I am also having trouble telling who E. E. Escultura is replying to in
each post, since he does not quote the person he replies to. Mr.
Escultura, it would be nice if you would quote the person you reply to
in your posts, so that when you say "you" it is easier to tell who you
are referring to.
Will do - EEE
In any case, the paper by Heyting on pgs. 52-61 of the book of
Benacerraf and Putnam is about an intuitionistic foundation for
mathematics, and the difficulties in constructing the real numbers
which he discusses there are only difficulties when using
intuitionistic logic (i.e., one is not allowed to use the argument
"the negation of P is not true implies that P is true" in one's
proofs). There are no contradictions to be found in this paper for
anyone who uses the more standard logic in which ~~P implies P.
The construction of the counterexamples by Heyting is valid regardless of his intuitionist perspective because it is based on the known properties of the rationals. I have also constructed my version of the counterexample in the paper, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84.
The Banach-Tarski paradox is likewise not a contradiction but merely a
counterintuitive fact, one which indicates the need for measure theory
(and the need to distinguish between measurable and non-measurable
sets) in order to have a usable theory of integration.
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This paradox is more than counterintuitive. It is a topological contradiction in R^3. Although it is blamed on the axiom of choice, the latter is incidental to it. It really stems from the inherent ambiguity of infinite set since not all its elements are known and any statement about is not verifiable. The proof of existence of nonmeasurable set is also based on the inherent ambiguity ambiguity of infinite set although, again, the axiom of choice is used. Among the sources of ambiguity aside from infinity are large and small numbers (due to limitation of our ability to compute even with advanced technology, e.e., a long string of digits cannot fit the computer screen) and vacuous concept; contradiction often hides in ambiguity.
Why are you claiming that these things represent contradictions in the
construction of the real numbers? They do not, unless you either
insist on using intuitionistic logic, or you insist on ignoring or
misusing measure theory.
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The source of all these is the inconsistency of the field axioms of the real numbers due to the false trichotomy axiom and the completeness axiom when applied to infinite set. Inconsistency of the axioms of a mathematical system collapses it since you can draw contradictory statements from it.
E. E. Escultura
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