C560 CREW HAD AN ALT OVERSHOT IN ZID CLASS A.

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AT APPROX 12000 FT AND 5 MINS INTO FLT; A CLB CLRNC TO AN UNDERSTOOD ALT OF FL220 WAS ISSUED. NEXT XMISSION TO US WAS TO POINT OUT 2 TARGETS TO THE W OF US AT HIGHER ALT; ONE OF THE TARGETS; 'A SLOW CLBING RJ'; WAS AT APPROX FL200. A TA WAS HEARD. ONE OF THE TARGETS WE HAD OUT-CLIMBED; AND THE OTHER; WHICH THE TCASII WAS ALERTING US TO; WAS 1500 FT ABOVE US; AND AT LEAST 15 MI HORIZ. AN ADJUSTED VERT SPD COMMAND WAS GIVEN BY THE TCASII PASSING THOUGH FL210 FT. I IMMEDIATELY LEVELED THE AIRPLANE OFF OF FL213. APPROX 8 SECONDS LATER; A DSND RA WAS ISSUED. A DSCNT WAS INITIATED TO APPROX FL210; AT WHICH POINT WE WERE TOLD BY ATC; SOUNDING CONFUSED AND UNSURE; TO DSND BACK DOWN TO FL200. A PHONE NUMBER WAS ISSUED BY ZID; WHICH WE WERE TOLD TO CALL IMMEDIATELY. WE LATER FOUND OUT FROM THE ATC TAPE; THAT OUR CLB CLRNC WAS TO FL200; NOT FL220; AND THAT THE FO HAD READ BACK FL200 TO ATC. UNFORTUNATELY; THE FO HAD DIALED FL220 IN THE ALT WINDOW; AND I DIDN'T CATCH THE MISTAKE. FATIGUE WAS A FACTOR. WE HAD A 12 HR DUTY DAY THE DAY BEFORE; HAD ONLY 10 HRS OFF IN BTWN; AND HAD FLOWN OVER 10 HRS IN THE PAST 24 HRS. FORTUNATELY; THE TCASII HELPED AVOID A NEAR MISS. UNFORTUNATELY; ATC WAS NOT MUCH HELP IN AVOIDING THIS INCIDENT; SINCE WE WERE THROUGH OUR ASSIGNED ALT FOR MORE THAN A MIN BEFORE ATC REALIZED THE CRITICAL SIT. SUPPLEMENTAL INFO FROM ACN 577564: ON CLB OUT OF COLUMBUS; OHIO; WE WERE GIVEN AN ALT AT WHICH TO LEVEL OFF. I UNDERSTOOD IT TO BE FL220. I SET FL220 IN THE ALT ALERTER. I BELIEVE I READ BACK THE INSTRUCTIONS FROM THE CTLR AS 'FL220.' ATC CTLR TOLD US TO DSND TO FL200. WHEN WE REACHED FL200; THE CTLR STATED THAT OUR ASSIGNED ALT WAS FL200. WE REPLIED THAT WE UNDERSTOOD OUR ASSIGNED ALT TO BE FL220. HE DID NOT AGREE; AND INFORMED US TO CALL ZID; AND GAVE US THE PHONE NUMBER.