Infinity by Division: The Real Number Arc and the Error of the Completed Infinite

For more than a century, the philosophy of the infinite has been divided between Cantorian actualism, which takes the real numbers as a completed totality with a transfinite magnitude, and Aristotelian potentialism, which treats infinity as an inexhaustible potential. Each pays a price. Actualism reads the cardinal of a set as its magnitude, so that a whole and its proper part can have the same magnitude and the whole is no greater than its part. The hierarchy of all sets, a definite domain, cannot be gathered into one completed set without contradiction. The forms of potentialism give up classical logic, weaken classical analysis, or decline to take the domain as one object. We offer a third alternative that keeps classical logic and every theorem of analysis and of ZFC, showing that an endless domain can be definite, fixed in advance, and taken as one object without having any number of terms. We take each real number as a ratio, expressed as a vertex angle on an arc bounded by the right angles where the tangent ratio ceases and numberhood ends. Central projection carries every ratio to the real number line, omitting the bounds, which are not numbers. Reversed, the projection carries the whole line onto the open half circle between the right angles, so the bound holds for the line as a limitation although no point of the line lies at it. On the arc the ratios are complete without any completed totality. Counting, not one-to-one correlation, constitutes a number of terms, so transfinite cardinals classify sets by one-to-one mapping and are not numbers of terms.