Christians say that Jesus is the Messiah spoken of in the Old Testament; Jews disagree. Why couldn't Jesus be the Messiah spoken of? Or could he possibly have been?
For a complete treatment of the subject, see a work like this. There are anti-missionary sites that certainly go through the various Biblical descriptions of the Messiah; these aren't hard to find if you google around a bit.
In brief, here's how Maimonides codifies the job description of the Jewish Messiah, in Chapter 11 of his Laws of Kings & their Wars; this is normative Jewish belief:
Suffice it to say that Jews believe Jesus failed to meet that description.
I used to be a Christian but converted to Orthodox Judaism more than 32 years ago. I have a website called: "A Primer: Why Jews Can't Believe in Jesus" that should more than adequately answer your question (although I actually designed it for Jewish education and not to combat missionaries). But let me just touch on the basics:
Not only isn't Jesus the Messiah, but it is obviously so that the Messiah has yet to come, although, G-d willing he will come soon. We know this definitively because we do not live in the Messianic Age. If we lived in the Messianic Age you would know it because:
If you have read the papers recently, you would know that none of these things has yet occurred.
Now, many missionaries tell me that Jesus will rise from the dead and do all the requisite things that Messiahs are supposed to do that he didn't get to. In golf, they call that a "mulligan" -- a do-over. There have been a lot of people who claimed to be, or were acclaimed to be the Messiah over the last 2000 years or so. If you give one guy a mulligan, they all have to get mulligans. So where are you now. You cannot say that Jesus gets exclusivity to the mulligan because he is the Messiah, because that is just circular.
You can tell me how Jesus raised the dead. Well, so did some of the rabbis who were almost his contemporaries, if you choose to believe in the Talmud's stories of the rabbis. My friends here cited some examples in response to this question.
You might tell me that Jesus suffered. He did? How long? Was it more than a half a day that he was crucified? Have you ever heard how ten of our sainted rabbis died at the hands of the Romans? See this. And do you know anything about people like the Klausenburg Rebbe (Rabbi Yekusiel Yehudah Halberstam (1905-1994)), zt'l, who were imprisoned by the Nazis. The Klausenburg lost his wife and 10 children who were killed in the death camps. For more than a year he was personally starved, beaten and tortured by the Nazis, and yet kept the commandments in the camps, even though no one had to under the circumstances. He even saved any rolls he received until the Sabbath eve when he would say the blessing over two rolls in honor of the Sabbath, and give that bread to the other inmates. When he was rescued at the end of the war, he weighed about 85 lbs. But he was undeterred. After the war, he went on to build schools for children and hospitals for the sick on three continents. Please read Judah Lifschitz's translation of the Klausenburg Rebbe's biography. You will see that although he had no claim to be the Messiah, he litterally fed the hungry, took care of little children and healed the sick, and over a far longer time than the Christian messiah.
And, if you tell me that he is the "suffering servant" defined in Isaiah 53, please go back to your copy of Isaiah and see who he defines as "the servant." No, it's not in chapter 53, but there are at least four verses in Isaiah that define Israel as G-d's "servant." (Is 44:2, 44:21, 45:4, and 49:3). Origen, the 3rd century Christian scholar, in Contra Celsum, admitted that this was the Jewish understanding of Isaiah 53, that the entire Jewish nation was G-d's servant and that they indeed had suffered for centuries before (and to come). But I will admit that Jewish liturgy and thought also suggest that Isaiah 53 does refer to the Messiah, among many others. The Messiah, like righteous people have in every generation, will accept upon himself suffering for the excessive sins of others that G-d chooses not to give to the sinner because then they might lose all faith in G-d. This belief, described at Talmud Bavli, Brachot 5a, is an explanation of why we see good people suffering for no apparent reason.
Please see my Primer for other points.
you have two completely different question here, one on the topic and other on the question body. so in order: 1) they have a tradition that goes ultimately to his first 12 students which believe because he said so. 2) the rambam brings several points on how to detect false prophets, see mishne tora. we have a tradition that the messias in a prophet. there are also several point that the messias need to fulfill, and this is basically how we would recognize him and how we can know which ones to reject (in plural, note that the one you mentioned was not the only one in the last ~3 thousand years).
I could make a list here but that would be the answers for a different question, one simple point that I believe anyone can check for himself is the temple, we simply have no temple (until this writting). you can come and check yourself.
