I don't think it's an argument from silence to consider Paul's overall perspective and then draw conclusions about a specific passage from that. For example, look at the New Testament teaching on eternal life and how that phrase is used in different places. A word study will reveal a larger concept centered around life in the coming age. Around here, we have coined the word aionic life: life as it will be in the age of ages.
Aionic life isn't merely an extension of this life that lasts forever and ever. It will be a world populated by human beings who have attained glorification and behave appropriately. They will love, honor, and obey God in every way. They will be dedicated to goodness, righteousness, and truth. And they will manifest the fruits of the Spirit.
When Jesus says that his sheep, his followers, have eternal life, he means that they have an entrance into Aionic life.
To put a sharper point on it, consider Paul