Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself.

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  • Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself. Gödel's incompleteness theorems show that Hilbert's program cannot be realized: If a consistent recursively enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic, then the theory cannot prove its own consistency. There are some technical caveats as to what requirements the formal statement representing the metamathematical statement "The theory is consistent" need to satisfy, but the outcome is that if a (sufficiently strong) theory can prove its own consistency then either there is no computable way of identifying whether a statement is even an axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, including false statements such as its own consistency). Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories. Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Two theories are equiconsistent if each one is consistent relative to the other.
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  • Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself.
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  • Equiconsistency
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