This question is not just about 'Who was, or will be, the Jewish messiah?' It's also about things like what 'the messiah' even means, and whether or not the whole religion of Christianity could also be true.
That would include the assertions that a man was an incarnation of God; that you need to believe in a messianic claimant in order to have forgiveness from and relationship with God; and that only Christian Jews are the fulfilment of Isaiah 59:21, while no other Jewish group is the place where God is speaking. Most communities of people who believe in Jesus adhere to one or more of those points.
It's also important to think about the starting points of the question. Why listen to Jesus/Yeshua, out of all the people who have claimed this? Should we take it seriously because of something in the Christian community or experience that stands out to us? Or because of prophecies in Tanach? Or because of a feeling of admiration for the personality of the messianic claimant described in the Christian gospels?
The divinity/incarnation claim is the most serious issue, because if it is not true then to worship a human as God is plain idolatry. For sure, a lot of Christians truly love and seek God, and He responds to them in their sincerity. But it's still destructive, and it's still off-limits to Jews and to anyone who knows better. At the heart of Tanach, and of Judaism, is the relationship between Creator and His creation. If you read Psalm 148 you can gain a beautifully clear sense of this attitude that pervades the biblical Israelite understanding. For us to treat something or someone around us in creation as if they were the one who made and deserves our hearts, or a power that we can completely trust and admire in and of themselves, is spiritual adultery. It's also humiliating to worship a fellow creature who owes worship to God as much as we do.
So clearly with this point, you have to be totally careful in light of the Torah and the prophets. When it comes to knowing God, it's good to cling to Him in faith even when our knowledge is small or unclear for a time. But when it comes to worshiping something that seems to be created, or possibly distinct from God, you can't allow something into your worship if you're not 100% sure that it's our Creator alone. The commandments about worship in Torah are clear and there's no room for ambiguity. A couple of questions arise. First, is it possible that God should become incarnate as a human? And also, how does the Torah expect anyone to test such a claim according to the standards of absolute carefulness regarding worship?
Most people who believe in divinity for Jesus think he was 'fully human and fully God'. In the New Testament, it's clear that Jesus prayed and that he had a relationship with God, calling Him Father. So one question would be, in this fundamental relationship between Creator and created, which side of the relationship was Jesus on? If 'Creator' then he was not one of us. If created, then he didn't deserve to be called God. We could talk about this for a long time, and it's not the deepest or clearest point here, but it is an important one. The most defining feature of 'what it means to be human' is our relationship of worship towards God, and the interesting thing is that (according to the gospels) Jesus portrayed himself as having the humility of a worshipper and a servant of God (in line with the idea of what the Davidic king 'should be'). If he was also going to ask for worship from us as well, then that is a fundamentally mixed signal.
The other issue on this point is the question of how Jews should test such a claim according to the Torah. For example, Torah expects that prophets will come and that there will be particular ways of testing them. Even miracles or fulfilled prophecies, though they might be a sign of a prophet, aren't a guarantee according to the Torah standards. In the commandment about tzitzit (Numbers 15:38-41), the Israelites were taught to hold onto the commandments so they wouldn't be led astray by their hearts or eyes. So, does the clarity of commandments in the Sinai revelation, where Israel met her God, point someone to expect that a new way of meeting with and worshipping God may one day be commanded? If so, what is the test; how can you know which incarnations are true, and which are false? Did God expect Israel to take any such claim seriously about a human they could meet or remember, or anything else with breath? Many Jews have loyally chosen to put up a wall around their worship with a door that will only ever be opened to God; they do not open it to Jesus because no reason to do so can match the carefulness taught in the Torah.
The question remains about whether Jesus might have been the messiah even if he was not God to be worshipped. It becomes difficult to ask this question, though, firstly because most of the Jews never accepted him, while most of the people who did ended up using him as an idol. Unless you're going to assume that the testimony of God could be held in Islam, but that's beyond the scope of your question and would only be considered for reasons quite different from the ones why people consider Christianity. Just quickly, a problem with Islam from the Jewish perspective is that Muhammed asked Jews in his time to follow him because of what they already knew of God and His Law, but Muslims insist that by that time the Jewish scriptures and traditions had been corrupted. To follow Muhammed would have been to violate Judaism as they understood it, so they would first have to stop trusting in the Jewish testimony; only then could they listen to Muhammed. There could be no continuity. If God set up a method of preservation in His covenant with Israel, what was it?
