I guess you want to use the footmisc
package so the footnote “numbers” are reset at each page. Here I define a maximum of four footnotes for a page, using more will raise an error.
\documentclass{article}
\usepackage[a6paper]{geometry} % just for getting small pictures
\usepackage[perpage,symbol*]{footmisc}
\usepackage{amssymb}
\DefineFNsymbols*{asterisks}{{*}{**}{***}{****}}
\setfnsymbol{asterisks}
\begin{document}
\begin{itemize}
\item Hello, I am the first sentence!\footnote{Ummh.. Looks good with a text!}
\item $\forall n \in \mathbb{N}$ $\exists m: n=2m\footnote{I am the second
footnote, and not a $m^2$ expression.} \lor n=2m+1$
\end{itemize}
\newpage
\begin{itemize}
\item Hello, I am the first sentence!\footnote{Ummh.. Looks good with a text!}
\item $\forall n \in \mathbb{N}$ $\exists m: n=2m\footnote{I am the second
footnote, and not a $m^2$ expression.} \lor n=2m+1$
\end{itemize}
\end{document}
The usage of geometry
is just to make small pictures for showing the output.

Side note. When you seem to need \<space>
in a math formula, ask yourself if you actually have two formulas, like in this case. And, please, don't inflict footnotes in math mode to your readers: they're very ambiguous.
\makeatletter, \makeatother, \@addtoreset
are, as much as why do you use\#
and*
for defining the command. The 5: counters are important, but it its an another question. – Kirill Oct 10 '16 at 17:39\#
and*
are arbitrary choices as mentioned in the answer. To use@
in control sequence names, one needs the\makeatletter/\makeatother
. The\@addtoreset
has a pretty self-explanatory name, and perhaps there is an interface without the@
. As pointed out in a comment, the resetting of footnote counter as the page changes sometimes has glitches, due to the way TeX ships out pages. – user4686 Oct 10 '16 at 20:10