Dear John Benavides,
1JB: "I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. "
1EB: My logic is the logic used by Euclid, Galileo, and many others. Where is the logic that defines and governs the basic notions in mathematics? Galileo compellingly concluded that the relations smaller, equal to, and larger do not apply for infinite quantities. Perhaps even Goedel was not able or not willing to accept that. He imagined the cardinality of continuum much larger than 2^aleph_0. Goedel proved CH does not contradict to ZF, and Cohen proved its opposite does also not contradict ZF. Hence CH is with Popper's words not falsifiable, in Pauli's language not even wrong. My logic tells me that the notion transfinite cardinality contradicts Galileo's conclusion. In other words, Georg Cantor misguided mathematics to leave the path of clean logic.
2JB:"I read your essay and what you see as problem is because you are unconsciously denying the excluded middle principle of classical logic that allows to abstract numbers with the limit point of the extension they represent."
2EB: I did definitely not unconsciously deny the TND. Read Fraenkel 1923. He admitted that there is what he called a fourth logical possibility besides the three above mentioned: not smaller, not equal to, not larger: incomparable. Brouwer clarified: The TND only applies within the realm of rational numbers. Unfortunately, Hilbert disserviced mathematics by ousting Brower. Hilbert was disappointed that his successor Weyl rejected much of set theory. Unfortunately Weyl emigrated soon. Mathematicians before Weierstrass and Cantor did not really bother treating the irrational numbers as if they were rational and accordingly subject to trichotomy.
Why do you mean classical logics dictates the TND, and why do mean every limit point belongs to the realm of countable numbers? The continuum as defined by Peirce cannot be split in single points.
3JB: " I agree with you extensions are more fundamental, but what you are ignoring is that if you say we should not identify extensions with its points, what are you saying is that we shouldn't use classical logic to describe the continuum, ...
3EB: My effort in this contest was not in vain if mathematicians like you agree on the reinstating of the Euclidean notion "number" as an extension. I suggested to identify the number with the extension between zero and an endpoint which is a limit from inside. I did not say we should not use classical logic to describe the continuum. On the contrary, I would like to make the mathematicians aware that Cantor cheated them.
4JB: ... that kind of misconceptions is what my essay is about. "
4EB: If only you did arrive at helpful conclusions.
5JB: "On the other hand you don't need a quantum computer to justify that quantum reality is ruled by a non classical logic, you just need a half-silvered-mirror and light to construct a machine that cannot be described using classical logic."
5EB: I am just an old engineer. Nonetheless I will try and hopefully find out on what possible mistakes the so far not functioning quantum computer is based. What experiment do you refer to? If something possibly very profitable does not work, this is a strong indication of something wrong. Isn't it?