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Why do we need to use 2 nominalizers もの and こと in the following sentence?

> 素数とは、1 より大きい自然数で、正の約数が 1 と自分自身のみである **もののこと** である。
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2017-12-31T18:54:23.947
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2018-01-01T04:42:13.563
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question
nominalization
Why do we need to use 2 nominalizers もの and こと in the following sentence?
> 「素数{そすう}とは、1 より大{おお}きい自然数{しぜんすう}で、正{せい}の約数{やくすう}が 1 と自分自身{じぶんじしん}のみである
> **もの** の **こと** である。」

**In this sentence, neither the 「もの」 nor 「こと」 is a nominalizer**.

The 「もの」 here just means "the **_ones_** " or " ** _those_** " and it refers
to "those/the ones among the natural numbers greater than 1 that have no
positive divisors other than 1 and themselves." That, of course, is the
definition of prime numbers.

To call a word a "nominalizer", it needs to be nominalizing a **_verb_**. 「もの」
is not nominalizing a verb here.

「～～のことである」 is a set phrase meaning "it is said that ~~~" or "it is defined
that ~~". Again, 「こと」 is not nominalizing a verb here.
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2017-12-31T22:39:03.763
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answer
This 「もの」 is not a nominalizer. It is a noun in the full sense of the word,
meaning "thing" or the pronominal "one" (in this case I think the latter is
more applicable), which is here modified by the relative clause 「1
より大きい自然数で、正の約数が 1 と自分自身のみである」.

So 「1 より大きい自然数で、正の約数が 1 と自分自身のみであるもの」 means:

> one (i.e. a number) which is a natural number greater than 1 and have no
> positive divisors other than one and themselves

As for the こと, it seems to be one indeed, according to
[デジタル大辞泉](https://dictionary.goo.ne.jp/jn/80340/meaning/m0u/).

Under the subhead 「2⃣他の語句をうけて、その語句の表す行為や事態を体言化する形式名詞。」（”a formal noun that
nominalizes the action or state expressed by another word or phrase”）, there's
this subentry:

> ㋕（「…のことだ」などの形で）ある言葉の指し示す対象である意を表す。「九郎判官とは源義経の事だ」

This says that the function of this particular use 「事/こと」, in formulations
like 「...のことだ」, is to state that a given word refers to some entity.

All in all, a translation of the whole sentence would be something like this:

> 素数とは1 より大きい自然数で、正の約数が 1 と自分自身のみであるものであるもの **のことである** 。
>
> "Prime number" **refers to** one (i.e. a number) that is a natural number
> greater than 1 and have no positive divisors other than 1 and themselves.
>
> (Compare: 素数とは...であるもの **である** 。→ A prime number **is** one (i.e. numbers)
> that is...)
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2020-06-17T08:18:27.500
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answer