FLC OF AN MD90 CLBED ABOVE AN INTERMEDIATE ASSIGNED ALT DURING INITIAL DEP DUE TO MISUNDERSTANDING OF THE CLRNC.

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AFTER TKOF; FLL TWR ADVISED US OF 'POP-UP' VFR TFC AT OUR 11 O'CLOCK AND 2 MI. WE WERE CLBING THROUGH 900 FT AND OUR ALT WAS ABOUT 1500 FT WHEN I SPOTTED THE TFC BELOW US. WHEN I NOTIFIED THE TWR THAT I HAD A VISUAL ON THE TFC; WE WERE CLRED TO CONTACT DEP. AT THIS TIME WE REACHED AND LEVELED OFF AT 3000 FT; WHICH WAS OUR ASSIGNED ALT. WHEN I SWITCHED FREQS; DEP WAS TALKING TO ANOTHER ACFT. AFTER; THE DEP CTLR WAS FINISHED; I CHKED IN 'LEVEL 3000 FT.' THE CTLR ACKNOWLEDGED OUR CALL SIGN AND CLRED US TO TURN TO A '310 DEG HDG AND MAINTAIN 13000 FT.' I READ BACK BOTH THE HDG AND ALT AND SET THE NEW ALT (13000 FT) IN THE ALT WINDOW ON THE PANEL. THERE WAS NO REPLY FROM THE CTLR. THE FO; WHO WAS FLYING THE ACFT; AND I USED THE 'DOUBLE POINT' PROC TO CONFIRM THE CORRECT ALT WAS ENTERED AND ARMED. WE BEGAN OUR CLB AND TURN AS INSTRUCTED. CLBING THROUGH 4100 FT; THE CTLR CALLED US AND STATED 'CALL SIGN; YOU'RE ASSIGNED ALT WAS 3000 FT; MAINTAIN 4000 FT.' I REPLIED; 'YOU ASSIGNED US 13000 FT; WHICH I READ BACK...WE ARE NOW DSNDING TO 4000 FT.' HE ANSWERED; 'I DIDN'T ASSIGN 13000 FT...MAINTAIN 5000 FT.' WE LEVELED AT 5000 FT. THE FLT CONTINUED WITH NO FURTHER INCIDENT NOR LOSS OF SEPARATION. OBSERVATION: DURING THIS HIGH ACTIVITY PERIOD OF INITIAL CLBOUT; AND WITH OTHER DISTRS CAUSED BY VFR TFC; COMS ARE VERY CRITICAL. THERE SHOULD BE NO REFS TO ALTS UNLESS THEY ARE DIFFERENT FROM WHAT IS BEING FLOWN; ESPECIALLY IF THERE IS NO TIME OR REQUIREMENT FOR A RESPONSE OR REPLY TO A 'READBACK.'