MDT CREW HAD CTLR MISSPEAK; AND ISSUE AN ALT HE DID NOT INTEND.

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I WAS FLYING THE ACFT. THE FO WAS OPERATING THE RADIOS. WE WERE TALKING TO ZNY ON FREQ 132.87. BOTH THE FO AND MYSELF HEARD THE CTLR CLR US TO CLB TO FL350. THE FO READ BACK THE CLRNC TO FL350. AND I SET THE ALT OF FL350 IN THE ALT PRESELECT OF THE FLT GUIDANCE SYS. AS WE WERE CLBING THROUGH FL316; THE CTLR STATED THAT WE HAD GONE/CLBED ABOVE OUR ASSIGNED ALT OF FL310. FO ADVISED THE CTLR THAT HE HAD CLRED US TO FL350. THE CTLR SAID HE DID NOT CLR US TO FL350. I THEN SAID TO THE CTLR THAT THERE WAS NO DOUBT THAT WE WERE INSTRUCTED BY HIM TO CLB TO FL350; THAT THE FO HAD READ BACK THE CLRNC AND I HEARD ALL OF THE COM AND FO AGREED. WE WERE ASKED WHAT DID WE WANT TO DO; 'CLB TO FL350 OR DSND BACK TO FL310?' BY THIS TIME WE HAD LEVELED AT FL320 AND WE DECIDED TO DSND TO FL310. ABOUT 7 MINS LATER THE CTR CTLR HANDED US OFF AND I ASKED HIM IF THERE WAS ANY 'PROB' REGARDING THE EARLIER OCCURRENCE. HE SAID THERE WAS 'NO PROB' FROM HIS POINT OF VIEW AND THAT HE 'MAY' HAVE TOLD US TO CLB TO FL350; BUT HE MEANT FL310. THERE WERE 'NO OTHER ACFT AROUND.' THERE WAS NO CONFLICT. HE (THE CTLR) LEFT IT THAT EVERYTHING WAS FINE. I KNOW THAT I AND MY CREW DID WHAT WE WERE INSTRUCTED TO DO AND WE WERE NOT AT FAULT IN ANY WAY. THE ATC CTLR MADE A MISTAKE. THE CTLR SHOULD HAVE CAUGHT HIS MISTAKE WHEN THE FO READ BACK THE CLRNC TO CLB TO FL350.