Two youtube videos by some anonymous person saying "Eternal Punishment is not Biblical" so it must be true. I think Abraham Lincoln once said "Don't believe everything you see online."
.....In the following twenty two verses αἰών and αἰώνιος are defined/described by association with other words and phrases as eternity, eternal, everlasting etc.: 1 Timothy 1:17, 2 Corinthians 4:17-18, 2 Corinthians 5:1, Hebrews 7:24, 1 Peter 1:23, 1 Timothy 6:16, Galatians 6:8, John 6:58, John 10:20, 1 John 2:17, 1 Peter 5:10, Romans 2:7, Luke 1:33, Revelation 14:11, John 10:28, John 3:15, John 3:16, John 5:24, Ephesians 3:21, Romans 1:20, Romans 16:26.
…..In the NT “aion/aionios” are used to refer to things which are not eternal but are never defined/described as meaning a period of time less than eternal as in the following verses.
[1]Romans 1:20
(20) For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal [ἀΐ́διος/aidios] power and Godhead; so that they are without excuse:
[2]Romans 16:26
(26) But now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting [αἰώνιος/aionios] God, made known to all nations for the obedience of faith:
In Romans 1:20 Paul refers to God’s power and Godhead as “aidios.” Scholars agree “aidios” unquestionably means eternal, everlasting, unending etc. In Rom 16:26 Paul refers to God as “aionios,” therefore Paul evidently considers “aidios” and “aionios” to be synonymous since he uses both words to refer to God's eternal nature.
[3]1 Timothy 1:17.
(17) Now unto the King eternal, [αἰών/aion] immortal, [ ̓́αφθαρτος/aphthartos] invisible, the only wise God, be honour and glory for ever [αἰών/aion] and ever [αἰώνιος/aionios]. Amen.
In this verse “aion” is paired with “immortal.” “Aion” cannot mean “age(s),” a finite period and be immortal at the same time. Thus “aion” by definition here means “eternal.”
[4]2 Corinthians 4:17-18
(17) For our light affliction, which is but for a moment, worketh for us a far more exceeding and eternal [αἰώνιος/aionios] weight of glory;
(18) While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal;[πρόσκαιρος/proskairos] but the things which are not seen are eternal [αἰώνιος/aionios]
In this passage “aionios” is contrasted with “for a moment,” vs. 4, and “temporal,” vs. 5. “Age(s)” a finite period, it is not the opposite of “for a moment”/”temporal/temporary.” “Eternal” is. “Aionios” by definition here means “eternal.”
[5]2 Corinthians 5:1
(1) For we know that if our earthly house of this tabernacle were dissolved, we have a building of God, an house not made with hands, eternal [αἰώνιος/aionios] in the heavens.
In this verse “aionios house” is contrasted with “earthly house which is destroyed.” Does the UR crowd think God is going to replace our destroyed earthly house with an ages long house which will also be destroyed at the end of an age? The aionios house is not destroyed, the opposite of “is destroyed.” Thus “aionios” by definition here means “eternal.”
[6]Hebrews 7:24 but because Jesus lives forever [αἰών/aion] he has an unchangeable [ἀπαράβατος/aparabatos] priesthood.
In this verse “aion” is paired with “unchangeable.” If “aion” means “age(s),” Jesus cannot continue “for a finite period” and be “unchangeable” at the same time. Thus “aion” by definition here means “eternal.”
[7]1 Peter 1:23
(23) For you have been born again, not of perishable seed, but of imperishable, [ ̓́αφθαρτος/aphthartos] through the living and enduring word of God. …
1 Peter 1:25
(25) but the word of the Lord endures forever.[αἰών/aion] " And this is the word that was preached to you.
In verse 23 “word of God” is paired with “imperishable.” In verse 25 the word of God “endures εις τον αιωνα unto eternity. ” Thus by definition “aion” here means “eternity.”
[8]1 Timothy 6:16
(16) Who only hath immortality, [ ̓́αφθαρτος/aphthartos] dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting[αἰώνιος/aionios]
In this verse “aionios” is paired with “immortality.” If “aionios” is only a finite period, God cannot be “immortal” and only exist for a finite period at the same time. Thus “aionios” by definition means “eternal.”
[9]Galatians 6:8
(8) For he that soweth to his flesh shall of the flesh reap corruption;[φθορά/fthora] but he that soweth to the Spirit shall of the Spirit reap life everlasting. [αἰώνιος/aionios]
In this verse “aionios” is contrasted with “corruption.” “Fleshly” people reap “corruption” but spiritual people reap “life aionios,” i.e. “not corruption.” “Age(s), a finite period, is not opposite of “corruption.” Thus “aionios life” by definition here means “eternal/everlasting life.”
[10]John 6:58
(58) This is the bread that came down from heaven. Your ancestors ate manna and died, but whoever feeds on this bread will live forever.[αἰώνιος/aionios]
In this verse “aionios life” is contrasted with “death.” If “live aionios” is only a finite period, a finite period is not opposite “death.” Thus “aionios” by definition here means “eternal.”
[11]John 10:28
(28) I give them eternal [αἰώνιος/aionios] life, and they shall never [αἰών/aion] perish; no one will snatch them out of my hand.
