On 18 Dec 2012, at 01:50, Stephen P. King wrote:

On 12/17/2012 4:31 PM, meekerdb wrote:
On 12/17/2012 1:15 PM, Quentin Anciaux wrote:
ISTM that consistency is the fact that you can't have contradiction.

In some logics you're allowed to have contradictions, but the rules of inference don't permit you to prove everything from a contradiction. I think they are then called 'para-consistent'.

Incompletness that you can't prove every proposition.

No, incompleteness is you can't prove every true proposition. Which implies there is some measure of 'true' other than 'provable'.

Brent


Is there a logic that does not recognize a proposition to be true or false unless there is an accessible proof for it? Accessible is hard for me to define canonically, but one could think of it as being able to build a model (via constructive or none constructive means) of the proposition with a theory (or some extension thereof) that includes the proposition.

I am trying to see if we can use the way that towers of theories are allowed by the incompleteness theorems...

This is studied in recursion theory. Turing shows that incompleteness continue to all effective transfinite tower, on the constructive ordinals.

Bruno



--
Onward!

Stephen


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