In response to problems in the foundations of mathematics such as *Russell’s paradox* (“consider the set of all sets which are not members of themselves; is it a member of itself?”), HILBERT [VI.63] proposed that the consistency of any given part of mathematics should be established by finitary methods that could not lead to a contradiction. Any part for which this had been done could then be used as a secure foundation for all of mathematics.

An example of a “part of mathematics” is the arithmetic of the natural numbers, which can be described in terms of FIRST-ORDER LOGIC [IV.23 §1]. We begin with symbols, both logical (connectives such as “not” and “implies,” quantifiers such as “for all,” the equality ...

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