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Author: Serge Lang
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I highly recommend the courses on brilliant.org. They are not axiomatic, but they heavily focus on building up deep insight rather than mechanical problem solving -- https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/03879...
The comment you replied to received loads of responses, and many of the answers ignored the actual request, and plugged their favorite math books instead. In case you missed it, there was at least one suggestion that gave a real answer to the question: 'Basic Mathematics' by Serge Lang. It's a streamlined but somewhat dry textbook that starts with the properties of arithmetic and ends with precalculus. You might try reading through the table of contents to get a better idea of whether or not it will solve your problem!
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