Sets, Logic, and Computation
Set theory brings remarkable clarity to the concept of infinity, but it shows infinity to be unexpectedly complicated–in fact, more complicated than set theory itself can describe.
Formal logic encompasses all known methods of proof, but at the same time it shows these methods to be incomplete. In particular, any reasonably strong system of logic cannot prove its own consistency.
Formal logic is the origin of the concept of computability, which gives a rigorous definition of an algorithmically solvable problem. However, some important problems turn out to be unsolvable.
KeywordsTuring Machine Computable Function Order Type Continuum Hypothesis Large Cardinal
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