A short discussion on the concepts
of discrete and continuum in mathematics,
and Zeno's paradoxes
I decided to publish this short discussion (a synopsis of the two long
papers published in Episteme N. 8, http://www.dipmat.unipg.it/~bartocci/ep8/ep8.htm)
after discovering - I must add: with surprise! - that this kind of comments
were not easily available neither in my Geometry, Algebra and Analysis
text books, nor in Internet. Thus I hope that my effort will be useful
to other people looking for the same kind of information.
* * * * *
When looking for the meaning of the two attributes of discrete and continuous in mathematics, when applied to some specific "structure" to be individuated (we do not mean, for instance, the concept of "continuous function" in topology), one usually finds, in actual mathematical text books:
1 - In set theory, one usually means that a discrete set is either a finite set, or an enumerable one. Moreover, one calls the power of the continuum the cardinality of the set of real numbers R.
2 - In topology, discrete is used in order to indicate a topological space in which each subset is open. A topological space is said to be a continuum, if it is a compact (implying Hausdorff separated) and connected (sometimes one adds: metrizable).
See for instance:
3 - In the theory of ordered fields, one calls continuum
the unique, up to isomorphisms, ordered complete archimedean field
of real numbers R.