Gemara

(Mishnah): Pesach is Pasul in the following cases:

The slaughter was Lo Lishmah, and also Kabalah, Holachah and Zerikah;

He intended Lishmah and then Lo Lishmah, e.g. l'Shem Pesach and then l'Shem Shelamim;

Question (Rav Papa): Does the Mishnah discuss different intents in one Avodah, or in different Avodos?

Zevachim 2b (Mishnah): Intent Lishmah followed by Lo Lishmah is Pasul. E.g. [regarding a Pesach], first he intended for Pesach, and then for Shelamim.

Inference: Had he first intended for Pesach, and then Stam, it would be Kosher. This shows that Stam is like Lishmah!

Rejection: Perhaps [normally,] Stam is not like Lishmah. Here it is Kosher, for one always does like his first intent.

Eruvin 96a (R. Elazar): If one finds dyed strips of Techeiles wool in the market, they are Pesulim. If he finds strings, they are Kesherim.

Question: Presumably, strips are Pesulim lest they were dyed for the sake of a garment [and not l'Shem Tzitzis]. Strings should be Pesulim for the same reason!

Answer: The case is, the strings are Shezurim (twined. It is not common to weave a garment with such strings.)

Question: People often twine threads to attach them on the edge of a garment!

Answer: The case is, they are cut [to the right size for Tzitzis].

Yoma 72a - Question: Perhaps the threads for the Choshen and Efod are twined with six gold threads?

Answer (Rav Acha bar Yakov): We learn from "v'Katzatz Pesilim." Pesil connotes two-fold, so Pesilim connotes four-fold.

Rishonim

Rambam (Hilchos Tzitzis 1:10): If one wants to twine white or Techeiles strings, he may. Even if a string was made of eight strings twined into one, it counts like only one string.

Rebuttal (Ra'avad): They must be twined. The Sifri learns a Gezeirah Shavah Techeiles-Techeiles from the Mishkan. Just like there the strings were twined, also here. Strips of Techeiles found in the market are Pesulim, but strings are Kesherim, i.e. if they are twined (Eruvin 96b).

Kesef Mishneh: Chachmei Lunil asked the Rambam from the Sifri. He answered "a Stam Sifri is like R. Shimon. I did not see anyone [else] say so, so I did not rule like him. If we learn from the Mishkan, we should require six strings twined together. I have no other support for this." How can the Rambam explain Eruvin 96b? He cannot say that it is like R. Shimon. An Amora, i.e. Rav Yehudah (our text - R. Elazar) taught it!

Suggestion: He (the Amora) holds like R. Shimon, and Chachamim disagree.

Rejection (Kesef Mishneh): Rav Yehudah said it Stam. This shows that he holds like it. We should rule like him. The Rambam must hold that even though we establish that the strings were twined, this does not mean that Tzitzis must be twined. Rather, people do not use twined strings for weaving at all, so surely these were for Tzitzis. Sometimes people twine strings for Tzitzis. In any case, we do not find anyone who argues with the Sifri. Why did he reject it from Halachah, just because if so, they should be twined six-fold? Also R. Shimon must explain this! He must have a Drashah to teach that it suffices to twine two strings together. We can say so according to everyone! Also, what is the Rambam's source that the Beraisa learns from the Mishkan, in order to ask that if so, they must be twined six-fold? It seems that since he realized the weakness of his opinion, he wrote "I have no other support for this."

Chasam Sofer: A Gezeirah Shavah is never partial. Eruvin 96b did not mention that the strings were twined six-fold. If they were, surely it was l'Shem Tzitzis! (We would not need to say that they were cut - PF.) This shows that the Halachah does not follow R. Shimon.

Migdal Oz: Everyone agreed with the Rambam, in particular the [Rif's] Halachos. This is correct according to Sod (hidden parts of Torah). The Sifri did not mean that twining is Me'akev. In Eruvin, twining did not show that they are for Tzitzis, rather, being cut to size.

Kesef Mishneh: Migdal Oz explains that the Rambam answered that the Sifri is R. Shimon, i.e. if it were Me'akev, the Gemara would have said so in some place. The Rambam does not connote like this. If so, why did he need to say that it is like R. Shimon? If it is not Me'akev, all can agree to it! The Rambam permits twining! Migdal Oz is wrong. The custom of all Yisrael is to twine the strings.

Rosh (Hilchos Tzitzis 20): The Sifri learns from "ha'Kenaf Pesil Techeiles" that the strings must be spun and twined. In Eruvin we say that twining shows that they were l'Shem Tzitzis.

Ran (Beitzah 7b DH mid'Oraisa): R. Tam says that Nuz (in Sha'atnez) means that two strings of each species (wool and linen) must be twined together [by themselves], and they are joined together. Kil'ayim 9:8 connotes like this. The Torah needed to permit Kil'ayim in Tzitzis because Tzitzis strings must be twined, like the Sifri says. Eruvin 96b connotes similarly, and also that Stam garments are not twined. Also the Rashba connotes like this, unlike those who say not to twine Tzitzis strings, for sometimes they become untwined, and then there are 16 strings. Really, they must be twined. If they become untwined, this is not a problem, for this is like remnants of Tzitzis, which are Kosher. They must remain twined for the Shi'ur of Kdei Anivah (enough to wrap around the other strings).

