The English word “oath” can be defined as “a solemn or formal declaration invoking God” as well as “an act of making such a declaration.”1 In Greek, these two definitions could be translated as ὅρκος and ὁρκωμοσία, respectively. ὅρκος, as far as I know,2 does not refer to the “act of making such a declaration.” That is, it does not share the double meaning like our English word “oath.” Instead, it is the Greek word ὁρκωμοσία that refers to the “act of making such a declaration.” This is the word we find in Hebrews 7:28.
Therefore, when the author writes «ὁ λόγος...τῆς ὁρκωμοσίας», this translates as “the word of the oath-taking.” The sense of the genitive is: “the statement made during the oath-taking.” This refers back to Hebrews 7:17, which refers back to Psalms 110:4, wherein it is written,
4 Yahveh swore and will not repent, “You are a priest forever after the order of Malki-Tzedek.”
The English verb “swore” means that Yahveh took an oath;3 likewise for the Hebrew verb שָׁבַע (shava)4 and the Greek verb ὀμνύω.5 The oath itself is indicated by the statement contained within the quotation marks, for an oath itself is a (verbal) declaration. Therefore, “the word of the oath-taking” is the oath: “You are a priest forever after the order of Malki-Tzedek.”
1 Oxford English Dictionary online, “oath,” 1., a.
2 LSJ, p. 1252, ὅρκος
3 OED, “swear,” 2.:
To promise or undertake something by an oath; to take an oath by way of a solemn promise or undertaking. (Const. as in 1; also const. dative or to the person to whom the promise is made.).
4 For the Hebrew verb שָׁבַע (in Psa. 110:4), see Gesenius, p. 802, שָׁבַע:
“to swear...to promise any thing by an oath to any one.”
5 For the Greek verb ὀμνύω (in Heb. 7:21 and Psa. 110:4 LXX), see LSJ, p. 1223, ὀμνύω:
swear to a thing, affirm or confirm by oath.
Gesenius, Heinrich Friedrich Wilhelm. Gesenius’s Hebrew and Chaldee Lexicon to the Old Testament Scriptures. Trans. Tregelles, Samuel Prideaux. London: Bagster, 1860.
Liddell, Henry George; Scott, Robert; et al. A Greek-English Lexicon. 9th ed. Oxford: Clarendon, 1940.