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What is the current status of Foundation-of-Mathematics programmes?

I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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What are the First Principles of Euclidean Geometry (Besides the Axioms)?

On first principles, Wikipedia says: A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of ...
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Is there an infinity of axioms in mathematics?

As I was trying to find a list of mathematical axioms used in modern branches of mathematics, I wondered if there's any meaning to the question of "how many mathematical axioms are there ?", and then ...
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