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Although people living in Eretz Yisrael usually only keep a one day holiday, on Rosh Hashanah everyone keeps two days, even those living in Eretz Yisrael. Furthermoreת these two days are considered like one long day and thus carry a number of Halachic stringencies that are not applicable by other Holidays. Admur is one of the few Poskim that properly delved into this subject explaining every side of it in a very clear and logical method. Below is a free translation [with some changes to allow flow of the text] of Admur 600/1-4. A summary of Admur is included below.
1. Understanding why two days is kept even in Eretz Yisrael:
Reason why in the Diaspora they keep two days by all Holidays: The reason why in the Diaspora they always keep two-day Holidays is because in previous times the new month was sanctified based on witnesses who saw the new moon. The new month was hence not dependent on any calculation but purely on the testimony of the sighting of the moon. [The first day of the Hebrew month would be announced by the Sanhedrin based on the witnesses or lack thereof. If witnesses came forward on the 30th day then the 30th day acted as the first day of the new month. If, however, witnesses did not come forward on the 30th day then the 31st day was sanctified as the 1st of the new month. Thus, some months lasted 29 days while others lasted 30 days and it was placed in the hands of the court to announce if the 30th day or the 31st day is the 1st of the next month. Now, due to the lack of technology, in most lands out of Israel they would only hear of this announcement much later on into the next month when a messenger was able to arrive.] Thus, the communities in the Diaspora did not have knowledge as to what day the new month was sanctified in Jerusalem [and they were always in doubt as to the current date]. Due to this uncertainty of the current date, when a Holiday would approach, they would keep a two day Holiday [in order to be certain that they kept the Holiday on the correct date]. Now, although today the date of each month is set in a calendar based on calculation [and not witnesses] and hence everyone knows on which day the new month began and hence when the coming festival falls out, nevertheless in the Diaspora they keep the original custom of their ancestors which was to always keep two days.
Reason why in Israel they keep only one day: [Even in previous times] the inhabitants of Israel which lived in proximity to Jerusalem would only keep a one day Holiday, as the Beis Din in Jerusalem would send out messengers in Nissan (Sivan) and Tishrei to notify them as to the day that the new month was sanctified by the Sanhedrin. Thus, all the communities of close enough proximity for the messengers to reach before Yom Tov celebrated only one day. Therefore, even today the descendants [of these communities] are not required to celebrate two days as we are expert in the calculation of the new moon and know what date is fit to set the new month and count the month from it.
Reason why on R”H even in Israel they keep two days: On Rosh Hashanah even those living in Eretz Yisrael keep two days as the entire reason that by other Holidays they keep one day is because even in previous times they only kept one day and hence today we also do so as we are expert in the calculation of the new moon and know what date is fit to set the new month and count the Holidays from it. This however only applied by the other Holidays. However, regarding Rosh Hashanah even in previous times, when the month was sanctified based on testimony, they would keep two days in Eretz Yisrael, and they are hence obligated to follow the custom of their ancestors [and keep two days of R”H even today]. The reason that even in previous times they kept two days is because the month was sanctified not based on calculation but based on the establishment of the Beis Din in Jerusalem which was founded on the testimony [of the witnesses] that saw the new moon. Now, although the Beis Din would send messengers to notify everyone, nevertheless on R”H, which is Yom Tov, the messengers could not exit the Techum [outside Jerusalem, and hence only those inhabitants within the Techum would receive the news]. [Thus, even in Eretz Yisrael, in the area outside the Techum of Jerusalem, there was doubt as to when the first of Tishrei-R”H-was established, and they hence had to keep two days due to doubt.]
When did they begin Yom Tov Sheiyni of Rosh Hashanah in Eretz Yisrael?
Some write that the two-day celebration even in Israel only began in the times of Rebbe Yochanon ben Zakaiy. Prior to this time, it was only celebrated as one day [in years applicable, as explained above]. Others however right at the two-day celebration in Israel began in the times of Ezra.
