I had intended to persue my interest in Doron's "Universal Reasoning" or "Complementary Logic" in his thread at philosophy-forums.com. I was able to register there but unable to post. And now the philosophy-forums has vanished from the web.
I noticed it is down. You should get a second email when your account is actually activated.
Buried beneath Doron's landfill of missused and discarded mathematical terminology is an actual idea. Once it's made clear, then it can be discussed and critiqued in an intelligent way. It rarely comes to that because Doron's way of thinking is counterintuitive, as if from some culture isolated from the rest of the world.
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As put forward by you, it seems an idea that could be pursued and actually turned into mathematics - that's the goal, isn't it? And this wouldn't tear down the vast building of mathematics in order to replace it, but be an extra wing built onto it. However, I doubt that your analysis is right - for reasons put forward below, and because there's already precedent that your interpretation of Doron's work turned out not to be what he meant, in previous threads, despite your good efforts.
If you get the idea, then there are a host of questions and critcisms that are not being addressed. I hope we can move on to these and not back to the years of forum wreckage and debacle Doron has behind him.
"Wreckage" is, IMHO, put mildly. There's 5.5 years of Doron posting on numerous fora. On various fora, e.g., physicsforums.com, IIDB, and here, he met people who were inclined to meticulously point out the various errors in his mathematical reasoning and gradually grew tired with his not listening to their arguments. In that time, he's shown no inclination whatsoever to actually learn mathematics; instead, he's grown ever bolder in statements like "Cantor was wrong" up to his laughably incorrect rendering here of Cantor's diagonalization argument in the proof of the uncountability of
R.
Would he honestly have some actual idea, and not only vague delusions, then methinks, he would have taken the effort to understand how mathematics works. He obviously hasn't. He doesn't know what an axiom is, how a definition looks like, etc., everything a freshman math student learns is alien to him. He doesn't take to heart criticism from others, but on questions posed in the best case regurgitates the same he's said before - in words or in links to earlier posts - or gives an answer that is even vaguer than what the question was about in the first place.
Would it be due to a lack of skill in English, then I honestly don't know what he's doing on English language boards. I note he's also active on haayal.co.il, a Hebrew-language board, and I obviously can't judge the clarity of his writings there, but I very much doubt it's better than his English writings - going on Skeptic's report of the wiki-discussion. And well, it's not that there's a lack of Hebrew-language highly-schooled mathematicians who could teach, coach or vet his work - apart from the questionable kindergarten teacher.
It will be more interesting if we can get to questions that are relevant to the very different way Doron thinks.
I'll gladly leave the floor to you to see if your interpretation of Doron's writings gets his approval.