On 2/19/2018 3:34 PM, John Clark wrote:
On Mon, Feb 19, 2018 at 3:55 PM, Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>>wrote:

    ​> ​
    Is there a non-constructive way to determine whether an axiomatic
    system (axioms+inference rules) must be Goedel incomplete?  Must
    one show that arithmetic is a model or is there some simpler method?


​
Any
​
axiomatic system
​
powerful enough to do arithmetic (and its not much use if it isn't)

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.  Of course for any finite system one can in principle simply write down all the theorems and see if there are any statements that are true but not among the theorems.  But then that raises the question of what 'true' means.  I think that in Goedel's incompleteness, 'true' doesn't mean corresponding to a fact.

Brent

can't be both self consistent (that is to say there is not a proof
​in the system​
 that something is true and also a proof that the same thing is false) and complete
​ (there are true statements that have no proof).
 And if
​the system​
 is self consistent there is no way to prove it is so while remaining in the system.

 ​John K Clark​




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