Revelation 13:8 (Greek)
καὶ προσκυνήσουσιν αὐτὸν πάντες οἱ κατοικοῦντες ἐπὶ τῆς γῆς οὗ οὐ γέγραπται τὸ ὄνομα αὐτοῦ ἐν τῷ βιβλίῳ τῆς ζωῆς τοῦ ἀρνίου τοῦ ἐσφαγμένου ἀπὸ καταβολῆς κόσμου
Revelation 13:8 (NASB)
All who dwell on the earth will worship him, everyone whose name has not been written from the foundation of the world
in the book of life of the Lamb who has been slain.
I will look at the portion highlighted in colour
1. ἐν 2. τῷ 3. βιβλίῳ
1. preposition [in] 2. definite article [the] 3. noun [book]
4. τῆς 5. ζωῆς
4. definite article [of-the] 5. noun [of-life]
6. τοῦ 7. ἀρνίου
6. definite article [of-the] 7. noun [of-lambkin]
The reason for the different versions of the definite article translated as "the" or "of-the" is because the definite article is an adjective, and its case, gender and number always match the word it modifies. Even though each example is the same word in root the English definite article may not fully represent what it is talking about.
τῷ βιβλίῳ
the and book are both dative, neuter, singular
τῆς ζωῆς
the and life are both genitive, feminine, singular
τοῦ ἀρνίου
the and lambkin are both genitive, neuter, singular
I think singular number everyone understands. In the case of gender feminine should be pretty clear. If not clear neuter gender is neither male nor female. Probably the biggest problems that people have is the "case" which the examples used are "dative" and "genitive".
Dative Case:
in general the dative marks the indirect object of a verb. There is no problems translating this as "the book" and the english communicates the indirect object simply from the position of the words
Genitive Case:
this marks possession and in English we would write "of" or an apostrophe "s" to indicate possession. So there is a problem translating this as "the life" or "the lambkin" as the English does not indicate the possession. Sometimes if you look at an interlinear bible you might see the noun translated as "of life" or your might see the article translated as "of the". In truth both words are marked with possession with the article agreeing with the noun. The correct translation in both of these cases are "of the life" and "of the lambkin"
The final translation of the area in questions is:
... in the book of the life of the lambkin ...
our names must not be written simply "in the life of the lambkin" as the text does not say this and the possesion is directly marked saying "
of the life
of the lambkin". What is also directly marked in the indirect object which is "the book". Your translation goes against the grammar directly presented in the text. Our names need to be written "from the foundation of the world" (object) "in the book" (indirect object) which is "of the life" (showing possession of the life) "of the lambkin" (showing possession of the lambkin). The position of the words grammatically represented in the indirect object and the possession are clear in the Greek and should be put no other way when translating.