Gemara

38a (Bnei R. Chiya): Remnants of Techeiles are Kosher.

Question: How much must remain?

Answer (Bar Hameduri): Kedei Anivah (enough to make a loop) must remain.

Question: Must one be able to loop (one string around) all the strings, or each by itself?

This question is not resolved.

(Rava): We learn from Bnei R. Chiya that one must tie the strings together after each Chulya (set of windings)

If this were not necessary, if only a small remnant remained, the last knot would become undone, and the whole Gedil (windings) would become undone!

Rejection: Perhaps it is not necessary. Bnei R. Chiya taught that remnants are Kosher only in a case in which the strings happened to be tied!

39a (Rav): If a string totally snapped, it is Pasul.

39b: Rabah bar bar Chanah expounds that the strings must be Gedil or Pesil (straight);

Rav requires Pesil in every case.

41b (Beraisa - Beis Shamai): Four finger-widths must be Meshulash (this will be explained);

Answer #2 (Beis Hillel): Three fingers are required;

(Rav Papa): The Halachah is, four fingers must be Meshulash.

Inference: The Beraisa implies that there is a Shi'ur (minimal length) of Tzitzis.

Contradiction (Beraisa #2): Tzitzis must extend somewhat past the corner. Mashehu (any amount) is Kosher;

Elders of Beis Hillel and Beis Shamai: There is no Shi'ur of Tzitzis. Similarly, there is no Shi'ur of a Lulav.

Suggestion: This means that there is no Shi'ur at all!

Answer: No. There is no upper Shi'ur, but there is a minimal length.

Support: It says 'similarly, there is no Shi'ur of a Lulav.' This cannot mean that there is no Shi'ur at all!

(Mishnah): If a Lulav is three Tefachim (and an additional Tefach) that rustles when shaken, it is Kosher.

We must explain that there is no maximum length of a Lulav, but there is a minimum; Likewise, there is no maximum length of Tzitzis, but there is a minimum!

Rishonim

Rambam (Hilchos Tzitzis 1:6): The length of the strings must be at least four fingers.

Rambam (9): The Gedil should be a third of the length of the strings, and the Pesil should be two thirds.

Rosh (Hilchos Tzitzis 7): The Gemara asked about whether one must be able to loop (one string around) all the strings, or each by itself. It did not resolve this. We are stringent, and require enough to loop around all the strings. Rava learned from Bnei R. Chiya that one must tie the strings together after each Chulya. This connotes that even if the Pesil was totally cut, and Kedei Anivah remained in the Gedil, it is Kosher. If we required Kedei Anivah in the Pesil, why did Rava say that if only a small remnant remained, the last knot would become undone, and the whole Gedil would become undone? Since there is Kedei Anivah in the Pesil, the knot will endure! Rashi said that Rava argues with Bar Hameduri, who requires Kedei Anivah. The Gemara connotes unlike this. The Sifri (Devarim 234) says that (initially the strings must be four fingers, but) the remnants can be any amount. This connotes that any amount of Gedil suffices.

Chazon Ish (OC 12:3): If the Gedil becomes undone, we do not say that now it is Anaf. The Anaf must be put in the Tzitzis to be Anaf.

Tosfos (41b DH Beis (2)): Rashi says that 'Meshulash' refers to the Pesil; four fingers must hang down straight. R. Tam says that it refers to the Gedil.

Mordechai (Hilchos Ketanos 939): We do not require Kedei Anivah and the entire Gedil. Rabah bar bar Chanah holds that Gedil or Pesil suffices! R. Tam requires that two full strings remain intact; 'in order to loop around all of them' means around all four (ends of) strings. According to the Ri, who says that all the strings can be cut, it means around all eight strings. I am unsure about 'each by itself.' The words connote (enough to wrap around) all the strings, each by itself. Or, perhaps it means that to loop one string alone. Semag connotes like this.

Mordechai (940, citing R. Shimshon): Rashi connotes that Kedei Anivah must be from the Pesil. This forced him to say that some of the Pesil was cut near the Gedil. This is like Rav, who requires Gedil and Pesil. Presumably, the Halachah follows Rav (against Rabah bar bar Chanah), for he was greater in Chachmah and number (of Talmidim), and all Amora'im who discussed the size of the Gedil and Pesil (39a) hold like him. I cannot be Machshir Kedei Anivah in the Gedil until Mashi'ach will come.

Poskim

Shulchan Aruch (OC 12:1): If all the strings on a corner snapped, and there remained Kedei Anivah enough to loop around all the snapped strings, it is Kosher.

Beis Yosef (DH v'Chosav ha'Rosh): The Rosh says that we are stringent (that the Shi'ur is to loop around all the strings). The Rif and Rambam did not mention this. It seems that they rule leniently, because the Shi'ur is mid'Rabanan, for mid'Oraisa any amount suffices. We hold like them when it is impossible, but when it is possible, it is good to be concerned for the Rosh's opinion.