Here “aionios” and “aion” are paired with “[not] snatch them out of my hand.” If “aion/aionios” means “age(s), a finite period,” that is not the opposite of “[not] snatch them out of my hand’” “Aionios life” by definition here means “eternal life.”
[12]1 John 2:17
(17) The world and its desires pass away, but whoever does the will of God lives forever. [αἰών/aion]
In this verse “aionios” is contrasted with “pass away,” “lives aionios” cannot mean a finite period, A “finite period” is not opposite of “pass away.” Thus “lives aionios” by definition here means “lives eternally.”
[13]1 Peter 5:10
(10) And the God of all grace, who called you to his eternal [αιωνιον/aionion] glory in Christ, after you have suffered a little while, [ολιγον/oligon] will himself restore you and make you strong, firm and steadfast.
In this verse “aionios” is contrasted with “little while” Does the UR crowd think that Jesus will give His followers an finite period of glory then they will eventually die? Thus “aionios” here means “eternal.”
[14]Romans 2:7
(7) To those who by persistence in doing good seek glory, honor and immortality, [ἀφθαρσία/apftharsia] he will give eternal [αἰώνιος/aionios] life.
In this verse “aionios” is paired with “immortality.” If “aionios” is only a finite period, believers cannot seek for “a finite period,” and “immortality” at the same time. But they can seek for “eternal life” and “immortality” at the same time. Thus by definition “aionios life” here means “eternal life.”
[15]Luke 1:33
(33) And he shall reign over the house of Jacob for ever; [αιωνας/aionas] and of his kingdom there shall be no end.[τελος/τελος]
In this verse “aionas” is paired with “without end.” “aionas” cannot be paired with “without end” if it means only “ages” a finite period. “Aionas” by definition here means eternal.
[16]Revelation 14:11
(11) And the smoke of their torment ascendeth up for ever and ever:[εις αιωνας αιωνων/eis aionas aionon] and they have no rest day nor night, who worship the beast and his image, and whosoever receiveth the mark of his name.
In this verse “aionas aionon torment” is paired with “no rest day or night.” If “aionas, aionon” means “a finite period” at some time they would rest, “Aionas, aionon” by definition here means “forever and forever.”
[17]John 10:28
(28) And I give unto them eternal [αιωνιον] life; and they shall never [εις τον αιωνα] perish, neither shall any man pluck them out of my hand.
In this verse “aionion” is paired with “[no man can] “pluck them out of my hand.” If “aionion” is only a finite period then at some time they could be plucked out. “Aionion” by definition here means eternal.
[18]John 3:15
(15) That whosoever believeth in him should not perish, but have eternal [αιωνιον] life.
In this verse “aionion” is paired with “shall not perish.” They could perish in a finite period, “aionion life” by definition here means eternal life.
[19]John 3:16
(16) For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting [αιωνιον] life.
In this verse “aionion” is paired with “shall not perish.” People could eventually perish in a finite period, “aionion life” by definition here means eternal life.
[20]John 5:24
(24) Verily, verily, I say unto you, He that heareth my word, and believeth on him that sent me, hath everlasting [αἰώνιος] life, and shall not come into condemnation; but is passed from death unto life.
In this verse “aionios” is paired with “shall not come into condemnation” and “passed from life unto death.” “Aionios” does not mean “a finite period,” by definition here it means “eternal,” unless Jesus lets His followers come into condemnation and pass into death.
[21]Romans 5:21
(21) That as sin hath reigned unto death, even so might grace reign through righteousness unto eternal [αἰώνιος] life by Jesus Christ our Lord.
In this verse “aionios life” is contrasted with death. “A finite period” is not opposite death, “eternal life” is. “Aionios life” by definition here means ‘eternal life.”
[22]Ephesians 3:21
(21) to him be glory in the church and in Christ Jesus throughout all generations, for ever [του αιωνος/tou aionios] and ever! [των αιωνων/ton aionion] Amen.
In this verse “tou aionios ton aionion” is paired with “throughout all generations.” "Age(s)" a finite period cannot refer to "all generations." By definition “tou aionios ton aionion” means forever and ever.
There have been erroneous claims that the above scriptural proofs which show that aion means eternity and aionios means eternal, everlasting have been refuted but that is not true. There have been some specious arguments presented which have been refuted.
.....See e.g. this claim quoted in the preceding post "OTOH here we see many examples where αἰών andαἰώνιος are defined/described as being of a finite duration:" There are no, zero, none verses in which aion/aionios are compared or contrasted with other adjectives or phrases which show that either is of finite duration.
.....I acknowledge that both words are sometimes used to refer to something that is finite as in Matt 13:39.
Matthew 13:39
(39) and the enemy who sows them is the devil. The harvest is the end of the age,[αἰών]and the harvesters are angels.
This verse refers to something which has an end but there are no other adjectives or phrases in the verse which further indicate that aion is finite as in the 22 vss. listed above.
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I truly consider him to be the 'anointed pen' of my life. I first started getting his monthly teachings in the early 80's. And I am continually amazed by the things that he shares. His 'Savior of the world' series was one of the early findings I had to help answer all the questions continuously asked by the ETer's here who can not hear the teaching which we have 'heard'. Just the other day my wife opened the mailbox and said, as she has every month, for years; "Your birthday present arrived." 
And then she hands me the latest monthly teaching from Preston.