 Admur 600:1; Michaber 601:2
 600:1; See also Levush 600; Beis Yosef 600; M”A 601:1; Rambam Hilchos Kiddush Hachodesh 5:4-12; Kaf Hachaim 601:5
 Contradiction from Admur 494:1 regarding if Shluchim were sent in Sivan: Tzaruch Iyun from Admur 494:1 who writes that the holiday of Shavuos is not dependent at all on the day of Rosh Chodesh Sivan, but rather from the count of the Omer. The above however implies that Shavuos is set to the 6th day of Sivan and not the 50th day of count, as otherwise, what necessity is there for them to know which day was the 1st of Sivan, either way they will keep Shavuos on the 50th day from the 16th of Nissan. See also Chasam Sofer 145 who emphasizes that by Shavuos there was never a true doubt of Yom Tov Sheiyni, as it is based on the 50th day of the count, and it was only done in order not to differentiate between it and the other Moadim. [See Likkutei Sichos 4:1030; 17 Sicha Beis Tazria]; Furthermore, in the Mishneh in Rosh Hashanah 18a it completely omits Sivan from the recorded dates that Shluchim were sent and Tosafus there explains that indeed Shluchim were not sent in Sivan due to the above-mentioned reason that it is not necessary being that it is dependent on the counting of the Omer. Vetzaruch Iyun Gadol! See however Kuntrus Hashulchan p. 53 who writes that this must be a printers error that it should be omitted, and was never written by the Alter Rebbe. However, see Toras Menachem 31 p. 57 [2nd day of Shavuos 5721] that the Rebbe indeed accepts this version of Admur that includes Sivan as one of the dates and explains that Shluchim were sent in Sivan in order so people know whether they should recite the words of Zeman Matan Toraseinu in Shemoneh Esrei of Shavuos, as it is only said when Shavuos falls on the 6th of Sivan. Now, the reason why the Mishneh does not mention it is because the Mishneh follows the opinion who holds that Shavuos always falls on the 6th of Sivan. The Rebbe, however, concludes there “I don’t know to how much the above is concrete, but nonetheless it could be the beginning of a subject for a Kinus Torah.”
 Vetzaruch Iyun Gadol as a) Even in Jerusalem they would have to keep two days in all years that witnesses did not come due to doubt that perhaps witnesses would arrive that day, and in the end if the witnesses did not arrive then the next day would be R”H. Thus, why are we saying the two days were kept only outside of the Techum? Perhaps however one can answer the above by saying that since there were years that in Jerusalem they kept only one day due to witnesses arriving and hence the next day was a weekday, the 2nd of Tishrei, and since the two days were not kept every year it cannot serve as a reason to keep two days everywhere in Eretz Yisrael, and hence Admur does not mention it. However, this leads to the second question: B) If witnesses did not arrive then it is understood that everyone, including the inhabitants of Jerusalem, kept two days due to the doubt. If witnesses did arrive then in Jerusalem, they kept one day and the next day was a weekday. If so, then why didn’t they send messengers immediately after the first day ended when there is no longer a prohibition of leaving the Techum as they know that it is a weekday. Thus there is no reason why the people outside the Techum of Jerusalem should keep two days of Yom Tov due to doubt in those years that witnesses arrived on the first day, as the messengers could easily arrive on their second day and tell them that it is no longer Yom Tov? Perhaps one can answer this question as follows: a) There were no cities close enough to Jerusalem that the messengers could arrive before the end of the second day if they did not start traveling already on the first day past the Techum. B) Alternatively since all the places outside the Techum had to at least begin the 2nd day following all the prohibitions of Yom Tov due to doubt, and many years they had to keep the entire second day due to lack of witnesses arriving, therefore also today the inhabitants of Jerusalem always keep two days. Vetzaruch Iyun!
 See Menachem Meishiv Nafshi Vol. 1 p. 85
 Hagahos Rav Chaim Vital
 Tosafus Rosh Hashanah 19b