# Gödel II and Löb's theorem

The ab­stract form of Gödel’s sec­ond in­com­plete­ness the­o­rem states that if $$P$$ is a prov­abil­ity pred­i­cate in a con­sis­tent, ax­iom­a­ti­z­able the­ory $$T$$ then $$T\not\vdash \neg P(\ulcorner S\urcorner)$$ for a dis­prov­able $$S$$.

On the other hand, Löb’s the­o­rem says that in the same con­di­tions and for ev­ery sen­tence $$X$$, if $$T\vdash P(\ulcorner X\urcorner)\rightarrow X$$, then $$T\vdash X$$.

It is easy to see how GII fol­lows from Löb’s. Just take $$X$$ to be $$\bot$$, and since $$T\vdash \neg \bot$$ (by defi­ni­tion of $$\bot$$), Löb’s the­o­rem tells that if $$T\vdash \neg P(\ulcorner \bot\urcorner)$$ then $$T\vdash \bot$$. Since we as­sumed $$T$$ to be con­sis­tent, then the con­se­quent is false, so we con­clude that $$T\neg\vdash \neg P(\ulcorner \bot\urcorner)$$.

The rest of this ar­ti­cle ex­poses how to de­duce Löb’s the­o­rem from GII.

Sup­pose that $$T\vdash P(\ulcorner X\urcorner)\rightarrow X$$.

Then $$T\vdash \neg X \rightarrow \neg P(\ulcorner X\urcorner)$$.

Which means that $$T + \neg X\vdash \neg P(\ulcorner X\urcorner)$$.

From Gödel’s sec­ond in­com­plete­ness the­o­rem, that means that $$T+\neg X$$ is in­con­sis­tent, since it proves $$\neg P(\ulcorner X\urcorner)$$ for a dis­prov­able $$X$$.

Since $$T$$ was con­sis­tent be­fore we in­tro­duced $$\neg X$$ as an ax­iom, then that means that $$X$$ is ac­tu­ally a con­se­quence of $$T$$. By com­plete­ness, that means that we should be able to prove $$X$$ from $$T$$’s ax­ioms, so $$T\vdash X$$ and the proof is done.

Parents:

• Löb's theorem

Löb’s the­o­rem

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.