The point of view of the book derives from Leopold Kronecker (1823-1891), whose views were quite consonant with those of MAP. Kronecker resisted the overthrow of the Aristotelian (and Gaussian) prohibition of completed infinites on the ground that the overthrow was unnecessary. In other words, he committed himself to doing mathematics without completed infinites, basing his work on algorithmic ...

It is argued that every measurement is made in a certain scale. The scale in which present measuments are made is called present scale which gives present knowledge. Quantities at the limits to present measurement may be observables in other scales. Cantor’s series of infinites is used to describe scales of measurement. Continuum Hypothesis and Schroedinger

We investigate order-theoretic properties of the nonstandard real line, by isolating the basic order-types involved, and the possible relations among them. We focus on order-isomorphisms between the infinitesimals, the infinites, and the whole hyperreal line. We also compute the main cardinal invariants of the order-topology of the nonstandard real line.

some contemporary logicians have introduced a new simple method of logical consequences and reduced all immediate arguments to conversion and obversion. by dint of positive and infinite terms, one of the logicians, reza akbari, increased the four traditional quantified propositions to 32 ones: the 4 well-known positives, 4 subject-infinites, 4 predicate-infinites, 4 two-sided-infinites, and the...

We will review the main episodes of the history of the infinite in mathematics, stressing the asymmetry between infinitesimals and infinites. While the actual infinite, after being avoided for so long, when it became a mathematical object it entered mathematics to stay, infinitesimals have had a troubled history of condemnations and resurrections. This has probably to do with the ambiguous basi...

The mixed fortunes of PAUL DU BOIS-REYMOND'S infinitary calculus and ideal boundary between convergence and divergence are traced from 1870 to 1914. CANTOR, DEDEKIND, PEANO, RUSSELL, PRINGSHEIM and others objected. STOLZ, BOREL, HARDY and others accepted, at least in part, and built further. HAUSDORFF to some extent effected a compromise. The differing attitudes of the different participants to...

As the a's approach certain limiting values, one or more of which is infinite, Ta may have a definite finite transformation T as a limit. The transformation T may be regarded as a transformation of the family, and, if equivalent to a transformation Tb with finite parameters, can be generated by an infinitesimal transformation of the family (namely, bxXx + •" + brXr), but not otherwise.* Let it ...

Research on automatic software repair is concerned with the development of systems that automatically detect and repair bugs. One well-known class of bugs is the infinite loop. Every computer programmer or user has, at least once, experienced this type of bug. We state the problem of repairing infinite loops in the context of test-suite based software repair: given a test suite with at least on...

Recently, it has been claimed that the back reaction of vacuum polarization on a black hole spacetime naturally regularizes infinities in the black hole entropy. We examine the back reaction calculation and find no such short-distance cut-off, in contradiction with these recent claims. Moreover, the intuitive expectation that the perturbative calculation breaks down near the event horizon is co...