The holy city of Yerushalayim isn't mentioned by name in the Torah. However it is alluded to: "And Malki Tzedek the king of Shalem brought out bread and wine, and he was a priest to G-d, the most high." (Bereshis 14:18) Onkeles translates: "The king of Yerushalayim" and the rest of the commentators explained that the Torah refers to Jerusalem (HaRav Sa'adya Gaon, Even Ezra, Ramban and the Radak).

Already in his name Malki Tzedek, we see that the issue of righteousness and justice is connected with Yerushalayim.

And the Ramban wrote:

And its king was called "Adoni Tzedek" (Yehoshua 10:1)… and Malki Tzedek (like) Adoni Tzedek – Yerushalayim is called Tzedek, as it says (Yishayahu 1:21): "Righteousness will dwell within it."

The Radak said regarding this verse:

And the kings of Jerusalem were called Malki Tzedek or Adoni Tzedek, and in the Book of Yehoshua is written "Adoni Tzedek the king of Yerushalayim" because Jerusalem is the abode of righteousness and peace, and will not indefinitely tolerate injustice, extortion and despicable behavior and therefore it vomits (expels) the sinners who dwell there. As it says: "So the land will not vomit you for defiling it, as it vomited the nation which dwelled there before you."  (Vayikra 18:25)

Avraham Avinu arrives at the city of righteousness after his war with the four kings, which broke out because they oppressed the population of the land and abducted Lot. Indeed, the Creator of the world attested of Avraham:  "Since I know him that he will command his sons and his house after him and they will keep the way of HaShem to do righteousness and justice in order that HaShem will bring on Avraham that which He spoke regarding him." (Bereshis 18:19)

Over sixty years will pass until Avraham will return to the same place, only this time it will receive an additional name: 

"And (G-d) said: Please take your son, your only one, whom you love, Yitzchak, and go to the land of the Moriah and offer him up as a sacrifice on one of the mountains which I will tell you." (Bereshis 22:2)

And the Gemara asks (Ta'anit 15a): What is (the meaning of) Har HaMoriah? Rabbi Levi Bar Hama and Rabbi Hanina differed: One said: The mountain that hora'ah (teaching) emanated from to Israel. And the other said: the mountain that mora (fear) emanated from to the nations of the world. 

In other words, Yerushalayim is unique in two fundamental ways. One: that righteousness and justice are its guiding light. And two: Torah study and teaching to all of Israel, as Rashi says:

The mountain from which teaching - Torah to Israel - goes out: "Since from Zion will go out Torah" (Yishayahu 2:3), "They will teach Your laws to Ya'akov" (Devarim 33:10), and Lishkat haGazit, the chamber of the Sanhedrin in the Temple, was where the prophets stood and rebuked Israel.

Now we can understand quite a few verses, words of the prophets, which tie Yerushalayim to righteousness.

The prophet Yishayahu call Yerushalayim, at the peak of its glory: "The faithful quarter (city), was filled with justice, righteousness dwelled in it." (Yishayahu 1:21) But when Israel sinned: "Your silver has become impure, your wine diluted with water. Your princes are rebellious and the companions of thieves, all love bribes and pursue payments. They will not judge the orphan and the dispute of the widow will not come before them." (ibid 22-23)

Perverted judgement and lack of justice lead to destruction.

And how will the future redemption come about? "And I will restore your judges as in the past and your counsellors as in the beginning, and afterwards you will be called the city of righteousness, the faithful quarter. Also: Zion will be redeemed through justice and her repentant will return through righteousness." (Yishayahu 1:26-27) And Rashi explains: "will be redeemed through justice – by virtue of having in it makers of justice."

Yirmiyahu also prophesized:

"Behold, days are coming, says HaShem, and I will enact the good thing which I spoke concerning the house of Israel and the house of Yehudah. In those days and at that time I will cause to grow for David a sprout of righteousness, and he shall do justice and righteousness in the land. In those days, Yehuda will be redeemed, and Yerushalayim will dwell securely and this is the name that He will call it: HaShem is our righteousness." (Yirmiyahu 33:14-16)

Therefore we can understand why the Torah instructed us that the place of the Sanhedrin is Yerushalayim:

"When something in law is baffling to you, between blood and blood, between judgement and judgement, between plague and plague, words of dispute within your gates, then you will arise and go up to the place which HaShem will choose. And you will come to the Kohanim the Leviim and to the judge who will be in those days, and you will inquire and they will tell you the words of judgement. And you will do according to what they will tell you… do not deviate from the word which they will tell you to the right or left." (Devarim 17:8-11)

At the same time that the judges make verdicts and pass judgements, the role of the Kohanim is to teach Torah:

"For the lips of the Kohen will keep wisdom, and Torah should be sought from his mouth, for he is an angel (emissary) of the Lord of Hosts." (Malachi 2:7)

Our joy over the redemption of Yerushalayim, and our prayer for its culmination in the building of Beit Ha Mikdash is not only in the physical sense. It is also directed to the great spirit of Torah learning in all its depth, as well as the supreme desire to walk in the path of Avraham Avinu, to keep the way of G-d and do righteousness and justice.

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