In the Sermon on the Mount, Jesus is correcting misinterpretations of the Law of Moses that He and others have been hearing in the synagogues for the past 30 years. In the gospel narratives, Jesus is described as traveling throughout Israel teaching in the synagogues (Matt. 4:23). "Large crowds" are now following Him (v. 25), and He has called His twelve disciples (5:1-2). After describing what a citizen of the kingdom is in The Beatitudes (5:3-12), and how citizens will be viewed by worldly people (vs.11-12), and how we are to view ourselves (vs. 13—16), Jesus immediately confronts the idea that He has "come to abolish the Law or the Prophets," stating that He has not come to abolish them, but to "fulfill" them (5:17-20).
I understand Matthew's usage of Jesus' words to mean that He is making plain what was a "mystery" in the OT (the classical argument of the OT conceals what the NT reveals) by contrasting what God actually meant with what the Pharisees and scribes had (mis)interpreted it to mean. Jesus and His apostles are "enemies of the state" being falsely accused of subverting God's nation (Mark 14:55-59; Acts 21:28). Immediately following Matt. 5:17-20, which concludes with the thought that "except your righteousness" (integrity, not moral superiority) "exceeds" (is more evident that you are real and not fake) that of the scribes and Pharisees, ye shall not enter into the (coming) kingdom" that He and John the Baptist have been announcing for about a year now--Jesus now begins correcting these misinterpretations of the Law which came as a result of these teachers being covetous hypocrites (Matt. 23:1ff).
The NIV divides Jesus' corrections conveniently into murder, adultery, divorce, oaths, etc. with paragraph headings. Each of these corrections begins with "You have heard…" or "It has been said…." These are the things that the people and Jesus had been taught all their lives in the synagogues. What Jesus is doing is correctly interpreting what the "scribes and Pharisees" had corrupted with their traditional interpretations and practices. Throughout the entire Sermon, Jesus is contrasting kingdom behavior with scribe and Pharisee behavior, especially throughout chapter 6.
Concerning divorce, Jesus is saying the exact same thing that Moses said. He is fulfilling it--not abolishing it. I noticed the KJV, ASV has "she may go and be another man's wife" (Deut. 24:2). This conveys the idea that "it is permissible" for her to go do it. This is not how the modern versions translate it (NIV, NKJV, ESV). They have "if she goes," or "when she goes." I believe this is still understandable in the KJV and ASV "she may go." The KJV and ASV are not saying, "it is permissible." They are saying 'if she goes'—i.e., hypothetically she 'may' go. The faulty "mis"understanding the religious teachers had was that "Moses gave us permission." Jesus is correcting their error in the Sermon (also see Matt. 19:1ff). They thought it was permissible to simply "give a certificate of divorce." Jesus is correcting their misunderstanding of Moses. Moses did not contradict Genesis 2.
The only command Moses gave in the scenario of Deuteronomy 24:1-4 was "that the first husband could not take her back because she was defiled." This would pollute the land. Jesus is explaining explicitly what Moses implied. Jesus is fulfilling = "filling full" (making plain) the Law and explaining to these "covetous" men (Luke 16:14-18) what God's real law of marriage said "from the beginning" (Gen. 2) = "What God has joined together, let not man separate." Moses was saying that the woman who was put away by the first husband "may go" and remarry, but when she does she becomes an "adulteress" and that is why her husband cannot take her back. She has been defiled by marrying another man while her husband lives. This is how Paul understood marriage (Rom. 7:1-3). Jesus is saying that anyone who puts away a wife other than for the cause of fornication "causes her to commit adultery" which is exactly what Deut. 24:2 says she "may go do" (She is not being given permission by God to do so--this was the misunderstanding as it is today). She becomes defiled when she does this, and the first husband cannot "remarry" her. Jesus says, 'Whoever marries the divorced woman commits adultery." This is the correct interpretation of the Law of Divorce that Jesus is correcting in the Sermon on the Mount (Matt. 5:31-32) and is the same thing he says later in Matthew 19:9, except he is talking about the "first husband" instead of the "wife."
When Jesus says, Moses permitted you "because of the hardness of your hearts," it still says nothing of re-marriage (like 1 Corinthians 7 says nothing about re-marriage). Is there marriage? Yes. Is there divorce? Yes. Is there approval for remarriage? No. And this is exactly what Moses says in Deuteronomy 24:1-4.
In Matt. 19:8, what I believe Jesus is saying to the Pharisees is: Are you now going to argue that "it's okay" to put away your wives, because "your hearts are hard?" So therefore, it's okay??!! Of course, this is an absurd argument, especially for "great religious teachers" the Pharisees thought themselves to be (Luke 18:9-14; Matt. 23:1ff). I think Jesus simply painted them into another corner with His argument like He did in the rest of Matthew (ch. 21-22).
May God bless our decisions, because we will answer for them personally (2 Cor. 5:10).