fact

facts are otherwise known as observations, tho the privileged information of
observations internal to individuals may be true or false with respect to
other observers

perhaps these mathematical logicians you refer to simply take a "fact" as a
statement that has been observed "to be 'true'", but i also think you confound
hardware logic and a truth in its circuitry, a purely mechanical thing, with
abstract truth in the general and with facts. a logic gate in a certain state
is a fact by itself but doesn't necessarily make another fact except by being
part of a representation of another observation 

--
tausyn

Relevant Pages

  • Re: Goedel - interesting problem?
    ... For obvious reasons, ... >no set of axioms such that we can tell what's an axiom and what isn't. ... If that person rejects "axioms" as the "true statements which are ... the basic logic sense of a logical truth condition. ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... >>any given axioms, there are true sentences NOT provable from that set. ... There are two reasons for this. ... for "truth of mathematics". ... there are true statements that are unprovable. ...
    (sci.logic)
  • Re: Godels incompleteness and formal language
    ... > there are true statements that are unprovable. ... That means that, for any such set of axioms, there are ... > Tr is the smallest set X of arithmetic sentences such tha t, ... > the best of my knowledge is again a definitional truth. ...
    (sci.math)
  • Re: Godels incompleteness and formal language
    ... Godel showed that any reasonable set of axioms for arithmetic will be ... "Tarski's Definition of Truth for Arithmetic: ... Tr is the smallest set X of arithmetic sentences such tha t, ... of facts as empirically confirmed predictions. ...
    (sci.logic)
  • Re: Godels incompleteness and formal language
    ... Godel showed that any reasonable set of axioms for arithmetic will be ... "Tarski's Definition of Truth for Arithmetic: ... Tr is the smallest set X of arithmetic sentences such tha t, ... of facts as empirically confirmed predictions. ...
    (sci.math)
Quantcast