Re: Robot Evolution

John Edser wrote:
Aatu Koskensilta aatu.koskensilta@xxxxxxxxx wrote:-

A formal theory T is said to be consistent if there is no sentence A
such that both A and the negation of A are formally provable in T.

To express what you wrote in just a simple way: contradictions are not
allowed within any consistent theory. Is this correct?

It's not a question of allowing or prohibiting. The definition of a
consistent theory in mathematical logic is that a consistent theory is
such that there is no sentence A such that both A and the negation of A
are formally provable in T.

Gödel's first incompleteness theorem shows that, for formal theories T
meeting certain criteria, it is possible to find a formula G_T with the
property that G_T is true just in case T is consistent, i.e. G_T is true
if T is consistent, and false if T is inconsistent.

It appears G is deducible from T but not the reverse.

No. If T is consistent, then G_T is unprovable but true, and if T is
inconsistent, then G_T is provable (since everything is provable in an
inconsistent theory) but false. Here G_T is a sentence in the formal
language of T, expressing, by means of certain technical coding tricks,
that "G_T is not formally provable in T".

IOW what Gödel was driving at as far as the epistemology of science was
concerned is that mathematics remains entirely deducible from non mathematics
and not the reverse.

What does it mean to say that "mathematics remains entirely deducible
from non mathematics and not the reverse"? It appears in any case to
have absolutely no connection to the mathematical content of the
incompletenss theorems. The same goes for your other remarks. Your
reflections might be immensely significant, but perhaps it would be
better to leave poort old Kurt out of it all?

I just wished to make the rather trivial observation that the
incompleteness theorem establishes an implication, "if T is consistent,
then the Gödel sentence of T is true but unprovable in T", ..

JE:-
As I understand this: If T contains no contradiction then it remains a
valid induction, the truth of which cannot be proven.

It makes no sense to say of a formal theory that "it remains a valid
induction". A formal theory T is just a bunch of expressions in a
mathematically defined language. Such a theory might, or might not,
have some connection to our actual mathematical practice or theories,
depending on the theory in question, but that calls for a further
argument in each specific case.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus



.

Relevant Pages

  • Re: Would it matter if ZF was inconsistent?
    ... mathematics. ... always a bigger input than an algorithm can handle, ... aren't consistent with standard methods. ... infinitesimals that aren't) haven't been found to accomplish that. ...
    (sci.logic)
  • Re: parallels
    ... I strongly suggest you replace your consistent rhetoric (previously I ... If you reverse them then you reverse their ... indefinitely becoming amazingly complex (as they are within mathematics). ... provide just a single axiom which is not a tautology. ...
    (sci.bio.evolution)
  • Re: Complex numbers in the plane, axiom or consequence?
    ... David C. Ullrich ha escrit: ... Actually saying that there's no way to show PA is consistent ... as a basis for mathematics are assumed to be consistent. ... with the other axioms". ...
    (sci.math)
  • Re: When is a logic a quantum logic?
    ... > mathematics we seek the simplest consistent axiom set, ... > many valued logic is a consistent mathematical structure by reducing it ... Boolean logic some of the expressive power of topological spaces. ... The "non-classical" logic that Sikorski refers to are modal logics, ...
    (sci.physics.research)
  • Re: infinity
    ... the most "real" of the infinite systems, as it is the one that is most ... widely used across different areas of mathematics. ... ideal point at infinity to give the real line topological equivalence ... for any arguments which support them, consistent or not). ...
    (sci.math)