According to the context of St. Matthew 25 and ONLY the context, please fill in the empty lines.
The foundation for “everlasting punishment” Matt. 25=
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"And these shall go away into everlasting punishment: but the righteous into life eternal."
1. For I was an hungred
2. I was thirsty
3. I was a stranger
4. Naked
5. Sick
It seems to boil down to this:
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And the King shall answer and say unto them, 'Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.'
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(Mt 25:40)
In this context, it seems to directly follow from Jesus's ministry in chapter 23 (if I'm not mistaken what He is saying in 25 is a continuation of 23), where He speaks of the greatest commandment.
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37 Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. 38 This is the first and great commandment. 39 And the second is like unto it, Thou shalt love thy neighbour as thyself. 40 On these two commandments hang all the law and the prophets.
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(Mt 22:37–40).
Not abiding by the greatest commandment (of which the whole of the law rests on), not loving our neighbor, (i.e. could be anyne, as perhaps a stranger in need we cross paths with perhaps only once (like in the Parable of the Good Samaritan)), seems to be the crux. It seems almost impossible to break the second commandment without breaking the first, we are called to love even our enemies (what power do they have over how much of Christ's Love I have in my heart - so I should guard it against the seeds of hatred instead of letting that be an excuse for me to hate more and be a part of the problem instead of the solution).
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7 Beloved, let us love one another: for love is of God; and every one that loveth is born of God, and knoweth God. 8 He that loveth not knoweth not God; for God is love.
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(1 Jn 4:7–8).
Edit: Here in this context we may see clearly that small acts of kindness for others means a great deal. Perhaps storing up the oil refers to our works of faith that have built our character.
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So, not loving God and/or one's brethren, everlasting punishment may be the consequence. I would like to refer to some commentary on the subject of everlasting punishment:
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(3) The last judgement.—The NT doctrine of the last judgment leads to the same conclusion. Two things seem plainly taught about this judgment: the first, that it proceeds on the matter of the present life—“the things done in the body” (Mt 25:31–46; 2 Cor 5:10; Rev 20:12); and the second, that it is decisive in its issues. Not a single suggestion is given of a reversal of its decisions in any future age. Such silence is inexplicable if the Scriptures meant to teach what the opponents of this doctrine so confidently maintain.
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2. Nature of Punishment
... The supreme penalty of sin is unquestionably the loss of God’s life and love—the being sinful. Environment, indeed, may be expected to correspond with character, but the hell is one the sinner essentially makes for himself, and, like the kingdom of God, is within.
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4. Gradation of Punishment
The fullest weight must further be given to what the Scripture so expressly says of gradation of punishment, even of the unsaved. It is not the case that the lot of all who fail of the eternal life in Christ is all of one grade. ... There is a vast area here for the Divine administration on which no light at all is afforded us.
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Orr, J. (1915). Punishment, Everlasting. In J. Orr, J. L. Nuelsen, E. Y. Mullins, & M. O. Evans (Eds.), The International Standard Bible Encyclopaedia (Vol. 1–5, pp. 2502–2504). Chicago: The Howard-Severance Company.
Everlasting punishment seems to entail separation from God somehow. After the judgment, there are no second chances. There seems to be degrees of punishment for the doomed (the punishment will fit the crime). To be in rebellion against God, nothing in His creation being of comfort - all being as vanity of vanities striving for the wind - with of course not even God to turn to, if destruction would come it would ensue as an act of mercy. As much as I sympathize for any soul that goes to hell, seeming like the most hard to accept truth out there that some end up utterly doomed to everlasting punishment, all there is to do is trust the scriptures and heed the stern warnings, seeking to enter into life via the narrow way. Even the damned will be kneeling to God before His Justice:
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9 Wherefore God also hath highly exalted him, and given him a name which is above every name: 10 That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth; 11 And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
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The Holy Bible: King James Version. (2009). (Electronic Edition of the 1900 Authorized Version., Php 2:9–11). Bellingham, WA: Logos Research Systems, Inc.