# Is the extension of every undecidable theory undecidable?

While it is not the case that the extension of every decidable theory is decidable, is it true that:

the extension of every undecidable theory undecidable?

In other words, given an undecidable theory A, is it enough to show

$$A \subseteq B$$

to prove that B is undecidable?

• Cross-posted to computer science and CS theory. Please do not do this. It is against site policy because it fragments answers and wastes people's time when they work on something that has already been answered elsewhere. Dec 3 '14 at 9:40
• Properties like undecidability tend to never be closed against sub- or superset. $\emptyset$ and $\Sigma^*$ are trivial counter-examples to any such claim.
– Raphael
Dec 3 '14 at 11:44