The verb ἀπελεύσονται (“will go away”) in Matthew 25:46 is decidedly in future tense - how does it interact with the testimony of John?
A gift has been given
In 1 John 5:11 ἔδωκεν (aorist indicative active) is straightforwardly rendered “has given”.
The gift of God’s son has indeed been given (see John 3:16). That it has been given does not mean it has been (fully) received.
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Born of the spirit
I have argued elsewhere that death is a separation and birth is a joining together. This definition offers substantial insight to Jesus' words to Nicodemus in John 3:
6 That which is born of the flesh is flesh; and that which is born of
the Spirit is spirit.
7 Marvel not that I said unto thee, Ye must be born again.
Jesus is not saying that we stop being physical entities and become only spirit; when we are physically born our self (more on that in the linked post) receives a physical body; when we are spiritually born we receive the Holy Ghost.
Birth does not imply, however, that development is complete. A biting but witty saying on mortal life is that birth is followed by a quarter-century of physical growth and then a half-century of physical decay (I’m over 25 so I can say that =) ).
We are not done developing when we are born of the Spirit.
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The Earnest of the Spirit
Paul taught the Ephesians of the earnest of the Spirit:
13 In whom [Christ] ye also trusted, after that ye heard the word of
truth, the gospel of your salvation: in whom also after that ye
believed, ye were sealed with that holy Spirit of promise,
14 Which is the earnest of our inheritance until the redemption of
the purchased possession, unto the praise of his glory (Ephesians
1:13-14).
He gave a very similar teaching to the church at Corinth:
21 Now he which stablisheth us with you in Christ, and hath anointed
us, is God;
22 Who hath also sealed us, and given the earnest of the Spirit
in our hearts (2 Cor. 1:21-22, see also 2 Cor. 5:5).
The word rendered “earnest” (or “pledge” in some renderings) is ἀρραβών (“arrabon”). In a financial transaction an earnest is:
earnest-money, a…part of the payment, given in advance as a security
that the whole will be paid afterwards (source)
A buyer gives earnest-money to demonstrate sincerity; a seller will sometimes require earnest-money in order to commit not to sell the possession to another prospective buyer.
The Biblical application of the word:
A pledge or security. The word thus translated is a commercial term
denoting the deposit paid by a buyer on entering into an agreement for
the purchase of anything. As used by Paul… it means that the Lord
gives us His Holy Spirit in this life as a foretaste of the joy of
eternal life. The Spirit is also the Lord’s surety that He will
fulfill His promise to give eternal life to the faithful.
(source)
God gives us a guarantee–a portion of His fullness–as a pledge for what is to come.
Let’s say (hypothetically) I sell God a house. We decide we want to make a deal, and we draw up the covenants. We agree to the terms, and God makes a pledge–a down-payment–earnest money–we now have a binding agreement. We have (present-tense) a deal.
There will still be an inspection before the key is handed over, and if the inspection finds that I have made misrepresentations, under some circumstances the contract could be invalidated (separate discussion).
One of the keys to this (admittedly mundane, earthly, imperfect) parable is that 1) the agreement is made and 2) the key is transferred – on distinct occasions, but it doesn’t make the covenants any less-binding.
We can rest assured that we’re working with someone who does business the right way–we can trust God’s promises:
I have spoken it, I will also bring it to pass; I have purposed it, I
will also do it (Isaiah 46:11).
This is what hope is, in the Biblical sense. It’s not “I hope it doesn’t rain tomorrow”, it’s something much more well-founded: “I am motivated by what God–whom I trust–has promised me”.
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Immortality
Immortality is not a uniquely theological word–in both religious & secular use, immortality means living forever–the focus is on time and an absence of decay, not on quality.
Immortality–through the resurrection–is promised to all the family of Adam (see 1 Cor. 15:22).
Eternal life, however, is only promised to the faithful (for example, see the passages in the OP). What then is eternal life?
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That they might know thee
As described in Matthew 25:34-46 (see also Rev. 20:12-13), giving an ultimate accounting to our Maker, standing on the right hand of God and entering into the kingdom of the Father, is well beyond the scope of our mortal experience.
The eternal life Jesus speaks of in John 17:3 is past the confines of our mortal comprehension:
And this is life eternal, that they might know thee the only true God,
and Jesus Christ, whom thou hast sent.
Truly knowing someone does not happen at a distance. In the days before caller ID there were a handful of people I knew so well, I could recognize their voice from a single word. At "hello" I knew exactly who I was talking to. Developing that profound, up-close familiarity with someone is a process, not an event.
This profound, up-close familiarity with God is maximized only after we receive a resurrected body capable of existing in that state of glory (see 1 Cor. 15:40-42).
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The Eternal Nature of God
In this mortal existence, everything we create, everything we experience, and everything we agree to, has a beginning and an end. Whether it’s human life, or the pyramids, or a mountain range, or even a star, none of it lasts forever.
Psalms 90 & 102 make this contrast explicit:
Before the mountains were brought forth, or ever thou hadst formed the
earth and the world, even from everlasting to everlasting, thou art
God (Psalm 90:2)
25 Of old hast thou laid the foundation of the earth: and the heavens
are the work of thy hands.
26 They shall perish, but thou shalt endure: yea, all of them shall
wax old like a garment; as a vesture shalt thou change them, and they
shall be changed:
27 But thou art the same, and thy years shall have no end (Psalm
102:25-27)
Whatever plans we make or agreements we enter into, once we are dead we will have no ability to carry them out or enforce them–even a last will and testament relies entirely on the living left behind to be put into effect (this is the point being made in Psalm 146:3-4, contrasted with God’s eternal power in v10 of the same Psalm)
Eternal, then, is an attribute of God. God has the power to speak and it will be done (e.g. Genesis 1:3, Isaiah 46:11); God can command and enforce with efficacy beyond the grave; God possesses the attribute of eternal in a way nothing we make or do ourselves does. A few example passages:
The eternal God is thy refuge, and underneath are the everlasting arms
(Deut. 33:27)
Thy throne is established of old: thou art from everlasting (Psalm
93:2)
Thou, O LORD, art our Father, Our Redeemer from everlasting is Thy
name (Isaiah 63:16)
Jesus Christ the same yesterday, and to day, and for ever (Hebrews
13:8)
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Endless in duration and godlike in quality
Eternal describes Deity in a way it doesn’t describe anything else.
I will suggest an interpretation here that will make some uncomfortable–if you’re feeling queasy, skip the next two paragraphs.
That which is eternal is of God; eternal is a descriptor of God. This interpretation is not unique to me but I offer it as something I have found persuasive:
Eternal fire is God’s fire. Eternal reward is God’s reward. Eternal punishment is God’s punishment. Eternal judgment is God’s judgment. Eternal life is God’s life.
As succinctly stated by Dr. Jason Carroll:
Eternal life is a life that is both endless in duration and godlike
in quality
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Conclusion
By covenant God offers eternal life, the kind of life that He has (see Romans 8:17). The investment God makes in who we are and what we know does not mature at spiritual rebirth.
Eternal life is pledged in this life (it's been given, we have an agreement); the realization of that blessing comes when we are ready for it, after resurrection & judgment.