As there are no Greek words for eternal or everlasting, the idea of eternal damnation and eternal life appears to falls apart on translation, apparently making the doctrine of eternal life and eternal punishment/damnation false?

''I understand the meaning of the word aionios (often appearing in genitive plural aionion) in Greek to carry the connotation of 'pertaining to the age' or 'age enduring.' The word is a form of the word we have borrowed into English from the Latin transliteration of the Greek as aeon or eon. The problems in interpreting it as the English “eternal” or “everlasting” are several.''

Adaptation and Syncretism
This pattern of starting with our cultural ideas, the looking for a Bible passage that could support it, leads us across the line into syncretism. There is very much unacknowledged syncretism in Western Christianity as a whole. This is one factor that makes western Christian forms different from the cultural and theological forms of non-western churches.

These ancient churches in other cultural setting have also tried to make the Gospel relevant in their cultures and histories. This process of interpreting in our cultural context is a natural way our brains work to learn. We start where we are and relate new input to the concepts we already have from experiences we have had up to that point.

Any change imposed upon a biblical word of concept is normally inadvertent and results from imperfect attempts to bridge the gap. We are cultural creatures. The abstract rational approach of the Modern mind, focused on the ability of Human Reason to grasp and understand Ultimate knowledge misleads us to thinking what we have understood up to a point is in fact the ultimate structure of reality and truth and God sees it! The original sin of wanting to know like God knows!

The Age of Ages
Now, with that background caution, back to the specific question. The English word “Everlasting” and the concept is entails is just wrong as an attempt to interpret what was entailed by the Greek term in its context. The New Testament terminology in Greek was interpreting the Hebrew messianic idea, as modified in the fulfillment claimed by the Christians, of the “Age” or “New Age” or “of the ages.” How literal or symbolic this was thought to be seems to vary.

The usage should be considered somewhat of an idiom, because of the way it is used in the biblical texts and as an interpretation of the messianic concepts we claim as believers in Christ. The main idiomatic feature for us would be the plural. Perhaps this is for emphasis, a common Semitic/Hebrew way of making emphasis''.

So in a nutshell, do the verses that threaten eternal damnation - "aionion" in Greek - actually refer to 'eternal' as we understand the word?