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How do the Jewish people get forgiveness without animal sacrifices?
To begin with, the animal sacrifices are not abolished. The Jewish people are not able to perform the laws pertaining to the animal sacrifices now, because we are not allowed to sacrifice here and there and everywhere, but only on the altar in the Temple courtyard. This restriction is commanded in the Torah: "If any man of the house of Israel kills an ox or a lamb or a goat in the camp, or kills it outside the camp, and does not bring it to the door of the tent of meeting, to offer it as a gift to the LORD before the tabernacle of the LORD, bloodguilt shall be imputed to that man; he has shed blood; and that man shall be cut off from among his people. This is to the end that the people of Israel may bring their sacrifices which they slay in the open field, that they may bring them to the LORD, to the priest at the door of the tent of meeting, and slay them as sacrifices of peace offerings to the LORD; And the priest shall sprinkle the blood on the altar of the LORD at the door of the tent of meeting, and burn the fat for a pleasing odor to the LORD." Leviticus 17:3-6.
“But when you go over the Jordan, and live in the land which the LORD your God gives you to inherit, and when he gives you rest from all your enemies round about, so that you live in safety, then to the place which the LORD your God will choose, to make his name dwell there, thither you shall bring all that I command you: your burnt offerings and your sacrifices, your tithes and the offering that you present, and all your votive offerings which you vow to the LORD. And you shall rejoice before the LORD your God, you and your sons and your daughters, your menservants and your maidservants, and the Levite that is within your towns, since he has no portion or inheritance with you. Take heed that you do not offer your burnt offerings at every place that you see; but at the place which the LORD will choose in one of your tribes, there you shall offer your burnt offerings, and there you shall do all that I am commanding you.” Deuteronomy 12:10-14.
So the Jewish people are only allowed to sacrifice on the altar in the Temple courtyard, and not any place else. And that is the reason that after the destruction of the Temple no more animal sacrifices are being brought. But the prophets tell us that after the building of the third and everlasting Temple the animal sacrifices will be reinstated again, as we can read in Zechariah 14:20, Malachi 3:1-4 and Ezekiel 43.
So how do we get forgiveness for sins now we don’t have a Temple and sacrifices anymore? The same way we got it before: Through repentance and returning to God and His law. The sacrifices in the Temple were, except for a few exceptions, only for unintentional sins. Look at Leviticus 4:2-3; “"Say to the Israelites: `When anyone sins unintentionally and does what is forbidden in any of the LORD's commands-- "`If the anointed priest sins, bringing guilt on the people, he must bring to the LORD a young bull without defect as a sin offering for the sin he has committed.”
Leviticus 4:13-14; “"`If the whole Israelite community sins unintentionally and does what is forbidden in any of the LORD's commands, even though the community is unaware of the matter, they are guilty. When they become aware of the sin they committed, the assembly must bring a young bull as a sin offering and present it before the Tent of Meeting.”
Leviticus 4:22-23; “"`When a leader sins unintentionally and does what is forbidden in any of the commands of the LORD his God, he is guilty. When he is made aware of the sin he committed, he must bring as his offering a male goat without defect.”
Leviticus 4:27-28; “If a member of the community sins unintentionally and does what is forbidden in any of the LORD's commands, he is guilty. When he is made aware of the sin he committed, he must bring as his offering for the sin he committed a female goat without defect.”
Leviticus 5:15 "When a person commits a violation and sins unintentionally in regard to any of the LORD's holy things, he is to bring to the LORD as a penalty a ram from the flock, one without defect and of the proper value in silver, according to the sanctuary shekel. It is a guilt offering.
Numbers 15:23-24; “"`Now if you unintentionally fail to keep any of these commands the LORD gave Moses-- any of the LORD's commands to you through him, from the day the LORD gave them and continuing through the generations to come-- and if this is done unintentionally without the community being aware of it, then the whole community is to offer a young bull for a burnt offering as an aroma pleasing to the LORD, along with its prescribed grain offering and drink offering, and a male goat for a sin offering.”
Numbers 15:27; "`But if just one person sins unintentionally, he must bring a year-old female goat for a sin offering.”
Very clear, sin offerings are for unintentional sins. But what happens when somebody sins willingly?
Numbers 15:30-31; “But the person who does anything with a high hand, whether he is native or a sojourner, reviles the LORD, and that person shall be cut off from among his people. Because he has despised the word of the LORD, and has broken his commandment, that person shall be utterly cut off; his iniquity shall be upon him.”
No sacrifice for this intentional sinner. That means that in this case forgiveness must be acquired through other means than a sacrifice. That is repentance and turning back to God and His law. Read the book of Jonah. God said to Jonah that he is going to destroy Nineveh because of the evil of it’s inhabitants. But they Nineveh repented and fasted and God decided not to destroy the city. No sacrifices here, only repentance, and God forgives.
The same held true for King David. He did what he did with the wife of his neighbor, and the prophet Nathan confronted him with it. After David realized what he had done he cried out: “I have sinned before God”, and the prophet Nathan said after that: "God has taken away your sin." II Samuel 12:9-13. No animal sacrifices, just repentance and God forgives.
HOWEVER, this does not mean that after we sinned, we just have to repent and God forgives and that's it. God does forgive every sin, after sincere regret and returning to His law, but this does not mean we get away unpunished. God forgave King David, but nevertheless he was severely punished for his deed. When the prophet Nathan confronted him, he gave him some severe curses: “Why have you despised the word of the LORD, to do what is evil in his sight? You have smitten Uri'ah the Hittite with the sword, and have taken his wife to be your wife, and have slain him with the sword of the Ammonites. Now therefore the sword shall never depart from your house, because you have despised me, and have taken the wife of Uri'ah the Hittite to be your wife. Thus says the LORD, 'Behold, I will raise up evil against you out of your own house; and I will take your wives before your eyes, and give them to your neighbor, and he shall lie with your wives in the sight of this sun. For you did it secretly; but I will do this thing before all Israel, and before the sun." David said to Nathan, "I have sinned against the LORD." And Nathan said to David, "The LORD also has put away your sin; you shall not die. Nevertheless, because by this deed you have utterly scorned the LORD, the child that is born to you shall die." And every one of these curses came true. We read in II Samuel 12 that the first child of David and Bathseba died. The son of David, Amnon, fell in love with Tamar, his half sister, and he raped her. Absolom was so angry about this that he killed Amnon. II Samuel 13. Later Absolom rose up against his father, and David had to run for his life. This was the evil rising up against David from his own household.
Then the adviser of Absolom told him to rape the concubines of his father in order to show the people that Absolom was in charge now. So a tent was pitched on top of the palace, and Absolom went into the tent with the wives of his father, fulfilling the curse that Davids wives would publicly be given to another. II Sam 16:20-22
So here we see how it works: When we sin we get punished, and not God, not the neighbor, but we ourselves. This same concept of everybody suffering for his own sin we also see in Isaiah 53. Of course it is much more convenient that we can do whatever we want and we are already forgiven in advance by the suffering of God, but that is not how it works.
For a deeper Biblical (that means non-Christian) outlook upon sin and punishment look here: Why did the holocaust happen?
For the main article: "Why don't the Jewish people recognize the New Testament?" LOOK HERE