1) "THE HALACHAH FOLLOWS THE STUDENT'S OPINION"
QUESTION: The Mishnah (11b) relates that a certain Talmid said in the name of Rebbi Yishmael that Beis Shamai and Beis Hillel agree that an entrance that is less than four Amos wide needs only a Lechi or a Korah to permit one to carry in a Mavoy. They disagree about an entrance that is larger than four (and less than ten) Amos wide; Beis Shamai requires that the Mavoy have both a Lechi and a Korah, and Beis Hillel requires that the Mavoy have either a Lechi or a Korah. Rebbi Akiva maintains that Beis Shamai and Beis Hillel disagree even about an entrance that is less than four Amos wide.
The Beraisa relates that Rebbi Akiva declared that Rebbi Yishmael himself could not have said that Beis Shamai and Beis Hillel agree about an entrance less than four Amos wide. Rather, the Talmid said it erroneously in the name of Rebbi Yishmael. Rebbi Akiva adds that the Halachah follows that Talmid.
The Gemara points out that Rebbi Akiva seems to contradict himself. Rebbi Akiva first says that the Talmid made a mistake when he quoted Rebbi Yishmael, and then he rules that the Halachah follows that Talmid!
Rav Yehudah in the name of Shmuel resolves the apparent contradiction. He says that Rebbi Akiva was not actually ruling in accordance with that Talmid. Rather, he was merely praising that Talmid for his sharp and innovative thinking in order to encourage his own students to think innovatively.
How could Rebbi Akiva state a falsity merely for the sake of sharpening his students?
ANSWER: RAV GEDALYAH RABINOWITZ zt"l (of Manchester) in GIDULEI HEKDESH explains that there are two parts to the Talmid's statement in the Mishnah. First, the Talmid quoted Rebbi Yishmael as saying that Beis Shamai and Beis Hillel do not argue about an entrance less than four Amos wide. Second, he added on his own what they do argue about -- an entrance that is between four and ten Amos wide.
Rebbi Akiva taught that Rebbi Yishmael did not say that Beis Shamai and Beis Hillel agree about an entrance less than four Amos wide (rather, they indeed argue in such a case). However, the Halachah does follow that Talmid's second statement, in which he said that they argue about an entrance that is between four and ten Amos wide. That is, the maximum width of a Mavoy is ten Amos, and not like Rebbi Yehudah (2b and 10a) who says that a Mavoy may be wider than ten Amos. (See RAV ELAZAR MOSHE HA'LEVI HOROWITZ and RAV YAKOV EMDEN for alternative approaches.)
2) THE VALUE IN BEING "METAHER" A "SHERETZ"
QUESTION: The Gemara relates that there was a certain very sharp Talmid who was able to present 150 arguments to convince others that a definitely Tamei Sheretz is Tahor. Ravina said that he himself could prove a Tamei Sheretz to be Tahor with a Kal v'Chomer: If a snake, which kills and thereby increases the Tum'ah in the world, is not Tamei itself when it dies, then certainly a Sheretz, which does not kill and does not increase Tum'ah in the world, should not be Tamei when it dies.
What is the value of constructing a Derashah which is not true?
ANSWER: The Derashah is true with regard to a certain element of the Tum'ah of the Sheretz. RABEINU TAM (cited by Tosfos) explains that we know that a Sheretz is Metamei only b'Maga (through contact) and not b'Masa (by carrying it). The Talmid and Ravina displayed their acumen by deriving a source for the law that a Sheretz -- even though it is a Chayah (to which Tum'as Neveilah applies and is normally Metamei b'Masa when it is dead) -- is not Metamei b'Masa when it is dead (just as a snake is not Metamei b'Masa).
How, though, did they know that a snake is not Metamei b'Masa with Tum'as Neveilah? According to Rabeinu Tam, they learned it from the Toras Kohanim (Shemini 10:3) which derives from the words "Kol Holech Al Gachon" (Vayikra 11:42) that a snake is excluded from Tum'as Neveilah.
3) AGADAH: 150 REASONS TO BE "METAHER" A "SHERETZ"
The Gemara relates that there was a certain very sharp Talmid who was able to present 150 arguments ("Ta'amim") to convince others that a definitely Tamei Sheretz is Tahor.
The VILNA GA'ON provides a novel explanation for this statement. When the Talmid said that he was able to be Metaher a Sheretz with 150 "Ta'amim," he did not mean that he had 150 reasons or ways to be Metaher a Sheretz. Rather, he meant that he had only one way, and that was the Kal v'Chomer that Ravina expressed. What, then, did he mean when he said that he had 150 "Ta'amim" to be Metaher a Sheretz?
The verse (Shemos 27:18) describes the dimensions of the Chatzer of the Mishkan as "a length of one hundred Amah, and a width of fifty by fifty." The names of the cantillation marks (Ta'amei ha'Mikra) on those words ("Me'ah ba'Amah v'Rochav Chamishim ba'Chamishim") are Kadma v'Azla, Munach Revi'i. The Talmid meant that by understanding the "Ta'amim" (cantillation notes) on the words describing the 100 x 50 Amos dimensions of the Chatzer, one can find an allusion to the source of the Kal v'Chomer. What is that allusion?
The Gemara in Sanhedrin (59b) teaches that originally the snake had legs and feet with which it walked ("Kadma v'Azla" -- "it went ahead and walked"). When the snake sinned, Hash-m took away its legs and made it lie crouched, close to the ground ("Munach Revi'i"), from where it would bite man and bring Tum'as Mes into the world. It is from the snake, that spreads Tum'ah but itself is Tahor, that a Kal v'Chomer may be derived to declare a Sheretz Tahor (regarding Tum'as Masa)!