Foundations of Mathematics.
A Neoclassical Approach to Infinity

First Edition, July 26 2015, ISBN 978 94 625422-0-4

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Preface, Table of Contents and Introduction (PDF)

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A core proof section is available as this paper (13 pages):
Two results on ZFC:
(1) If ZFC is consistent then it is deductively incomplete,
(2) ZFC is inconsistent,

June 18 - July 27 2015

This book relies on A Logic of Exceptions (ALOE)

Russell’s Paradox and Logicomix

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Cover text

Foundations of Mathematics. A Neoclassical Approach to Infinity is for (1) students interested in methodology and the foundations of mathematics – e.g. studying physics, engineering, economics, psychology, thus a broad group who use mathematics – and (2) teachers of mathematics who are sympathetic to the idea of bringing set theory and number theory into mathematics education.

The book presents:
(A) Constructivism with Abstraction, as a methodology of science.
(B) Particulars about infinity and number theory, within foundations and set theory.
(C) Correction of errors within mathematics on (B), caused by neglect of (A).

Other readers are (3) research mathematicians, who would benefit from last correction, but who must mend for that they are not in the prime target groups.

Set theory and number theory would be important for a better educational programme:
(i) They greatly enhance competence and confidence.
(ii) They open up the mind to logical structure and calculation also in other subjects.
(iii) They are fundamental for learning and teaching themselves.

The axiomatic system for set theory ZFC is shown to be inconsistent. Mathematics has been in error since Cantor 1874 because of neglecting above methodology of science.

MSC2010 codes for FMNAI
In bold the three major ones if required to select
00A30 Philosophy of mathematics
Mathematics education research
Methodology of mathematics, didactics
97D20 Mathematics Education - Philosophical and theoretical contributions (maths didactics)
97E60 Mathematics Education - Foundations: Sets, relations, set theory
97C70 Mathematics Education - Research: Teaching-learning processes
97B10 Mathematics Education - Educational research and planning
Mathematics itself
03B50 Many-valued logic
03E30 Axiomatics of classical set theory and its fragments
03E35 Consistency and independence results
03E70 Nonclassical and second-order set theories
03F65 Other constructive mathematics

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Warning: This book proves radical new ideas and must be read with care. If you spend more than a cursory glance on this summary then it is advisable to get the hardcopy and continue reading from paper. I tend to focus my research on misconceptions that lead society away from common sense, and then I select pivots that cause crucially different points of view depending on how the argument is resolved. Such a pivot only works well if the argumentation gets proper attention.

Legacy papers

CCPO-PCWA and PV-RP-CDA-ZFC are superseded by FMNAI above. The discussions and reports remain relevant for documentation.

Paper Contra Cantor Pro Occam - Proper constructivism with abstraction (March 2012, March 2013) (CCPO-PCWA)
PM. Legacy paper Contra Cantor Pro Occam (August 2011, update March 2012) superseded in 2012, but still with a bit on infinitesimals.
PM. Reply to Jan van Rongen (March 2012). 

ZFC is inconsistent. A condition by Paul of Venice (1369-1429) solves Russell's Paradox, blocks Cantor's Diagonal Argument, and provides a challenge to ZFC,
November 14 2014 - June 26 2015. (PV-RP-CDA-ZFC).
(1) Since June 17 2015 the phrase "ZFC is inconsistent" has become the main title, so that the reference to Paul of Venice has become the subtitle. The part on ZFC itself gave cause to split off the "Two results" paper. The Paul of Venice article still is important for the alternative axioms, and the question what to do with Cantor's Theorem and the transfinites.
(2) Till May 27 2015 the focus was on asking questions to users of ZFC, but then I decided to do some proofs myself.
(3) See the earlier versions. Version 1 has 6 pages and still is fine. Later versions got somewhat cluttered. June 17 2015 is streamlined again, but 30 pages.
(4) Chance had - if it exists - that K.P. Hart (TU Delft) published in NAW on the history of the diagonal argument. I sent this criticism to NAW, May 5, updated June 19 2015.
(4a) The criticism was included as Appendix B in the May 1 - June 12 versions. Those also contained Appendix C on another proof by Hart. Here is Hart's response to Appendix B and my rejoinder. You have to look in the June 12 version to find these Appendices B and C, since those have been replaced and removed in the June 17 version. Hart's proof in that Appendix C now has become Theorem 1.3.B in the "Two results" paper below, and it appears to provide a transparant vehicle to show that ZFC is inconsistent.
(4b) Background supporting documentation for Appendix B in the paper on Paul of Venice, May 20 2015
(4c) Here is the email exchange Colignatus - Hart 2011-2015.
(5) The paper refers to: Memo for Bas Edixhoven and Jan Bergstra (both KNAW), part Dutch part English on CCPO-PCWA and Paul of Venice, October 29 & November 10 2014.
(6) See also a breach to integrity.
(7) See a NAW Referee Report of June 11 2015 with my comments of June 15 2015.