But for all of those communities who do believe in Tanach (as currently preserved) and who also believe that Jesus was messiah, regardless of whether they think he was God, the thing that they are lacking most is the sign of the covenant between God and Israel. I'm not saying that the rabbinic/Orthodox Jewish community is infallible or perfect, by any stretch, but there's no other claimant to being the 'righteous remnant' of Israel who can say that in every generation, there were people among them keeping Shabbos, keeping testimonial signs, mitzvot, and observances, and passing them on to their children along with the messsage of their God who brought them out of Egypt. That's a big problem for them: how can Jesus be the messiah if in some generations, all the people keeping the signs of the remnant of God, and living righteously according to the Torah and prophets, didn't accept him? No matter what feelings, miracles, or beautiful expressions of community and love for God exist in Christianity or in 'Messianic Jewish' (following Jesus) groups, how can they claim that they are the only Jewish remnant living by faith, and that all other Jews have no concept of grace or of coming close to God in love and His forgiveness?
When you look at the proof texts that are brought by Christians to support the idea of Jesus being the messiah, you can see from the context that most of them are not even clearly about the messiah, the/an anointed one, or the king who will restore David's line. A lot of them can also be read legitimately in a number of different ways. The description of someone suffering is not necessarily a reference to Jesus(!), and there are also some translation issues in Christian Bibles. I can't go into them all because it would take a long time, but suffice to say that these parallels seem much bigger to someone who thinks 'messiah'='Jesus'; who equates the idea of moshiach with death and resurrection, with incarnation, or with a sacrifice for forgiveness; and who has truly known God in the Christian community and setting while reading the 'Old Testament' with a structure that highlights their belief in (and thankfulness for) Jesus' 'sacrifice'. They (and Tanach as a whole) can always be read very legitimately, and more naturally, in different ways. And you should remember that the people who wrote the New Testament knew Tanach and the imagery of the hope of Israel, so some parallels could be because of their understanding of what happened (influenced by biblical imagery) rather than the reality that everyone else could see.
The idea of needing forgiveness for sin is a non-issue. God clearly promised this forgiveness to the Jewish people and already built it in to the Torah. It's a gift beyond words that we can't understand, but Isaiah 55:6-9 makes it clear that we don't need to understand how it happens. Christianity looks at it retrospectively and says that they can now explain it, but that in itself is no proof for anything. Also see Deuteronomy 30 for the way in which God taught Israel to return to Him and be forgiven, even in the last generation, and Ezekiel 33:10-16.
A good parshah to give foundations for considering all the claims that people make about Jesus is Va'etchanan (Deuteronomy 3:23-7:11). This is where the caution and loyalty, which Christians often perceive as pride or stubbornness, is coming from. Remember also how important the community of the righteous remnant is in the prophetic understanding of God's program in the world concerning Israel. If Jews who love God and rely on His mercy while keeping His commandments carefully are really holding His light in our world, then to ignore them and listen to another cult or sect or religion is to ignore the word of God in our midst and to cause a lot of pain and isolation to members of the real firstborn-son and servant of God (Israel) who are holding on to their testimony and to His Law, which is their life and the method of preserving the covenant. It's true that they're still waiting for the restoration of many things, but what is the standard in Tanach for defining the righteous among Israel? Doesn't it involve justice, faithfulness to God, and mercy for people who are suffering? Nothing is said about accepting a messianic claimant in the time before he has fulfilled any actual messianic prophecies, or else losing your relationship with God, especially if the trusted leaders and the righteous community of Israel (whom we trust as the ones who recognised the prophets) don't recognise a person as such. No one can tell you that you owe this, and there is no clear reason to draw the circle of 'salvation' so small as to take the words of God away from the people who are so loyally and lovingly, obediently guarding them.
Some Christians also believe that the Torah is no longer valid in the 'messianic age' of Jesus, and that Jews who try to follow it are rejecting grace. I don't think that this was the message in most, if not all, of the New Testament. But for groups who believe it, there are some very severe biblical issues with accepting an idea like this as well. Obviously this comes from a lot of misconceptions about how grace and a law 'on your hearts' are already a foundational part of the Jewish experience of Torah.
Basically, the question of whether Jesus was / will be moshiach and the one of whether Christianity (or Messianic Judaism) is true are two very different questions. But I can see no reason to seriously consider or accept either claim, and in each case there's a lot at stake in doing so if you're wrong.