7:3

without genealogy: Like other humans, Melchizedek must have had parents, and he may have had offspring. Therefore, Melchizedek would have had a genealogy, or family tree. But because the Bible does not record any details of Melchizedek’s ancestors or descendants, Paul could say that Melchizedek was “without genealogy.” The Syriac Peshitta (a translation in use from the fifth century C.E. onward) renders Heb 7:3 as follows: “It is not written in the genealogies who his father and mother were, when he was born, or when he died. Instead, his priesthood stands eternally in the same way as it does for God’s Son.” Jesus’ priesthood does not depend on human genealogy. He was not born in the priestly tribe of Levi. Rather, like Melchizedek, he became “a high priest” through a direct appointment by God. (Heb 5:10; 7:15, 16) In contrast, the laws regulating the Levitical priesthood were based on “fleshly descent,” so it was crucial that those priests keep accurate genealogical records.​—Heb 7:16; Nu 3:10, 15, 16; Ne 7:63, 64.

having neither a beginning of days nor an end of life: Paul does not imply that either Melchizedek or Jesus was literally without any beginning. Melchizedek was born in the natural way. Jesus was created as a spirit creature, “the firstborn of all creation” and “the beginning of the creation by God.” (Col 1:15 and study note; Re 3:14) The point is simply that Melchizedek​—like Jesus​—did not owe his priesthood to any inheritance conferred on him by his birth. Also, because Melchizedek’s “end of life” went unrecorded, Paul could use that fact to illustrate that Jesus’ priesthood would be an eternal one.

being made like the Son of God: Paul does not say that God made his Son like Melchizedek but that, in a sense, he made Melchizedek like his Son. Jehovah did so by seeing to it that the Genesis account about king-priest Melchizedek contained no details about his background, birth, family, or death. (Ge 14:18-20) That carefully worded account made it possible for Melchizedek to serve as a prophetic pattern​—a priest who was appointed directly by God.

he remains a priest for all time: The Bible contains no indication that Melchizedek had either a predecessor or a successor in his priesthood. That is why he can be called a priest “for all time,” or “perpetually.” “In the manner of Melchizedek,” Jesus had no predecessor. (Heb 5:5, 6, 10; 6:20; 7:15-17) Also, the Bible shows that Jesus’ “priesthood has no successors.”​—Heb 7:24 and study note.