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I have just read the set theory definition of natural numbers:

\begin{align} 0 &=\emptyset\\ 1 &=\{0\}\\ 2 &=\{0, 1\}\\ 3 &=\{0, 1,2\}\\ \end{align}

However, this doesn't tell me how I should represent sequences of natural numbers. It seems that there is no difference between the number $n$ and the sequence $\{0,1,\ldots,n-1\}$ as used in the metalanguage. For example, I can write $2\in n$ just like I would write $2\in \{0,\ldots,n-1\}$.

Is there a more general rule to represent the interval $[m,n]$?
Perhaps:

$$(n\cup \{n\}) \setminus m$$

For example for $\{3,4,5\}$:

$$(5\cup \{5\}) \setminus 3 = (\{0,1,2,3,4\}\cup \{5\}) \setminus 3 = \{0,1,2,3,4,5\} \setminus \{0,1,2 \} = \{3,4,5\} $$

Is that correct?

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Yes, under the von Neumann definition of the natural numbers, you have $$ (n\cup \{n\})\setminus m=(n+1)\setminus m=\{m,m+1,\dots,n\} $$ However, I would advise against using this notation to refer to intervals of natural numbers in practice. People do not think of natural numbers as sets of previous natural numbers, so you will likely confuse people with this notation. The point of mathematical writing is not to be technically correct, it is to communicate well.


One final comment: there is a distinction between sequences of numbers and sets of numbers. Sequences are ordered, and may have repeated entries. Sets do not have an order, and cannot have repeated elements. Above, $(n+1)\setminus m$ is a set of numbers.

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