  An alternative mathematics based on qualitative plurality of finiteness is
developed to make non-standard mathematics independent of infinite set theory.
The vague concept "accessibility" is used coherently within finite set theory
whose separation axiom is restricted to definite objective conditions. The weak
equivalence relations are defined as binary relations with sorites phenomena.
Continua are collection with weak equivalence relations called
indistinguishability. The points of continua are the proper classes of mutually
indistinguishable elements and have identities with sorites paradox. Four
continua formed by huge binary words are examined as a new type of continua.
Ascoli-Arzela type theorem is given as an example indicating the feasibility of
treating function spaces.
  The real numbers are defined to be the points on the linear continuum and
have indefiniteness. Exponentiation is introduced by the Eulerian style and
basic properties are established. Basic calculus is developed and the
differentiability is captured by the behavior on a point. Main tools of
Lebesgue measure theory is obtained in a similar way as Loeb measure.
  Differences from the current mathematics are examined, such as the
indefiniteness of natural numbers, qualitative plurality of finiteness,
mathematical usage of vague concepts, the continuum as a primary inexhaustible
entity and the hitherto disregarded aspect of "internal measurement" in
mathematics.
