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At the end of davening, when we introduce the daily psalm, for Sunday, we use the term - יום ראשוז בשבת - The first day in the week. (I'm translating the word שבת as "week", here. It's a valid translation, as my rav explained to me.)

However, in a ketuvah (marriage contract) or a get (divorce document) written on Sunday, we use the term באחד בשבת - On day one in the week.

I suspect that the use of the term אחד instead of ראשון is to emulate/allude to Genesis 1:5:

וַיִּקְרָ֨א אֱלֹהִ֤ים ׀ לָאוֹר֙ י֔וֹם וְלַחֹ֖שֶׁךְ קָ֣רָא לָ֑יְלָה וַֽיְהִי־עֶ֥רֶב וַֽיְהִי־בֹ֖קֶר י֥וֹם אֶחָֽד

And God called the light Day, and the darkness He called Night. And there was evening and there was morning, one day.

What's the difference between the different contexts? Why in a ketuvah and get is the Torah term from Genesis for the 1st day אחד used, but in the daily psalm the more normal sounding term ראשון is used?

Note: I have used the example focusing only on the 1st day of the week, as this is the only place where we notice a term difference.

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