Big Drew said in post #1:
This tells me that if the unbelieving partner leaves then the believer is no longer bound to that marriage, and are free to remarry.
1 Corinthians 7:15 means that a believing spouse isn't under the bondage of having to keep together a valid marriage to an unbeliever when the unbeliever is determined to get a divorce. But 1 Corinthians 7:15 doesn't mean that a believing wife, after being divorced by an unbelieving, yet valid, husband, can then marry someone else. For if a man marries a woman divorced from a valid husband, he is committing adultery (Luke 16:18b). But the scriptures don't forbid a man divorced from a valid wife to marry a second, single, woman who isn't divorced from a valid husband, so long as it was his first wife (whether an unbeliever or believer) who divorced him. But then in God's eyes, he will be married to two women at the same time (so long as both remain alive), which, while no scripture requires is a sin in itself, because it isn't the best situation, it disqualifies him from taking any leadership positions in the church (1 Timothy 3:2,12), based on the basic idea of 1 Timothy 3:5.
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A husband is not to divorce his wife (1 Corinthians 7:11b), and a wife is not to divorce her husband (1 Corinthians 7:10). If a wife does divorce her husband, she must remain unmarried or be reconciled to her husband (1 Corinthians 7:11). Regarding becoming reconciled, a Christian must always completely forgive everyone who has wronged him or her in any way (Mark 11:25), no matter how great the wrong and no matter how many times a wrong has been committed (Matthew 18:21-35). For if a Christian refuses to forgive anyone for anything, God will refuse to forgive that Christian for his or her own sins (Mark 11:26).
If a husband divorces a valid wife and marries another woman, he is committing adultery (Mark 10:11). And if a wife divorces a valid husband and marries another man, she is committing adultery (Mark 10:12). The exception for fornication (as distinguished from adultery) in Matthew 19:9 permits a husband to divorce a valid wife for having had pre-marital sex, and to marry another woman without his committing adultery. But this applies only to cases where a husband doesn't discover until after he's married that his newlywed wife isn't a virgin (cf. Deuteronomy 22:14, Matthew 1:19). There is no such pre-marital-sex exception granted to a wife. Also, there is no pre-marital-sex exception granted to a man who marries a divorced woman. If a man marries a woman divorced from a valid husband for any reason, he is committing adultery (Luke 16:18b).
The now-abolished letter of the Old Covenant Mosaic law (Ephesians 2:15-16, Colossians 2:14-17, Romans 7:6) permitted a divorced woman to marry someone else (Deuteronomy 24:2). But if her second marriage ended, the letter of the Old Covenant Mosaic law forbade her to remarry her first husband (Deuteronomy 24:4). The New Covenant rules turn this on its head. For now a woman divorced from a valid husband can't marry anyone else (Mark 10:12, Luke 16:18b), but she can remarry her valid husband (1 Corinthians 7:11). It was because the letter of the Old Covenant Mosaic law permitted a divorced woman to marry someone else, that Jesus, while the letter of the Old Covenant Mosaic law was still in effect, could acknowledge the woman of Samaria's five marriages (John 4:18, assuming that all five did not end in the death of her husband: cf. Luke 20:29-31). The New Covenant rules forbidding a woman divorced from a valid husband to marry anyone else didn't come into legal effect until Jesus' death on the Cross brought the New Covenant into legal effect (Hebrews 9:16-17, Matthew 26:28) and abolished the letter of the Old Covenant Mosaic law (Ephesians 2:15-16, Colossians 2:14-17, Romans 7:6).
God never said that marriage would be easy. And he has set such strict, New Covenant rules regarding divorce and second marriages (Matthew 19:9, Mark 10:12) that the apostles said it is better not to get married at all (Matthew 19:10). Jesus answered them by saying that whoever can accept not getting married, and remaining celibate, should accept it (Matthew 19:11-12). The apostle Paul said the same thing, that unmarried celibacy is the best thing for a Christian if he or she can handle it (1 Corinthians 7:1,7-8,32-35). But if someone who hasn't been married can't contain himself or herself sexually, then he or she should get married in order to avoid fornication (1 Corinthians 7:2,9).
The strict New Covenant rules regarding divorce and second marriages cut both ways, in that if Christians find themselves in a miserable marriage which is an adulterous affair in God's eyes (Mark 10:11-12), then they can escape their misery and their unrepentant sin at the same time by divorcing their invalid spouse. But if they find themselves in a very pleasant marriage which is an adulterous affair in God's eyes, then they have to be willing to give it up in order to escape their unrepentant sin, and thereby avoid ultimately losing their salvation due to unrepentant sin (Hebrews 10:26-29, Galatians 5:19-21, Luke 12:45-46).
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Big Drew said in post #9:
Then do you view remarriage as an unpardonable sin?
The only unforgivable sin is blaspheming the Holy Spirit (Mark 3:28-29), such as ascribing a work of the Holy Spirit to Satan (Mark 3:22-30). Any other sin can be forgiven if it's repented from and confessed to God (1 John 1:9). Just as if believers find themselves living in the sin of an adulterous affair, they can't continue on in that sin, so if they find themselves living in the sin of second marriage adultery (Mark 10:12, Matthew 19:9), they can't continue on in that sin (or any other sin) and expect God's grace to forgive them (Hebrews 10:26-29, Galatians 5:19-21; 1 Corinthians 6:9-10). Instead, they must break off with the second, invalid spouse, even if they have had children with the second spouse, just as married people must break off an adulterous affair even if they have had children as a result of that affair.
After breaking off an adulterous second marriage, a wife must remain unmarried or be reconciled to her first, valid husband (1 Corinthians 7:11), if she has one. She can't marry someone else, even if, for example, that would help her and her children to escape poverty. For just as escaping poverty wouldn't justify the wife continuing in the sin of an adulterous affair with a man who financially supports her and her children (or wouldn't justify the sin of her becoming and remaining a well-paid prostitute), so escaping poverty wouldn't justify the sin of her entering into another case of second marriage adultery (Mark 10:12) with a man who financially supports her and her children.
Romans 3:31 means that Christians establish the Old Covenant Mosaic law not in its letter, but in its spirit (Romans 7:6), by loving others (Romans 13:8-10, Galatians 5:14, Matthew 7:12). Part of loving others is warning them if they are living in sin (Revelation 3:19; 2 Thessalonians 3:15, Hebrews 3:13, James 5:19-20). The worst thing that a Christian can do is to coddle people who are living in sin, instead of sharing with them the hard (yet saving) truths of God's Word (2 Timothy 4:2-4, cf. Jeremiah 23:14,22,29). Telling the truth to people can sometimes hurt them, but that is better than deceiving them with something which makes them feel good (Proverbs 27:6, Proverbs 28:23). The reason that second marriage adultery (or any other sin) is so common in the church today is because so much of the church has stopped teaching and believing the hard truths of God's Word (2 Timothy 4:2-4, cf. Jeremiah 23:14,22,29).