Sometimes it is said that the number of DoF of a system means how many real numbers have to be used at least to describe the system. But we know from set theory that the cardinality of any tuple of reals is the same as the reals, so all the information that is in a 3-tuple of reals, can be represented in just one real number. Of course this representation may not be very useful as the properties of the system will be non-continuous functions of this variable, but still it is enough to describe the state of the system. So how could this confusing thing be resolved?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
While $R^n$ and $R^m$ may have the same cardinality for all positive integer $n,m$ dimensions, they are homeomorphic topological spaces if and only if $n=m$, a result due to Brouwer.
So one cannot say with assurance that a parameterization by $n$ real values is equivalent to one by $m$ real values, if the continuity of the parameterization plays a role (as it most often will).
Describing a system using a (sufficiently badly) discontinuous parameterization is useless for e.g. any scientific endeavor. Given that it is impossible to measure real-world parameters to infinite precision, all you ever have are approximations, and if you cannot relate the behavior of two similar states of your system using continuity how can you ever make any predictions about it?
Cardinality itself is a very bare notion of size. It disregards any structure given on the sets. Equivalently you could represent each tuple with a Borel set; a continuous function; etc. or any member of a collection of size continuum. Essentially everything can be represented with almost everything else. Representation is simply a way for us to think about one object in terms of another.
When exchanging one object by another you need to figure out whether or not such representation is useful for anything. For example, is there coherence between a naturally defined operation on the tuples and naturally defined operations on the real numbers? How simple is the representation itself, does it just "exists out there" or can we describe it in a relatively definitive way?
Doing "highly discontinuous" things has very little advantage, since continuity assures some degree of coherence between two structures (continuity is not the only thing which matters though).