Re: Godel did not destroy the Hilbert Frege Russell programme
- From: Rupert <rupertmccallum@xxxxxxxxx>
- Date: Tue, 11 Mar 2008 20:41:55 -0700 (PDT)
On Mar 11, 7:14 pm, Newberry <newberr...@xxxxxxxxx> wrote:
On Mar 10, 8:38 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 11, 10:41 am, Newberry <newberr...@xxxxxxxxx> wrote:
On Mar 9, 11:01 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 10, 1:02 pm, Newberry <newberr...@xxxxxxxxx> wrote:
On Mar 9, 9:25 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Mar 10, 6:11 am, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
The Australian philosopher colin leslie dean shows that
Godel did not destroy the Hilbert Frege Russell programme to create a
unitary deductive system in which all mathematical truths can can be
deduced from a handful of axioms
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
Godel is said to have shattered this programme in his paper called "On
formally undecidable propositions of Principia Mathematica and related
systems"
This paper certainly shows how to prove in Z_2 that the set of first-
order arithmetical truths is not recursively enumerable. That's pretty
good evidence that you can't have a formal deductive system for all
mathematical truths.
(It was already known that the halting problem was recursively
unsolvable, but to deduce the further consequence that the set of
first-order arithmetical truths is not recursvely enumerable requires
Goedel's representation theorem, which appears as Proposition VII in
the paper).
but this paper it turns out had nothing to do with Principia Mathematica
and related systems" but instead with a completly artificial system
called P Godel uses axioms which where not in his version of PM thus his
proof/theorem cannot apply to PM thus he cannot have destroyed the
Hilbert Frege Russell programme and also his system P is artificial and
applies
to no system anyways
In "Principia Mathematica", Russell and Whitehead did not precisely
define their formal system. Today it is usually understood to be the
ramified theory of types with the axiom of reducibility (which is
easily seen to be interpretable in simple type theory). If we adjoin
the axiom of infinity to PM, in the form "there exist denumerably many
individuals", then the resulting system is bi-interpretable with P in
the following sense. There exists a one-one birecursive function f
from the set of well-formed formulas in the language of PM onto a
recursive subset of the set of well-formed formulas of the language
of P, and a one-one birecursive function g from the set of well-formed
formulas in the language of P onto a recursive subset of the set of
well-formed formulas in the language of PM. Given any sentence S in
the language of PM, S is a theorem of PM plus the axiom of infinity if
and only if f(S) is a theorem of P, and g(f(S)) is equivalent to S in
PM plus the axiom of infinity. Given any sentence T in the language of
P, T is a theorem of P if and only if g(T) is a theorem of PM plus the
axiom of infinity, and f(g(T)) is equivalent to T in P. In this sense
the two systems are equivalent. Goedel certainly shows, in his paper,
how to construct a sentence which cannot be proved or disproved in PM
plus the axiom of infinity, assuming that this system is omega-
consistent. His proof that the sentence has the properties in question
is finitary.
Obviously it is a fallacy to say that Goedel's proof must only apply
to P. He makes it clear in the paper that it applies to a large class
of systems. He presents some sufficient conditions for the theorem to
hold. It is now known that it holds for any consistent recursively
enumerable extension of Robinson Arithmetic.
Most people thought the axiom of reducibility was problematic given
that Russell's purpose was to vindicate logicism, and Russell agreed.
However, almost all working mathematicians today make free use of
impredicative comprehension. A handful of people, like Soloman
Feferman, think it should be avoided. This debate has no bearing on
the merits or interest of Goedel's proof, which goes through in a weak
fragment of finitary number theory, and applies to many object
theories, finitary, predicative, and impredicative.
The system in the second edition of PM is the same as the one in the
first edition and includes the axiom of reducibility, but this really
doesn't matter. Goedel's theorem applies to many systems which don't
use the axiom of reducibility. Even if this were not the case, this
would not rob the theorem of all interest. You tell us a theory you
like, then we'll discuss whether Goedel's argument applies to it..
Right here:http://xnewberry.tripod.com/IBL_2006_11_11.html
I'm not sure this is the best use of company time, but I started to
have a look. I haven't got up to the definition of your theory yet but
you start by talking about propositional logic. Goedel's result does
not apply to propositional logics. Perhaps you will move on to higher
languages later.
In fact I have moved to predicate calculus in the very first
paragraph.
Now, first of all, with regard to your expressions (1) and (2), am I
to understand those as specific formulas, or as schemas instantiated
by infinitely many different formulas?
P, Q, R are propositional variables of atomic formulas.
Second, (3) is not a subformula of (1) under the usual definition of
"subformula". Do you want to work with a different definition?
Look at the forrest not at the trees.
Obviously I can't evaluate this work if you refuse to tell me what the
definitions of your terms are. I'm not going to bother to give you
feedback unless you answer my questions.- Hide quoted text -
- Show quoted text -
Here is the relevant definition:
A set S of propositional variables of a formula A is truth determining
if the value of A is determined for all assignments of the truth
values to the set S.
I read this as follows.
Let A be a formula in a propositional language L with denumerably many
propositional variables. Let S be a subset of the set of propositional
variables of L. We say that S is truth-determining for A if there
exists a function f from the set of valuations of S to the set of
truth values {t,f}, such that, for all valuations v of S, the truth-
value of A under any Boolean valuation of the set of formulas of L
which agrees with v on S equals f(v).
In particular, the empty set would be truth-determining for every
tautology.
Have I read you correctly?
A variable P is truth redundant irrelevant) in A
with respect to S if P is not in S. Otherwise it is truth relevant (t-
relevant.)
With respect to S.
If an irrelevant variable exists in A for some set S then
the formula is not t-relevant.
By the above, that would be true for all tautologies.
"Subformula" is not a part of the formal exposition.
Do not misuse your company time.
I won't.
.
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