Re:Gödel, Escher, Bach: an in-depth explainer
by skoodge on Wednesday September 14, 2022 @04:10AM
(#32834476)
That's only true for systems where the principle of explosion actually holds though, isn't it? So it wouldn't apply to paraconsistent systems.
In the end, Gödel is actually giving us a choice: Either accept incompleteness or accept inconsistency. Of course it's true that historically incompleteness has been perceived as the only viable choice, but at least a few paraconsistent logicians like Graham Priest have argued for (non-explosive) inconsistency instead.
Somewhat oddly, this is actually technically correct - a "sufficiently rich" system is one that can distinguish true versus false statements of Peano Arithmetic, and a inconsistent system, by the principle of explosion, can prove any statement, so cannot distinguish statements of Peano Arithmetic (eg, 1+1=2 and 1+1=3 are both provable), and hence is not sufficiently rich.
That's a somewhat unintuitive way of looking at it in practice, though.