Re: A simple paradox in Godels incompleteness theorem that invalidat

On Oct 4, 11:52 pm, Peter_Smith <ps...@xxxxxxxxx> wrote:
On 5 Oct, 06:47, Newberry <newberr...@xxxxxxxxx> wrote:





On Oct 4, 11:52 am, Peter_Smith <ps...@xxxxxxxxx> wrote:

On 4 Oct, 15:52, Newberry <newberr...@xxxxxxxxx> wrote:

On Oct 3, 11:23 pm, Peter_Smith <ps...@xxxxxxxxx> wrote:

On 4 Oct, 04:29, Newberry <newberr...@xxxxxxxxx> wrote:

On Oct 3, 8:21 am, Peter_Smith <ps...@xxxxxxxxx> wrote:

Sigh. More idiocy.

Gödel's "holds for a very wide class of formal systems" is entirely
different from your "independent of the nature of the formal system".
The theorem holds for those those formal systems whose nature is such
that they can encode enough arithmetic. In other words, the
applicability of the theorem depends crucially on the nature of the
formal system under consideration.

I have a problem with this vagueness. Goedel proved a number of things
about one particular system:

1) It is syntatctically incomplete
2) It is semantically incomplete
3) It is omega incomplete
4) It cannot prove its own consistency

Specifically which of these holds for ANY system that can encode
enough arithmetic??

Prescinding from standard worries about exactly what we mean by
"cannot prove its own consistency", then being a primitively
recursively axiomatized theory that contains Q (Robinson arithmetic)
gives us all of (1) to (4). [Errrr .... is that the sort of answer you
wanted??]- Hide quoted text -

- Show quoted text -

No, I want to know which of 1, 2, 3, 4 holds for ANY system that can
encode sufficient arithmetic. For example I believe that 1 does. How
about the rest of them?

What exactly do you mean by "system", I wonder? After all if we take
the theory standardly called "True Arithmetic", whose axioms are the
truths of the language of first-order arithmetic, it trivially
contains "enough arithmetic" by any standard, but none of 1 to 4 are
true of it. So presumably you mean (primitively) recursively
axiomatized system?- Hide quoted text -

- Show quoted text -
"This situation does not depend upon the

special nature of the constructed
systems". NB "special". Repeat, NB "special". In other words,
incompleteness isn't due to some quirky specific feature of PM or
some
quirky specific feature of ZF. But of course that doesn't mean that
incompleteness doesn't depend on the nature of these two theories.
Indeed, later in the paper Gödel tells us about general character of
a
theory T to which the incompleteness theorem does -- a character
shared by PM and ZF. <<

As I said Goedel proved 1,2,3,4 about PM. Does 1,2,3,4 apply to all
the theories T to which 1 applies?

Depends a bit exactly how you spell out 1. (After all there are
negation incomplete theories for which the concept of omega-
completeness doesn't get to apply.) But if 1 is short for something
like "Any (primitively) recursively axiomatized theory which can
represent all (primitive) recursive functions is negation-incomplete"
or more or less equivalent "Any (primitively) recursively axiomatized
theory which extends Q is negation-incomplete", then yes the
corresponding versions of 2,3 and 4 apply.

I have never seen any proof of this. I also remember that some time
ago Rupert wrote that if the theory has canonical valuation then it
will also be semantically incomplete. It implies that if the theory
does not have canonical valuation then it will not necessarily be
semantically incomplete.



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