Equiconsistency

In

In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.

In mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency.

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.

When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, or ZF+AD, set theory with the axiom of determinacy), the notions described above are adapted accordingly. Thus, ZF is equiconsistent with ZFC, as shown by Gödel.

The consistency strength of numerous combinatorial statements can be calibrated by large cardinals. For example:

• the negation of Kurepa's hypothesis is equiconsistent with the existence of an inaccessible cardinal,
• the non-existence of special ${\displaystyle \omega _{2}}$-Aronszajn trees is equiconsistent with the existence of a Mahlo cardinal,
• the non-existence of ${\displaystyle \omega _{2}}$-Aronszajn trees is equiconsistent with the existence of a weakly compact cardinal.^[1]

• Large cardinal property