Dora Bas Rivka Silver O'H
The angel did Lot a favor and did not destroy the suburb so that Lot might flee there. The town became known as Tzoar ("insignificant") because Lot had said that it was small enough for them to overlook. Lot's family headed there while G-d rained fire and brimstone down on Sodom and Gomorrah and the other cities. Lot's wife, however, disregarded the instructions notto look back at the city. For her violation, she was turned into a pillar of salt.
Avraham was watching from a respectable distance. From his vantage point, the smoldering plain looked like a smoking kiln.
Lot was too scared to remain in Tzoar, so he and his daughters ended up relocating to the mountains as the angel had originally instructed. There, they sought refuge in a cave. As far as they knew, everything outside had been destroyed and they were the last people left alive. In order to perpetuate the human race, the girls conspired to get their father drunk and conceive children with him; this they successfully accomplished. They had sons from their father, whom they named Moav ("from father") and Ben-Ami ("son of my people" or, in this context, "son of my family member"). These boys are the ancestors of the nations Moav and Ammon.
Meanwhile, Avraham and Sarah were on the move again. They moved to the Negev and Avraham often conducted business in the land of Gerar. As he had done in Egypt, Avraham publicized that Sarah was sister, so Avimelech, the king of Gerar, did not hesitate to take Sarah for himself. That night, G-d appeared to Avimelech and told him that he would die for abducting a married woman. Avimelech complained that Avraham had entrapped him by claiming Sarah was his sister; anything Avimelech had done wrong was without malice aforethought. G-d told Avimelech that He knew this, which is why He prevented him from touching Sarah. G-d told Avimelech that if he wanted to live, he must return Sarah and ask Avraham to pray for him.
Avimelech went to Avraham and asked why he felt the need to deceive them. Avraham replied that he could see that Gerar was a "godless" place and, besides, the claim wasn't false since Sarah was the daughter of Avraham's brother. (In this sense, "sister" in Hebrew means a close relative.) When they travel, Avraham said, they only publicize their blood relationship. Avimelech gave Avraham gifts and told him that the entirety of the land was at his disposal. He also paid a hefty fine to compensate them for their trouble; this would serve as a sign to people that Avraham and Sarah were innocent in the matter.
When Avimelech abducted Sarah, G-d had sealed all the wombs of his household. (He possibly sealed all the bodily orifices, which would have even greater repercussions.) Avraham prayed for them and they were healed. Avimelech's wife and servant girls were now able to have children.
As long as G-d was healing women and allowing them to conceive, He acted on His promise to Sarah. She conceived and had a son, as G-d had said, and they named him Yitzchak (Isaac). On the eighth day, Yitzchak was circumcised, as G-d had commanded of Avraham's descendants.
Now, back to the matter of Lot's daughters. These women are generally criticized for giving their sons names that none-too-subtly reveal that they were impregnated by their own father (albeit without his knowledge or consent). In 1921, Lot's daughters finally got their "day in court," to explain their actions. At that time, Rav Moshe Feinstein was summoned to the home of a congregant, who was on his deathbed. With his dying strength, the man explained that he was being punished for disparaging Lot's daughters in a lecture a week earlier. That night, two old women appeared to him in a dream. They identified themselves as Lot's daughters and said that the reason they gave their sons such names is as follows:
As Lot's daughters, they were part of Avraham's family and, in his merit, they were saved from the destruction of Sodom. They impregnated themselves through their father because they thought the world was otherwise destroyed. When they exited the cave, they learned this was incorrect. They could have claimed – or others might have assumed – that they miraculously conceived without the benefit of a man. They gave their offspring "shameful" names in order to publicize the truth and to prevent their sons being deified as having Divine parentage. For this selfless act, they merited being ancestors of Moshiach. (David is descended from Moav and Shlomo's wife, the mother of Rechavam, is descended from Ammon.)
This story is recorded in the biography of Rav Moshe found in the introduction to Igros Moshe volume 8 (p. 15).