A Jealous God Trying to understanding multi-generational punishment

(4) You shall not make for yourself a sculptured image, or any likeness of what is in the heavens above, or on the earth below, or in the waters under the earth.(5) You shall not bow down to them or serve them. For I the LORD your God am an impassioned God, visiting the guilt of the parents upon the children, upon the third and upon the fourth generations of those who reject Me,(6) but showing kindness to the thousandth generation of those who love Me and keep My commandments.

(5) לשנאי OF THEM THAT HATE ME — This must be explained in the same sense as the Targum takes it: when they retain in their hands (follow the example of) the evil doings of their ancestors (Sanhedrin 27b); and He keeps (stores up) the mercy which a person does to give a reward for it to the thousand generations of that person’s descendants. It follows, therefore, that the measure of good (reward) is greater than the measure of punishment in the proportion of one to five hundred, for the former is threatened only to four generations whilst the latter is bestowed upon thousands (two thousands at least) (Tosefta Sotah 4:1; see Rashi on Exodus 34:7).

And the definition of pekida is "remembered." For example, "And God pakad Sarah," i.e. "God remembered." God extends the time [it takes God to punish] the wicked one since perhaps the wicked will turn back from sin and will give birth to a child that is better than the wicked . And then if the child, grandchild and great-grandchild follow in the footsteps of the parent God will no longer extend mercy. God hates all of them. In the fourth generation their memory will be erased [they will be forgotten] since God remembers what the first, second and third generations did. Therefore God does not have mercy for the fourth generation.

But their explanation is not correct. Scripture mentions God's remembrance of all of them equally, and it does not specify that the vengance is exactly [only] in the end, i.e. on the fourth generation. Beside, the term p'kidah in conjunction with the work "on" (visiting the iniquity on the third and fourth generations) -- is not used in connection with remembrance, but rather signifies vengeance (or punishment). Thus:"And on the day that I punish, I will punish them for their sins," "in that day God will punish with his sore and great and strong sword leviathan the slain serpent, and leviathan the tortuous serpent, and God will slay the dragon that is in the sea," "God will punish the host of the heavens on high." All of these are expression of vengeance and punishment.

The correct interpretation thus appears to me to be that the scripture is stating that God visits the iniquity, which the parent perpetrated, upon the children and excises them on account of the iniquity of their father, something like it is said, "Prepare your slaughter for the children for the iniquity of the parents."

(6) The LORD passed before him and proclaimed: “The LORD! the LORD! a God compassionate and gracious, slow to anger, abounding in kindness and faithfulness,(7) extending kindness to the thousandth generation, forgiving iniquity, transgression, and sin; yet He does not remit all punishment, but visits the iniquity of parents upon children and children’s children, upon the third and fourth generations.”

(16) Parents shall not be put to death for children, nor children be put to death for parents: a person shall be put to death only for his own crime.

(20) The person who sins, he alone shall die. A child shall not share the burden of a parent’s guilt, nor shall a parent share the burden of a child’s guilt; the righteousness of the righteous shall be accounted to him alone, and the wickedness of the wicked shall be accounted to him alone.

(16) לא יומתו אבות על בנים, even if the crime involved was insurrection against the established authority, king, etc. It had been a widely accepted rule that kings would also execute the family members of rebels in order to ensure that the surviving relatives of such executed rebels would not form an active opposition to the king as a result of their grievance. Such matters are referred to in Isaiah 14,21 “prepare a slaughtering block for his sons because of the guilt of their father. Let them not arise to possess the earth! Then the world’s face shall be covered with towns.” The meaning of the word soneim [lit. haters] which I used earlier is to be understood as in Samuel I 28,17 “he became a foe.” In any event, the Torah prohibits Jewish kings from killing one person as a substitute for another; this is part of G’d’s pity for members of His people. Amatziah, King of Yehudah, abided by this legislation when he executed the servants who had killed his father but did not touch their children (Chronicles II 25, 3-4)

This is as it is taught in a baraita: The verse: “And also in the iniquities of their fathers shall they pine away with them” (Leviticus 26:39), is referring to a case where they adopt the actions of their ancestors as their own. Do you say that it is referring specifically to a case where they adopt the actions of their ancestors, or perhaps it applies even where they do not adopt their ancestors’ actions? When the verse states: “Every man shall be put to death for his own sin,” the Torah explicates that one is not put to death if he did not sin. Therefore, the verse in Leviticus is clearly referring to a case where they adopt the actions of their ancestors as their own.The Gemara asks: And are descendants not punished for the sins of their ancestors unless they adopt their behavior? But isn’t it written: “And they shall stumble one upon another” (Leviticus 26:37)? This verse is homiletically interpreted to mean that the Jewish people shall stumble, one due to the iniquity of another, i.e., they are punished for each other’s sins, which teaches that all Jews are considered guarantors, i.e., responsible, for one another.The Gemara answers: There, in the verse in Leviticus, the reference is to a case where others had the ability to protest the sin but they did not protest. Consequently, they are punished for not protesting, regardless of any familial relationship they may have with the sinner.