the decidable / recognizable / co-recognizable picture

Once the dust settles, computability theory organizes every language into a small, clean map worth memorizing. There are three nested or overlapping zones. Decidable (recursive) languages are the friendliest: a machine always halts with a correct yes or no. Recognizable (recursively enumerable) languages are looser: a machine halts-and-accepts on every member, but may loop forever on non-members. Co-recognizable languages are the mirror image: their complement is recognizable, so a machine halts-and-rejects on every non-member but may loop on members.

The zones fit together by one elegant theorem: a language is decidable if and only if it is BOTH recognizable AND co-recognizable. Geometrically, decidable is exactly the intersection of recognizable and co-recognizable. A_TM lives in recognizable but not co-recognizable (you can confirm acceptance but never confirm non-acceptance). Its complement co-A_TM lives in co-recognizable but not recognizable. Their overlap — what is both — is the decidable region, and A_TM, being undecidable, must sit strictly inside recognizable only. Outside everything sits the vast majority: languages that are neither recognizable nor co-recognizable, which the counting argument guarantees are the norm.

This picture is the payoff of the whole undecidability story, and it dissolves a lot of confusion. 'Decidable' and 'recognizable' are not synonyms: recognizable is strictly weaker, because looping on non-members is allowed. 'Recognizable' and 'co-recognizable' are not the same either; A_TM and co-A_TM split them apart. And being undecidable still leaves room for being recognizable. Holding the three zones and their one connecting theorem in your head lets you place any language correctly and predict at a glance what kind of machine, if any, could ever handle it.