The package nomencl relies on makeindex for sorting and massaging the nomenclature entries.
For makeindex the character ! is special so it should be quoted in material passed to it. What's “quoting” in makeindex?
The makeindex style file, in this case nomencl.ist, may contain a line such as
quote "%"
If no quote line is present in the style file, the default for makeindex is " and you could type in your entry as
\nomenclature{\(\mathbb{N}_{r}\)}{Set of all positive integers greater than or equal to $r"!$}
This happened to annoy German users, because babel makes " into a shorthand character, so starting from nomencl version 2.5 the nomencl.ist file contains
quote '%'
(double or single quotes can be used there for isolating the string). Thus you can input your entry as
\nomenclature{\(\mathbb{N}_{r}\)}{Set of all positive integers greater than or equal to $r%!$}
but, of course, this breaks the syntax coloring/checking of editors (in particular the one in Overleaf that tries to check syntax on the fly). The LaTeX run would be good even if the point is marked “wrong”.
What we could do is to replace the special characters in the arguments before passing them to \nomenclature. You can judge if this is worth the pain. In the code [\!\@] stands for the list of special characters we want to quote with a %.
% arara: pdflatex
% arara: nomencl
% arara: pdflatex
\documentclass{article}
\usepackage[intoc]{nomencl}
\usepackage{amssymb}
\makenomenclature
\renewcommand{\nomname}{List of Abbreviations}
\ExplSyntaxOn
\NewCommandCopy{\originalnomenclature}{\nomenclature}
\RenewDocumentCommand{\nomenclature}{O{}mm}
{
\shaya_nomencl:nnn { #1 } { #2 } { #3 }
}
\tl_new:N \l_shaya_nomencl_one_tl
\tl_new:N \l_shaya_nomencl_two_tl
\cs_new_protected:Nn \shaya_nomencl:nnn
{
\tl_set:Nn \l_shaya_nomencl_one_tl { #2 }
\tl_set:Nn \l_shaya_nomencl_two_tl { #3 }
\regex_replace_all:nnN { [!@] } { \cO%\0 } \l_shaya_nomencl_one_tl
\regex_replace_all:nnN { [!@] } { \cO%\0 } \l_shaya_nomencl_two_tl
__shaya_nomencl:nVV { #1 } \l_shaya_nomencl_one_tl \l_shaya_nomencl_two_tl
}
\cs_new_protected:Nn __shaya_nomencl:nnn
{
\originalnomenclature[#1]{#2}{#3}
}
\cs_generate_variant:Nn __shaya_nomencl:nnn { nVV }
\ExplSyntaxOff
\begin{document}
\tableofcontents
\section{Introduction}
Some text
\nomenclature{(\mathbb{N}_{r})}{Set of all positive integers greater than or equal to $r!$}
\printnomenclature
\end{document}
Please, note that \mathbb N is not the best way to input the symbol and it should be \mathbb{N}. Also, if you want to insert the nomenclature in the table of contents, use the intoc option, rather than doing it manually.

\beginand\end{document}and removing the extra backslashes, so I can't reproduce your error. Could you please edit your question to include a minimal working example? That is, it should be possible to copy-paste your code and see the issue without needing further alteration. – gz839918 Mar 27 '24 at 01:55\documentclass, a\begin{document}, and an\end{document}. The link in my comment has more details about what you should include in your question. There may also be other packages that are affecting your code. Could you please edit your question to include these details? Thanks! – gz839918 Mar 27 '24 at 03:23