Mathematics took major strides in the 19 th century, but it also became increasingly clear at that time that there was reason to be concerned about the very foundations of the enterprise. For example, it is a key principle of analysis that every bounded set of reals has a supremum, a principle that is often accepted at face value. But is there any precise argument that can be made in support of this principle? Pushing a bit further, is there a reasonable way to describe natural numbers, integers and rationals that does not simply appeal to intuition? One might argue there is no need for this, since everything is absolutely clear from intuition. But Euclidean geometry is also clear from intuition and was found to be less than universal in the 1830’s. By the end of the century, there were enough problems to cause great alarm in certain quarters and declare a foundational crisis (Grundlagenkrise). In particular David Hilbert and his school felt the need to get serious about putting mathematics on a solid foundation. There are two important parts to this enterprise: Analyze Reasoning More precisely, analyze reasoning as it pertains to mathematics. This is the mathematical logic part and may seem rather straightforward, but it turns out to be quite challenging. Analyze Mathematical Foundations The hope here is to find a few elementary mathematical concepts that can be used together with logic to account for all of mathematics in an entirely precise and rigorous way
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