We are up to Mishnah 2 in the fourth chapter of Maseches Gittin, which contains two halachos instituted "for the betterment of the world": one concerning the cancellation of a get, and one concerning the names of the husband and wife written in it.
Cancelling the get - the situation before the enactment:
Originally, a husband who had sent a get to his wife was permitted to convene a beis din somewhere else and declare before it that the get was null and void, and that cancellation took effect. The Gemara records a dispute as to how many judges are required for such a cancellation, and this dispute has a direct bearing on how we understand the rest of the Mishnah.
The first opinion: Even two are sufficient. Although a beis din normally consists of three, we find that two also serve as a beis din in a case that involves nothing more than a statement.
The second opinion: Specifically three are required, so that word will spread that the husband cancelled the get. Accordingly, a cancellation made before two is not an effective cancellation and is not valid.
The enactment "for the betterment of the world":
Later an enactment was instituted that a husband may no longer convene a beis din somewhere else and cancel the get before it, and all of this for the betterment of the world.
What is that "betterment of the world," and what was the enactment meant to correct? The Gemara offers two answers:
To prevent mamzerim: So that mamzerim should not be born. This opinion follows the view that the cancellation is made before only two, and therefore word does not spread. The wife does not necessarily hear that the get was cancelled, and she may go on to marry another man on the strength of it; and since her original get was cancelled, her marriage to the other man is a forbidden union, and her child is a mamzer.
To prevent a woman being left chained: So that women should not be left unable to remarry and unable to receive their get. That is, we want to make it harder for the husband to cancel, so that he cannot do so before some chance beis din but only in his wife's presence. This is the opinion that holds that even if the beis din consisted of three - and that was the original form of cancellation - the cancellation is nevertheless no longer valid according to the enactment of Rabbi Akiva, since our aim is to make cancellation difficult for the husband.
The names of the husband and wife in the get:
A further halachah in the Mishnah: originally, when the name of the couple's city was pronounced one way in their home town and a different way in the place where the get was written, the get would be written according to the form used in the place of writing, even though this was not the form used in their own place.
Later an enactment was instituted that they should write in the get "ish peloni v'chol shem sheyesh lo" - "so-and-so and any other name he has" - and "ishah plonis v'chol shem sheyesh lah" - "such-and-such a woman and any other name she has" - for the betterment of the world. Here the point of the enactment is not to cast aspersions on the get: that nobody should find fault with the get out of a concern that it was not written with the correct name, and treat it as an invalid get.
Two ways of understanding the wording of the Mishnah were offered:
Some understand that one actually writes out the other, additional names of the man and the woman.
Others understand that one writes the general statement itself, literally in the words "so-and-so and any other name he has."
In summary: In this Mishnah we learned two enactments instituted for the betterment of the world: cancelling a get before a beis din somewhere else, regarding which there is a dispute whether it is done before two or before three, and a dispute as to the reason for the enactment - whether to prevent mamzerim or to prevent a woman being left chained; and writing the names of the couple in the get with the formula "and any other name he has," so that no one should cast aspersions on the get and invalidate it.