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We introduce a measure on strict plane partitions that is an analog of the uniform measure on plane partitions. We describe this measure in terms of a Pfaffian point process and compute its bulk limit when partitions become large. The above measure is a special case of the shifted Schur process, which generalizes the shifted Schur measure introduced by Tracy and Widom and is an analog of the Schur process introduced by Okounkov and Reshetikhin. We prove that the shifted Schur process is a Pfaffian point process and calculate its correlation kernel. Along the way we also obtain a one parameter generalization of Macmahon's formula for the generating function of plane partitions.