Three things were said in the presence of Rabbi Akiva: two in the name of Rabbi Eliezer, who came from Beis Shammai, and one in the name of Rabbi Yehoshua, who was from the side of Beis Hillel. Both of them were teachers of Rabbi Akiva.
The first matter: going out with an 'ir shel zahav' on Shabbos:
The first thing said in the name of Rabbi Eliezer is that a woman is permitted to go out on Shabbos wearing the ornament known as "ir shel zahav" - a city of gold (a Mishnah in Maseches Shabbos). The underlying principle: one may not carry an object four amos in a public domain, and whoever does so desecrates Shabbos. However, a person is permitted to wear clothing, since a garment is considered subordinate to the body, and for the same reason one may also wear jewelry that serves as an adornment for the body - but the Sages decreed that women should not wear jewelry in public, out of concern that a woman would remove her ornament in order to show it to her friends, and once it is in her hand it is no longer subordinate to her body, and she would carry it four amos in the public domain.
According to Rabbi Eliezer, the 'ir shel zahav' is excluded from the decree. This ornament is a sort of headband or crown made of gold, large and heavy, bearing an image of the skyline of Jerusalem. Only a very wealthy and prominent woman - that is, a distinguished woman - would own such an expensive ornament, and a woman like that would never remove her jewelry in a public domain in order to show it to her friends, since that is not a dignified thing to do. That being so, there is no reason to extend the decree and forbid wearing an 'ir shel zahav' in a public domain.
The halachah:
The Sages did not accept this view, and according to the seforim a woman may not go out with an 'ir shel zahav' even if she is wealthy, prominent and distinguished, out of concern that she may remove the ornament and carry it in the public domain. Nevertheless, although it seems clear that women may not wear jewelry in a public domain, the practice of Jewish women for a long time now has in fact been to wear jewelry in public. A great deal of ink has been spilled in reconciling this, but it is certainly the widespread custom today and the accepted practice in halachah.
The second matter: those who race birds are disqualified from testifying:
The second thing said in the name of Rabbi Eliezer is that those who race after birds, doves or turtledoves, are disqualified from testifying in beis din. This subject has already been learned twice, in Maseches Rosh Hashanah and in Maseches Sanhedrin, and there are two explanations of it:
It refers to those who race birds against each other, the way horses are raced nowadays, and they are therefore gamblers.
It refers to those who train their birds to draw a neighbor's birds away from his dovecote, where he raises and cares for them, and there is a kind of theft in this.
Regarding gambling:
A precise point must be made: gamblers are disqualified from testifying by rabbinic law. By Torah law a person must be a robber or a thief in order to be disqualified from testimony, or wicked in a general sense - anyone who does deeds that render him wicked, theft among them. Gambling is not actual theft, but a kind of mild theft, since the loser does not expect to pay, and even when he does pay he does not truly want to, so that the money leaves his hand against his will.
A second approach to the rabbinic disqualification of gamblers is that the Sages said that gamblers are not engaged in the settling of the world: they take no part in the general economy, in advancing society or in building civilization. For that reason they are considered outsiders, who do not share in the norms of society and do not care about what happens in it, and therefore they are not to be used as witnesses.
According to this last approach, which is the accepted halachah, there is a major practical difference: someone who gambles only part time and holds a steady job - for example, a person who works during the week as a dentist and gambles only on weekends - is engaged in settling the world and contributing to society, and is therefore fit to testify.
Regarding driving birds out of a neighbor's dovecote:
According to the second explanation, this refers to someone who trains his bird to chase after his fellow's birds or to lure them out of their coop. In the time of the Mishnah, raising birds was a major business: the birds were used for eggs, for fertilizer, as carrier pigeons, for sacrifices and for meat. And how does one acquire birds? You put out food for them and they come. It turns out that legally the birds resting in the backyard are not the property of the owner of the yard at all; they are like wild birds that came to rest in his yard because he put out food for them. Therefore, one who lures them away is not stealing his fellow's birds, since they are not really his birds.
Even so, the Sages said that this is a recipe for very bad neighborly behavior, and for the sake of peaceful relations they forbade a person to lure his neighbor's birds away, because it leads to quarreling. And one who does so is penalized by being considered wicked and disqualified from testifying. Both of these laws are cited in the Shulchan Aruch.
The third matter, in the name of Rabbi Yehoshua: "HaSheretz b'fi chuldah":
The third matter stated before Rabbi Akiva was said in the name of Rabbi Yehoshua, and this is its wording: "HaSheretz b'fi chuldah um'haleches al gabei kikaro shel Terumah" - a dead creeping creature in the mouth of a weasel that is walking over a loaf of Terumah. First we will translate, and then we will explain.
A sheretz is one of the eight small creatures listed by the verses in the book of Vayikra, which, when they are dead, transmit tumah through direct contact. The precise identity of these eight is not clear, and they are known as "the eight sheratzim."
The case is this: one of these dead creeping creatures, such as a dead mouse, is in the mouth of a weasel. The weasel itself is also one of the creeping creatures, but it is alive, and therefore it does not transmit tumah. In its mouth, since it is a predator, is a dead mouse, which is certainly a source of tumah - except that creeping creatures transmit tumah only through direct contact. And the weasel, with the dead mouse in its mouth, is walking over loaves of Terumah bread.
If the mouse touched the loaf of bread, the bread is certainly tamei, and it may not be eaten but must be burned. But how do we know that this is what happened? We do not know, and this is a case of doubt: we do not know whether the dead creeping creature came into direct contact with the Terumah. About this the Mishnah says: "safek naga safek lo naga - sfeiko tahor" - it is doubtful whether it touched or did not touch, and the doubtful case is pure. In other words, if we do not know whether there was direct contact or not, we rule that all the loaves of bread are pure, and the kohen may eat them.
The usual rule in cases of doubtful tumah:
The usual rule in a situation of uncertainty about the spread of tumah is this: a doubtful case of tumah in a public domain is pure. If the question arises in a public domain, meaning a place where people are found, three people or more - as we already learned earlier in the tractate - we rule that it is pure. But a doubtful case of tumah in a private domain, meaning a place where three people are not normally found, is decided as tamei. This is the rule that governs most cases of doubtful tumah.
The overriding rule: tumah that has not come to rest:
But there is a higher principle at work here that overrides the usual rule, and in fact there are several such principles. The rule the Mishnah teaches here is that even in a private domain, where doubts are generally resolved stringently, if the source of tumah never came to rest but was in constant motion, we rule leniently. And that is precisely our situation: the dead sheretz is in the mouth of a weasel that is moving about, so the source of tumah is in motion and never at rest. The rule is that if the tumah did not come to rest, we assume that no direct contact took place, and the doubt is resolved in favor of purity. This is the third case in our Mishnah, and it too is accepted as the halachah.