Overview

"Axiomatic Algebra Without Actual Infinity --- This book broadens applied mathematics from Group Theory to Axiomatic Set Theory by removing actual infinity, because removing actual infinity removes paralyzing holes in logic. Transitioning Axiomatic Set Theory from temporary, experimental pure mathematics into useful applied mathematics completes the formalization of real numbers started by Cantor 150 years ago. -- Modify Cantor's Continuum Hypothesis by replacing countable infinity with an ever-increasing finite counting natural number and by replacing uncountable infinity with reciprocal of zero, so that positive actual infinity cannot be a quantity of a set. -- Replace ""Don't Divide by Zero"" of rational numbers with two axioms that permit 1/0=7+1/0, 1/0=7*1/0, and 2^1/0=3^1/0=1/0 or 0, but not 0/0 that leads to errors, and have 1/0 the quantity of a set. -- Remove Axiom of Choice so that only bulk operations, like truncation to form an integer, apply to 1/0 quantities. Without axiom of choice, no individual operations can happen beyond a count of the largest number yet counted-to. -- Real Numbers have only a finite count of knowable place-value digits both before and after the decimal point. Left of the string of zeros unknowable digits form large-scale imprecision which is akin to infinity (although it can't be a quantity of a set) and can be analyzed using finite numbers. -- Special Relativity. A Lorentz Transformation adds large-scale imprecision to the time-space hyperbolic angle to derive Maxwell's Equations from the Dirac Equation. Particle properties calculated from the Dirac Spinor conform to measured electromagnetic field force and energy. -- That successful application of the axioms to discover a quantum model for 100+ year-old electromagnetism substantiates the new axiomatic algebra is correct and is available for modern in-development theories of physics. --- Although the math being depicted is quite deep and fundamental, symbols used in the book are nothing beyond what high school students learn in their most advanced math classes. --- Perhaps what is published in this book should be placed in a technical journal. The book isn't in a journal because the author is not familiar with journals. Paul C Daiber is an engineer and not associated with academia. As a hobby, he has dug into infinity, a feature of math and science that greatly troubled him like it has troubled many of us, and he thinks he has fixed the trouble by restructuring infinity so that it fits into logic. He placed the fix into this book to share it. On Amazon's free pages please read a summary of what he discovered, and if you want more details there's the rest of the book. \\//,"

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