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I am trying to resolve what appears to be a conflict in several sources.

Avot D'rav Natan 2:3 (excerpt)

איזו סייג שעשה משה לדבריו?

הרי הוא אומר: (שמות יט) "ויאמר ה' אל משה, לך אל העם וקדשתם היום ומחר". לא רצה משה הצדיק לומר להם לישראל כדרך שאמר לו הקדוש ברוך הוא, אלא כך אמר להם: "היו נכונים לשלשת ימים אל תגשו אל אישה", והוסיף להם משה יום אחד מעצמו

My translation:

What type of "fence" (precaution to prevent a violation of a mitzvah) did Moses do for G-d's commandment?

Behold it says (Exodus 19:10) "G-d said to Moses, go to the people and sanctify them today and tomorrow." But Moses did not want to tell B'nai Yisra'el exactly what G-d told them, but instead, he said to them (Ex. 19:15) "Be ready for three days; Don't touch a woman," and he added an extra day on his own.

In summary, the commandment G-d told Moses was to be ready for the 3rd day, meaning that they were to prepare for 2 days, and the Torah would be given on the 3rd day.

Moses says, to be ready (prepare yourself) for 3 days, meaning that the Torah would be given on the 4th day.

Yet, we see then when the "sound and light show" occurred, Exodus 19:16 (Sefaria) says:

וַיְהִי֩ בַיּ֨וֹם הַשְּׁלִישִׁ֜י בִּֽהְיֹ֣ת הַבֹּ֗קֶר וַיְהִי֩ קֹלֹ֨ת וּבְרָקִ֜ים וְעָנָ֤ן כָּבֵד֙ עַל־הָהָ֔ר וְקֹ֥ל שֹׁפָ֖ר חָזָ֣ק מְאֹ֑ד וַיֶּחֱרַ֥ד כָּל־הָעָ֖ם אֲשֶׁ֥ר בַּֽמַּחֲנֶֽה׃

And it came to pass on the third day, when it was morning, that there were thunders and lightnings and a thick cloud upon the mount, and the voice of a horn exceeding loud; and all the people that were in the camp trembled.

It says that this happened on the 3rd day, itself! So, how does Avot D'Rav Natan deduce that Moses added an extra day, since it seems that the event should have occurred on the 4th day, not the 3rd.

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According to Rashi this is the machlokes between Rabbi Yose and the Rabbanan in Maseches Shabbos 87a. However, Rashi says that "on the third day" according to Rabbi Yose means the same as the statement by Moshe Rabbeinu to "separate for three days"

Art Scroll 87a1 note 6 says that

In the Rabbi's opinion there were only two days of abstinence before the Revelation (Thursday and Friday), whereas Rabbi Yose maintains that there were three (Wednesday through Friday).

Thus Rabbi Yose would state that "on the third day in the morning" means in the morning after the third day (the fourth day). There is a concept that "miksas yom kekulo" (part of the day is like the entire day). Thus, the Rabbis would say that when the third day morning came, it was like three days had passed (as far as the semen was concerned - see Rashi below) and the revelation could occur on the third day.

Rashi gives this machlokes in 19:15

וַיֹּאמֶר אֶל הָעָם הֱיוּ נְכֹנִים לִשְׁלשֶׁת יָמִים אַל תִּגְּשׁוּ אֶל אִשָּׁה:

He said to the people, "Be ready for three days; do not go near a woman."

היו נכנים לשלשת ימים: לסוף שלושה ימים, הוא יום רביעי שהוסיף משה יום אחד מדעתו, כדברי רבי יוסי, ולדברי האומר בששה בחדש ניתנו עשרת הדברות לא הוסיף משה כלום. לשלשת ימים, כמו (פסוק יא) ליום השלישי:

Be ready for three days: For the end of three days. That is the fourth day, for Moses added one day of his own volition. This is the view of Rabbi Jose [who says that the Torah was given on the seventh of Sivan]. According to the one who says that the Ten Commandments were given on the sixth of the month, however, Moses did not add anything, and “for three days” has the same meaning as “for the third day.” [from Shab. 87a]

אל תגשו אל אשה: כל שלושת ימים הללו כדי שיהו הנשים טובלות ליום השלישי ותהיינה טהורות לקבל תורה שאם ישמשו תוך שלשת ימים, שמא תפלוט האשה שכבת זרע לאחר טבילתה ותחזור ותטמא, אבל מששהתה שלושה ימים כבר הזרע מסריח ואינו ראוי להזריע וטהור מלטמא את הפולטת:

do not go near a woman: [to have intimacy with her] for all these three days [of preparation], in order that the women may immerse themselves on the third day and be pure to receive the Torah. If they have intercourse within the three days, the woman could [involuntarily] emit semen after her immersion and become unclean again. After three days have elapsed [since intercourse], however, the semen has already become putrid and is no longer capable of fertilization, so it is pure from contaminating the [woman] who emits it. — [from Shab. 86a]

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