Mathematics and Its History pp 525551  Cite as
Sets, Logic, and Computation
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In the 19th century, perennial concerns about the role of infinity in mathematics were finally addressed by the development of set theory and formal logic. Set theory was proposed as a mathematical theory of infinity and formal logic was proposed as a mathematical theory of proof (partly to avoid the paradoxes that seem to arise when reasoning about infinity). In this chapter we discuss these two developments, whose interaction led to mindbending consequences in the 20th century. Both set theory and logic throw completely new light on the question, “What is mathematics?” But they turn out to be doubleedged swords.
It might be thought that the limits of formal proof are too remote to be of interest to ordinary mathematicians. But in the next chapter we will show how these limits are now being reached in one of the most downtoearth fields of mathematics: combinatorics.

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.
Keywords
Turing Machine Computable Function Order Type Continuum Hypothesis Large Cardinal
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