Kesef Mishneh: The Nimukei Yosef holds like R. Tam, that if they were initially twined nicely, even if they became untwined and are now 16, this is like remnants of Tzitzis, which are Kosher. Also Bahag says so. Ba'al ha'Itur did not say [like the Ran] that Kdei Anivah must remain twined. However, perhaps he agrees, but did not see a need to say so explicitly.

Tosfos (Zevachim 2b DH Ha): Rashi explains that "l'Shem Pesach and Stam" is when he began Shechitah l'Shem Pesach, and finished it Stam. This is astounding. Must one speak [intent Lishmah] during the entire Shechitah?! Rather, it discusses two different Avodos, e.g. Shechitah l'Shem Pesach and Kabalah Stam. However, in Pesachim we ask whether the intents were in one Avodah or two. The case in one Avodah is like Rashi! I answer that even if l'Shem Pesach and Stam is Pasul, one need not talk the entire time. It suffices to say at the beginning "I do the entire Shechitah l'Shem Pesach."

Tosfos (Yevamos 4b DH v'Amar): Why did the Torah need to permit Kil'ayim in Tzitzis? One could make Techeiles strings that are not hackled, spun and Nuz, which R. Tam says is twined! I answer that the Sifri obligates Tzitzis to be spun and twined, and similarly hackled. Our Stam garments are hackled, spun and Nuz.

Poskim

Shulchan Aruch (OC 11:2): Tzitzis strings must be twined.

Mishnah Berurah (14): Twining is doubling the string after spinning, and twisting them together. If one doubled each string to eight and twined them, it is not a problem. The Sifri obligates spun and twined [like the Mishkan. Alternatively, Stam Pesilah is twined.]

Bi'ur Halachah (DH u'Tzerichin): The Sifri learns from "Pesil Techeiles" that the strings must be spun and twined like a wick (Hagahos Maimoniyos). In Yoma 72a, we expound four-fold strings from "Pesilim". Rashi and Tosfos explain that this is through cutting them. If they were twined, this would make eight! Perhaps they are well twined, and this is considered twined even without doubling the string. This is a Chidush. It seems that the Ra'avad's text of the Sifri learns from the Mishkan. This solves the difficulty. Stam wicks are not twined. All the Poskim say that it suffices to twine two-fold strings. Eretz ha'Chayim says that it is Hidur Mitzvah to twine each string eight-fold. If they were twined two-fold Lo Lishmah, and later twined into eight Lishmah, or vice-versa, I am unsure whether it depends on the first twining or the latter.

Shulchan Aruch (ibid.): They must be twined Lishmah.

Magen Avraham (3): If they were not twined Lishmah, it is Pasul even b'Di'eved.

Gra (DH vshe'Yiyhu): This is like we expound in Sukah 9a, that Asiyah must be Lishmah. Shmuel surely requires Lishmah from spinning and onwards.

Mishnah Berurah (15): The Gra connotes like the Magen Avraham, that twining Lishmah is Me'akev. Some are lenient b'Di'eved even without twining.

Bi'ur Halachah (DH Lishman): The Gra and Pri Megadim require twining Lishmah. The Bach, Nachalas Tzvi and Eliyahu Rabah are lenient b'Di'eved even without twining. One may not rely on them, for many disagree, i.e. the Ran, Nimukei Yosef, Shulchan Aruch, Olas Tamid, Chayei Adam, Derech ha'Chayim, Shulchan Shlomo, and Eretz ha'Chayim. The Pri Megadim refuted Eliyahu Rabah's proofs. If they were twined Lo Lishmah, the Olas Tamid, Magen Avraham, Gra, Chayei Adam, and Shulchan Shlomo connote that it is Pasul. Perhaps one may rely on the lenient opinion in this case. Derech ha'Chayim and Eretz ha'Chayim are lenient about Lishmah, and the Pri Megadim was unsure. Even if we require Lishmah, if they are twined Stam, it seems that we rely on "one always does like his first intent" (spinning, which was Lishmah), even though this is a separate action. I learn from Tosfos (Zevachim 2b). He explains that the Gemara discusses intents in two Avodos, and it concludes that one always does like his first intent. The same applies here. Tosfos said that one could explain that we discuss intents in one Avodah, but not because "one always does..." does not apply two Avodos. The Magen Avraham disqualified "if they were not twined Lishmah." He did not write "if they were twined Lo Lishmah." He connotes that even Stam is Pasul. Perhaps he was not precise. Alternatively, this is when the one who twined did not spin the strings, so "one always does like his first intent" does not apply.

Kaf ha'Chayim (11): Machazik Berachah says that if one cannot find twined strings, in pressed circumstances he may rely on the lenient opinion. Lev Chayim disagrees. According to the Beis Yosef, he transgresses wearing the garment without Tzitzis! I say that the same applies if the only Tzitzis available were not twined Lishmah.