Seeing that this question has been reopened, I’ve thought that, after all, I could elaborate on this comment and offer a solution that relies only on the amsthm package.
As Barbara Beeton has already remarked in her answer, the *-form of the \newtheorem command does not create a counter associated with the environment being defined, and so yields theorems (or lemmas, propositions, definitions…) that are not numbered, but are labeled with the text specified in the second argument of the \newtheorem* command, unadorned. For example, it is typical to see in a paper the following declaration:
\newtheorem*{MT}{Main Theorem}
This creates an environment named MT that one uses to state the main result of the paper:
\begin{MT}
My research deserves to be funded.
\end{MT}
As already noted by Barbara, this has the drawback (but is it really a drawback?) that you need to define a different environment for each “named” theorem (or axiom, or…) that you want to mention.
Bernard asks in a comment whether it is possible to define — using only amsthm’s features — an environment for a “generic” unnumbered theorem, that uses, for the label, not the text specified in the second argument of the \newtheorem* command, but the contents of the optional argument of the environment itself, i.e., the argument meant to pass an “optional note” for the theorem header. The answer is affirmative, but you need to define a new “theorem style” by means of the \newtheoremstyle declaration.
The \newtheoremstyle command is described in Subsection 4.3 of the manual of the amsthm package. In its ninth argument you can indicate how the header that labels the theorem/axiom/… is built from the following three pieces:
The theorem’s name (e.g., “Theorem”, “Lemma”, “Axiom”…):
this is the text specified in the second argument
of the \newtheorem (or \newtheorem*) command.
The theorem’s number: this is the current value of the counter
associated to the environment (more precisely, it is the
expansion of, say, \theaxiom, if axiom is the name
of the counter, which is the same as that of the environment).
The theorem’s optional note: these are the contents
of the optional argument of the environment.
More precisely (quoting from amsclass.dtx), the ninth argument of \newtheoremstyle…
…is interpreted as the replacement text for an internal three-argument
function \thmhead, i.e., as if you were defining
\renewcommand{\thmhead}[3]{...#1...#2...#3...}
but omitting the initial \renewcommand{\thmhead}[3].
The three arguments that will be supplied to \thmhead
are the name, number, and optional note component.
For more information, see the file amsclass.dtx. The above information suffices to see that, for example,
\newtheoremstyle{emptyplain}
{} % default space above
{} % default space below
{\itshape} % body font
{} % no indent
{\bfseries} % theorem head font
{.} % punctuation after theorem head
{ } % space after theorem head (normal interword space)
{#3} % using "\thmnote{#3}" is redundant, isn't it?
\theoremstyle{emptyplain}
\newtheorem*{namedaxiom}{}
does what Bernard asked for: it defines a “generic unnumbered theorem-like” environment that uses, for the the theorem head, the sheer contents of the optional note.
MWE that recaps all what we have being saying:
% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly
% declare the paper format.
\usepackage[T1]{fontenc} % Not always necessary, but recommended.
% End of standard header. What follows pertains to the problem at hand.
\usepackage{amsmath}
\usepackage{amsthm}
% I deem axioms should typeset like plain theorems...
% \theoremstyle{definition}
\newtheorem{axiom}{Axiom}
\newtheorem*{AC} {Axiom of Choice}
\newtheorem*{GCH}{Generalized Continuum Hypothesis}
\newtheoremstyle{emptyplain}
{} % default space above
{} % default space below
{\itshape} % body font
{} % no indent
{\bfseries} % theorem head font
{.} % punctuation after theorem head
{ } % space after theorem head (normal interword space)
{#3} % using "\thmnote{#3}" is redundant, isn't it?
\theoremstyle{emptyplain}
\newtheorem*{namedaxiom}{}
\begin{document}
A few ``generic'' axioms:
\begin{axiom}[Union set]
If $x$ and~$y$ are sets, then $x\cup y$ is also a set.
\end{axiom}
\begin{axiom}[Power set]
If $x$ is a set, then $\wp(x)$ is also a set.
\end{axiom}
A named axiom:
\begin{AC}
Blah blah blah\ldots
\end{AC}
Another one:
\begin{GCH}[a.k.a.\ Cantor's Aleph Hypothesis]
For every ordinal~$\alpha$,
\[ \aleph_{\alpha+1} = 2^{\aleph_{\alpha}} \]
\end{GCH}
With the \texttt{emptyplain} theorem style
(and a statement that spans two lines,
so you can check that there are no stray spaces):
\begin{namedaxiom}[Francesco's Axiom of Axioms]
All axioms are important, but some axioms are more important than others.
\end{namedaxiom}
\begin{axiom}[benchmark]
This is an axiom.
\end{axiom}
But:
\begin{namedaxiom}
This axiom is not correctly typeset.
\end{namedaxiom}
\end{document}
The output is:
