Don't buy the ainios, ainion argument from the UR folks.
I am a Bible scholar with a degree in Bible literature including Greek studies. I could go into great detail as to a study on this word, but I won't unless asked.
Suffice it to say the UR folks are correct in stating the word has a literal translation of "age". However the word is not defined simply as "age". But it does not actually mean a defined period of time. An age could be a period of time, or it could be the last age that never ends. It could be a period of years or reference an eternal space such as the age of God. It is one of those Greek words that is actually defined by how it is used.
For example this word is used to in Romans 16.
Now to him that is able to establish you according to my gospel and the preaching of Jesus Christ, according to the revelation of the mystery which hath been kept in silence through times eternal,but now is manifested, and by the scriptures of the prophets, according to the commandment of the eternal God, is made known unto all the nations unto obedience of faith: - Romans 16:25-26
Bible Gateway passage: Romans 16:25-26 - American Standard Version
This is a temporal or time listing of the word. An age.
But then it is used for something this IS eternal or forever.
Fight the good fight of the faith, lay hold on the life eternal, whereunto thou wast called, and didst confess the good confession in the sight of many witnesses.I charge thee in the sight of God, who giveth life to all things, and of Christ Jesus, who before Pontius Pilate witnessed the good confession;that thou keep the commandment, without spot, without reproach, until the appearing of our Lord Jesus Christ:which in its own times he shall show, who is the blessed and only Potentate, the King of kings, and Lord of lords;who only hath immortality, dwelling in light unapproachable; whom no man hath seen, nor can see: to whom be honor and power eternal. Amen. - 1 Timothy 6:12-16
Bible Gateway passage: 1 Timothy 6:12-16 - American Standard Version
Honor and power eternal certainly would not be a limited amount of time. Is Jesus Christ power and honor limited to a certain period of time? Of course not.
Context helps us see what this word means in the concept of time. It is not limited time. Sometimes it is and sometimes it isn't. A last example of this is.
Now to him that is able to establish you according to my gospel and the preaching of Jesus Christ, according to the revelation of the mystery which hath been kept in silence through times eternal,but now is manifested, and by the scriptures of the prophets, according to the commandment of the eternal God, is made known unto all the nations unto obedience of faith:to the only wise God, through Jesus Christ, to whom be the glory for ever. Amen. - Romans 16:25-27
Bible Gateway passage: Romans 16:25-27 - American Standard Version
The word is used to describe the eternality of God.
The word is used to describe our life after accepting Christ as eternal life. We won't be saved only for a certain period of time. It's forever.
The problem with UR doctrine is that it refuses to accept eternal damnation therefore they will not accept that ainion can mean forever. When clearly in scripture it does.
By the way, it is not the only Greek word that is like this. The meaning of the Ancient Greek words are not always one definition, but have broader definitions depending on what they are talking about.
It's like our word love. Love is often defined in context. Depending on how we are using it. I use the word love to describe my feelings for summer or a good breakfast. I use the word love for my kids, friends and wife. But it's not the same thing. You know what kind of love I am talking about by context.
That is aionion(s)