  Development of new algorithms in the area of machine learning, especially
clustering, comparative studies of such algorithms as well as testing according
to software engineering principles requires availability of labeled data sets.
While standard benchmarks are made available, a broader range of such data sets
is necessary in order to avoid the problem of overfitting. In this context,
theoretical works on axiomatization of clustering algorithms, especially axioms
on clustering preserving transformations are quite a cheap way to produce
labeled data sets from existing ones. However, the frequently cited axiomatic
system of Kleinberg:2002, as we show in this paper, is not applicable for
finite dimensional Euclidean spaces, in which many algorithms like $k$-means,
operate. In particular, the so-called outer-consistency axiom fails upon making
small changes in datapoint positions and inner-consistency axiom is valid only
for identity transformation in general settings.
  Hence we propose an alternative axiomatic system, in which Kleinberg's inner
consistency axiom is replaced by a centric consistency axiom and outer
consistency axiom is replaced by motion consistency axiom. We demonstrate that
the new system is satisfiable for a hierarchical version of $k$-means with
auto-adjusted $k$, hence it is not contradictory. Additionally, as $k$-means
creates convex clusters only, we demonstrate that it is possible to create a
version detecting concave clusters and still the axiomatic system can be
satisfied. The practical application area of such an axiomatic system may be
the generation of new labeled test data from existent ones for clustering
algorithm testing. %We propose the gravitational consistency as a replacement
which does not have this deficiency.
