Without father (540) (apator from a = without + pater = father) means fatherless and was a term used in secular Greek to describe children who were orphaned, abandoned, estranged, or born out of wedlock. TDNT adds that apator had the following secular uses... Humans. When used of humans, apator can mean “orphan,” “foundling,” “bastard,” “of unequal parentage,” “disinherited,” or “of nonnoble or unknown origin.” In Judaism converted pagans are said to be “without father,” and Judaism also applies the sense of “unnamed” to Esther, who is an orphan in Esther 2:7. Deities. Such deities as Athena, Hephaestus, and Aphrodite are said to be without father or mother. God has no father in Orphic, Gnostic, and mystic works. The point is that he has no origin, or is uncreated. (Referring to the sole NT use in Hebrews 7:3) This says of Melchizedek that he has no father, mother, nor genealogy. The point is that he does not fall into the sequence of the Levitical priesthood. As the promise precedes the law in Paul, so this priesthood precedes the Levitical priesthood in Hebrews. Similarly, as the reference of the promise is to Christ, so the reference of Melchizedek's priesthood is to the high priesthood of Christ. (Kittel, G., Friedrich, G., & Bromiley, G. W. Theological Dictionary of the New Testament. Eerdmans) Without mother (282) (ametor) is motherless. Barclay comments on apator and ametor noting that... These words are very interesting. They have certain uses in secular Greek. They are the regular description of waifs and strays and of people of low pedigree. They contemptuously dismiss a man as having no ancestry. More, apator has a technical legal use in the contemporary Greek of the papyri. It is the word which is used on legal documents, especially on birth certificates, for father unknown and, therefore, illegitimate. So, for instance, there is a papyrus which speaks of: “ChairÄ"mōn, apator, father unknown, whose mother is ThasÄ"s.” Without genealogy (35) (agenealogetos from a = without + genealogeo = to trace a genealogy) is literally without a traceable genealogy and so without pedigree or descent. Barclay comments that this is a word that, so far as we know, no Greek writer ever used before. Vincent writes that... The meaning is that there is no record concerning his parentage. This is significant as indicating a different type of priesthood from the Levitical, in which genealogy was of prime importance. No man might exercise priestly functions who was not of the lineage of Aaron. The absence of traceable lineage is significant because it indicates that this order refers to one that is distinctly different from than the Levitical priesthood, wherein it was of critical importance to know one's genealogy. If one was not of the lineage of Aaron, they could not exercise priestly functions. As someone has said, the result of this emphasis on lineage was that priests were often more concerned about pedigree than holiness. Hughes writes that... The point is, Jesus’ priesthood, like Melchizedek’s, was based solely on the call of God, not on heredity. Jesus and Melchizedek were both appointed as “priests of God Most High.” (Hughes, R. K. Hebrews: An Anchor for the Soul. Volume 1. Crossway Books; Volume 2 or Logos) The silence of the Genesis narrative concerning Melchizedek’s parents or line of priestly descent was significant because of the contrast it posed with the Levitical priesthood, where recorded line of descent was required for accession to the priestly office Expositors Greek Testament writes that the terms "Without father, without mother, without genealogy" indicate that... he stands in Scripture alone, no mention being made of an illustrious father or mother from whom he could have inherited power and dignity, (and) still less can his priestly office and service be ascribed to his belonging to a priestly family. It is by virtue of his own personality he is what he is; his office derives no sanction from priestly lineage or hereditary rights; and in this respect he is made like to the Son of God. Of course it is not meant that in point of fact he had neither father nor mother, but that as he appears in Scripture his is without (such lineage). Having neither beginning...nor end - This does not mean that he came from nowhere. It simply means that in the Old Testament record nothing is said of his parents or origin. In practical terms the point the writer is seeking to get across to his readers with this statement about Melchizedek's having no beginning or end is that he was shadow the substance of which was realized in Messiah's eternal priesthood. As Vincent says... That is to say, history is silent concerning his birth and death. As an aside notable that some who believe in reincarnation appeal to this section of Hebrews to support their premise that Jesus is a reincarnation of Melchizedek. Clearly this is not a valid interpretation because the writer says Melchizedek was only “made like” Jesus, not that Jesus was Melchizedek. The writer also states in Hebrews 7:11 that Christ was a priest “according to the order of” Melchizedek and not that He was Melchizedek. BUT MADE LIKE THE SON OF GOD HE ABIDES A PRIEST PERPETUALLY: aphomoiomenos (RPPMSN) de to huio tou theou menei (3SPAI) hiereus: (Heb 7:17,23, 24, 25, 26, 27, 28) Made like - Note carefully that the Son of God is not made like Melchizedek, but he is made like the Son of God, and thus Melchizedek is presented in Scripture in such a way that he points to the truth about the Son of God. Leon Morris adds that... And it is the Son of God Who is the standard, not the ancient priest—king. The writer says that Melchizedek is “made like” (aphomoiomenos) the Son of God, not that the Son of God is like Melchizedek. Thus it is not that Melchizedek sets the pattern and Jesus follows it. Rather, the record about Melchizedek is so arranged that it brings out certain truths that apply far more fully to Jesus than they do to Melchizedek. With the latter, these truths are simply a matter of record; but with Jesus they are not only historically true, they also have significant spiritual dimensions. The writer is, of course, speaking of the Son’s eternal nature, not of his appearance in the Incarnation. (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing) Guthrie commenting on Melchizedek's priesthood wrote that... Any priesthood is evaluated according to the status of the deity who is served, which means that Melchizedek’s must have been of a highly exalted kind. Guzik commenting on "made like the Son of God" writing that... It really isn’t that Jesus has Melchizedek’s kind of priesthood. Instead, Melchizedek has Jesus’ kind of priesthood. (Ibid) "Copy and paste the address below into your web browser in order to go to the original page which will allow you to access live links related to the material on this page - these links include Scriptures (which can be read in context), Scripture pop-ups on mouse over, and a variety of related resources such as Bible dictionary articles, commentaries, sermon notes and theological journal articles related to the topic under discussion." http://www.preceptaustin.org/hebrews_71-3.htm#father