Beis Yosef (DH u'Mah): It seems that 'in order to loop around all of them' means around all of the snapped strings. I.e. one can loop them together. Some require enough to loop all eight together. This is unreasonable. Why should we require that the snapped strings can loop around the whole strings?! The Mordechai connotes like this. 'Each by itself' connotes enough to wrap the string around itself. Semag connotes that it means enough to wrap around another string. This is called 'each by itself', for this opinion does not require around all the strings.

Magen Avraham (12:2): Rashi explains that 'all of them together' is the bigger Shi'ur. The Mordechai explains that 'each by itself' means enough to loop around all the others individually (i.e. the Shi'ur to loop around one string, multiplied by the number of strings). This is a bigger Shi'ur. One should be stringent like this. The Beis Yosef explained differently; what I wrote seems correct.

Kaf ha'Chayim (1): Seemingly, it should suffice to be big enough to wrap around four strings, for we require only four ends with Kedei Anivah. However, we can say that even so, the Shi'ur Kedei Anivah is to loop around all eight ends (Levush). Zechor l'Avraham requires enough to loop around all eight ends even if only one snapped. R. Zalman is stringent like the Magen Avraham.

Chayei Adam (11:20): The Shi'ur for Tzitzis is mid'Rabanan. (However, the Torah requires at least Kedei Anivah). This is why in pressed circumstances, if only the Gedil remains, one may wear the garment. One may bless and wear it in Reshus ha'Rabim if four fingers remain. The Gemara and Poskim say Kedei Anivah; surely this is less than four fingers. (I require four fingers because we do not know how much is Kedei Anivah.) The Ri says that Kedei Anivah is from the Gedil, and the entire Gedil is only four fingers! Therefore, this is enough even according to Rashi. Even if all snapped, if he cannot get others, or he would miss Tefilah or similar concerns, he may wear it, but without a Berachah.

Bi'ur Halachah (DH Im): Even if the Magen Avraham's Perush is correct, Rashi, Semag and the Rosh disagree. The Mordechai himself seemed unsure about his Perush! We are not concerned for his Safek, since all the others were certain. Maharam Banat supported the Beis Yosef's Perush of the Mordechai. One need not be more stringent than the Shulchan Aruch. The Chayei Adam requires four fingers for Kedei Anivah. The Gemara calls it 'any amount.' Rashi holds that the entire Pesil is four fingers, so surely Kedei Anivah is not more than two fingers! Surely one need not be more stringent than than this.

Kaf ha'Chayim (1): Surely Kedei Anivah is not more than two and two thirds fingers! The Rambam says that strings must be at least four fingers long, and the Pesil should be two thirds of this. (The Shi'ur for remnants cannot be more than the minimal l'Chatchilah length of the Pesil!) Even though the Mechaber says that the strings must be 12 fingers, there is no reason to say that he argues about Kedei Anivah. Pischei Olam says that one need not be more stringent than two fingers.

Shulchan Aruch (3): Rashi holds that Kedei Anivah is from the Pesil. The Ri is Machshir even if all the Pesil was cut, and Kedei Anivah remains only from the Gedil. The custom is like Rashi. When it is not possible, one may rely on the Ri.

Beis Yosef (DH v'Gam): Why didn't the Tur mention that the Rosh agreed with the Ri? Also Semag and R. Yerucham hold like the Ri. Semak brought both opinions and did not decide. At first the Mordechai wrote like the Ri, but afterwards he cited R. Shimshon, who rejected the Ri's proofs. The custom is like Rashi. When one cannot fix the Tzitzis, he may rely on the Ri and the Rosh, for the Sifri and the Sugya support them.

Mishnah Berurah (4,13): In Sa'if 1, the Mechaber said that when possible, is good to be concerned for R. Tam's opinion, i.e. when it is easy to obtain other Tzitzis. One need not (toil to) pursue them if they are not readily available, or miss Tefilah b'Tzibur due to this. Here he wrote 'when it is not possible, one may rely on the Ri', i.e. to wear the garment in a place where Tzitzis are not available at all. One may not bless on them or wear it in Reshus ha'Rabim (if there is not Kedei Anivah according to Rashi). In pressed circumstances one may be Machshir also if part of the Gedil itself was cut. However, if the string was cut where it goes through the hole, even the Ri disqualifies. One must remove the garment. It does not help to stand still within four Amos, or to fix it while wearing it. He removes it and fixes it. Therefore, the Taz says that one with Yir'as Shamayim should check (before putting on the Tzitzis) also at the hole to ensure that no string snapped there.

Kaf ha'Chayim (16): Shalmei Tzibur is stringent not to go even in Karmelis with it. Even though the Shulchan Aruch, Birkei Yosef and Ma'amar Mordechai permit blessing on it, we are stringent about Safek Berachos, even against the Mechaber and a majority.

Kaf ha'Chayim (17): If only Gedil remains and one cannot get another string, he should unravel part so he will have Gedil and Pesil until he gets another string. Hilchos Ketanos (2:118) says that since it was Kosher when he made it (and it is still Kosher according to the Ri), this is not Pasul due to 'Ta'aseh v'Lo Min ha'Asuy.' (The Tzitzis must be Kosher and the garment must be obligated at the time he ties them on the corner.) Mishbetzos Zahav (11:3) disagrees. He must totally unravel it, and then make Gedil and Pesil.