it is said

http://en.wikipedia.org/wiki/Robinson_arithmetic

robinson arithmetic like PA is incomplete and incompletable in the sense
of Gödel's incompleteness Theorems, and essentially undecidable

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem

Gödel's first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:

    For any consistent formal, recursively enumerable theory that proves
basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.

but godel does not tell us what makes a statement true

until he does we cant tell if robinson arithmetic is incomplete

we need to know what makes a statement true in order to see that it cant
be proven

godel could have said there are gibbly statemets which cant be proven

but untill we know what makes a statement gibbly we have no way of knowing
what his theorem  is talking about

so untill he tells us what makes a statement true robinson arthmetic cant
be said to be incomplete

But since you have never studied this stuff,
you didn't know that.

Right.

i say
rubbish unless i know what a gibbly i cant not know what his theorem is
talking about

george say

 we DON'TNEED to know WHICH, or what "truth" "is",  in order to know to
know that theory is incomplete.

rubbish until i know what makes a statement true i cant identify a true
statement thus his theorem is meaningless as with out knowing what a true
statement is i cant say there are true statements which are not provable

george says

This whole endeavor DOES NOT EVEN CARE about
what's TRUE: it CARES about what's PROVABLE.
And we all HAVE told you that BEING AN AXIOM makes
a statement provable, AS does being INFERRABLE from
the axioms ACCORDING to the INFERENCE RULES of
the LOGIC.

i say
rubbish until i know what makes a statement true i cant identify a true
statement thus his theorem is meaningless as with out knowing what a true
statement is i cant say there are true statements which are not provable

goerge says

This whole endeavor DOES NOT EVEN CARE about
what's TRUE:

 i say rubbish it depends on there being identifibale true statements if i
cant tell what a true statement is the theorem collapses into
meaninglessness as i cant never find a true statement which i will know is
not provable

if there are no such thing as true statement the theorenm is meaningless
and
if there are true statement but i dont know what the hell they are then
again the theorem is meaningless as i cant make any identifications about
anything ie unprovable statments

Perhaps colin leslie dean would like to prove either (Ax)(0 + x = x)
or its negation in Robinson Arithmetic as defined e.g. in Boolos/
Burgess/Jeffrey edn 4, or in my book.

The theories have a "standard"
model, where 0 is zero, s(0) or (0+1) is 1, s(s(0)) or 1+1 is
two, ad inf.  All those numbers (and many different numeration
systems for them, in addition to the two mentioned HERE)
were ALREADY OUT THERE BEFORE Godel started
proving things, and EVERYbody was legitimately expected
to know what made 43*69=3426 true (or false), WITHOUT
any help from Godel or 1st-order logic.  So the point is,
WE DO ALREADY KNOW what "makes" these statements
true or false.  

i say
rubbish

you say

43*69=3426 true
it is only true via a proof
but godel is said to have made a distinction between true and proven
you are just giving the old discredired idea of truth
ie the hilbert idea that true statements are proven from axioms
you need to tell us why 43*69=3426 is true proof

quote
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system

(1) Robinson Arithmetic is complete
(2) Robinson Arithmetic has not been proved to be incomplete.

You started by saying (1), but you end up only arguing for (2). Do you
want to go with (1)? Do you think you can give some sort of
completeness proof for Robinson Arithmetic? Or do you just want to
stick with the weaker claim that it hasn't been proved to be
incomplete?

(1) We do have a definition of truth, due to Tarski
(2) In any case, Gödel's argument can be done without using the notion
of truth, there is a purely syntactic version of the argument which
only needs the assumption that Robinson Arithmetic is consistent

Cheers.

Perhaps colin leslie dean would like to prove either (Ax)(0 + x = x)
or its negation in Robinson Arithmetic as defined e.g. in Boolos/
Burgess/Jeffrey edn 4, or in my boo

i say rubbish
 godel made a distinction between true and proof
true is independent of a proof
so tell us why Ax)(0 + x = x) is true independent of a proof

In addition, from at least the time of Hilbert's program at the turn of
the twentieth century to the proof of Gödel's theorem and the development
of the Church-Turing thesis in the early part of that century, true
statements in mathematics were generally assumed to be those statements
which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system

You said Robinson Arithmetic is not incomplete. So you must think that
either this sentence or its negation can be proved in Robinson
Arithmetic. He's challenging you to actually do it.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

Who's that Colin Leslie anyway...? Why is he so frustrated by the
incompleteness theorem? weird....

Why do his followers keep claiming that Godel's theorem is false?

They keep claiming that there is a vicious circle in the demonstration
but there's none. There's no faulty circularity at all.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

You said Robinson Arithmetic is not incomplete. So you must think that
either this sentence or its negation can be proved in Robinson
Arithmetic. He's challenging you to actually do it.

i say

i could not care less
robinson arithmetic has not been proved to be   incomplete untill  we are
not told what make a statement true

the theorem depend on two things consistency and the presence of true
statements
but if you cant tell us what makes a statement true-independent of proof-
then the theorem is meaningles
regardless if (Ax)(0 + x = x)
or its negation in Robinson Arithmetic  can be prooved or not

and even if it was proved
what makes it true anyway

i say
not good enough tell s us what would makes us know GC is true even if it
is unprovable

tell us what would make GC true

And, with regard to your argument, as I was saying we do have a
definition of truth, but we can give an argument that Robinson
Arithmetic is incomplete without using the notion of truth at all.

Something about a formal system - as an axiom set - that has seemed to escape
our attention: it's supposed to reflect the entire underlying set of concepts;
hence the assumption of its consistency is a must however *assumed* it might
be. Consequently, any meta assertion that *must depend on the consistency*
must clearly state so to be a valid meta theorem!

There's actually nothing "sacred" about using Q as the encoding theory,
or about assuming N as the standard arithmetic model of Q, in GIT.
One could equally *assumes* ZFC be consistent, and perform certain
"set-ization", instead of "numerization", and still arrive at GIT,
as a hypothetical meta theorem. "Hypothetical" because of the assumption
ZFC be consistent in this case. It's only a matter of how explicit or implicit
we'd make this assumption for the encoding theory!

So what?