TITLE:
Extensions of the Constructivist Real Number System
AUTHORS:
E. E. Escultura
KEYWORDS:
Constructivism, Dark Number, Fermat’s Conjecture, g-Norm, g-Sequence, g-Limit, Goldbach’s Conjecture, Truncation, Vector Operators j and hθ
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.8,
August
20,
2018
ABSTRACT: The paper reviews the most consequential defects and
rectification of traditional mathematics and its foundations. While this work
is only the tip of the iceberg, so to speak, it gives us a totally different
picture of mathematics from what we have known for a long time. This journey
started with two teasers posted in SciMath in 1997: 1) The equation 1 = 0.99…
does not make sense. 2) The concept does not
exist. The first statement sparked a debate that raged over a decade. Both
statements generated a series of publications that continues to grow to this day. Among the new findings are: 3) There does not exist
nondenumerable set. 4) There does not exist non-measurable set. 5) Cantor’s
diagonal method is flawed. 6) The real numbers are discrete and countable. 7)
Formal logic does not apply to mathematics. The unfinished debate between
logicism, intuitionism-constructivism and formalism is resolved. The resolution
is the constructivist foundations of mathematics with a summary of all the
rectification undertaken in 2015, 2016 and in this paper. The extensions of the
constructivist real number system include the complex vector plane and
transcendental functions. Two important results in the 2015 are noted: The
solution and resolution of Hilbert’s 23 problems that includes the resolution of Fermat’s last theorem and proof Goldbach’s conjecture.