On Mon, Feb 19, 2018 at 7:41 PM, Brent Meeker <meeke...@verizon.net> wrote:

​> ​
> There might be a lot of axiomatic systems that can't model arithmetic.
> For one thing they need to be countably infinite to do so.  So that's why
> I'm asking whether there is some other, more general critereon to show a
> system must be incomplete


Well if you had a axiomatic systems
​
that was inconsistent it would contain proofs of all true mathematical
statement, the trouble is it would have proofs of false statements too.
​
​
Being complete isn't the only virtue, being free from self contradictions
is even more important, and any system that is free of self contradictions
must be incomplete. And no
​
axiomatic system
​
can prove itself to be free from self contradictions unless it is very very
weak.
​

​
In 1930 in his PhD dissertation, one year before his famous incompleteness
theorem
​,​
Godel published his less famous *Completeness theorem
​"​
; he proved that at least one logical system
​
called
​ ​
"first order logic"
​
 is absolutely consistent and free from contradictions. The trouble is
first order logic is a toy logical system
​
, its OK for stuff like "All men are mortal-Socrates is a man-therefore
Socrates is mortal", but its too weak for arithmetic or much else.


The very next year Godel proved that ANY logical system with a finite
number of axioms and powerful enough to perform arithmetic (which to me
seems like a very VERY low bar to pass) is inconsistent or incomplete or
both. According to Godel in any logical system worth a damn an infinite
number of statements are true but un-provable, if the Goldbach Conjecture
is one of these it means that it's true so we'll never find a
counterexample to prove it wrong, and it means we'll never find a proof to
show it's correct. For a few years after Godel made his discovery it was
hoped that we could at least always identify statements that were either
false or true but had no proof. If we could
​
at least
​
do that then we would know we were wasting our time looking for a proof and
we could move on to other things, but in 1935 Turing proved that sometimes
even that
​
is
​
impossible.

​> ​
> I think that in Goedel's incompleteness, 'true' doesn't mean corresponding
> to a fact.
>

​
No, Godel thought things like the the
​
Goldbach conjecture
​
were either true or they were not, the question Godel explored is can we
ever know if it or things like it are true or not. Truth and fact are the
same, but proof does not necessarily correspond to truth if the axiomatic
systems
​
used is inconsistent
​
; and a lack of a proof does
​
not necessarily correspond to falsehood.

​ ​
John K Clark

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