Equiconsistency

Inmathematical logic,twotheoriesareequiconsistentif theconsistencyof one theory implies the consistency of the other theory, andvice versa.In this case, they are, roughly speaking, "as consistent as each other".

In general, it is not possible to prove the absolute consistency of a theoryT.Instead we usually take a theoryS,believed to be consistent, and try to prove the weaker statement that ifSis consistent thenTmust also be consistent—if we can do this we say thatTisconsistent relative to S.IfSis also consistent relative toTthen we say thatSandTareequiconsistent.

Consistency

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

Hilbertproposed aprogramat 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 toarithmetic,the program quickly became the establishment of the consistency of arithmetic by methods formalizable within arithmetic itself.

Gödel'sincompleteness theoremsshow that Hilbert's program cannot be realized: if a consistentcomputably enumerabletheory is strong enough to formalize its ownmetamathematics(whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmeticsuffices), 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" needs 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 anaxiomof 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: LetSandTbe formal theories. Assume thatSis a consistent theory. Does it follow thatTis consistent? If so, thenT is consistent relative to S.Two theories are equiconsistent if each one is consistent relative to the other.

Consistency strength

IfTis consistent relative toS,butSis not known to be consistent relative toT,then we say thatShas greaterconsistency strengththanT.When discussing these issues of consistency strength, the metatheory in which the discussion takes places needs to be carefully addressed. For theories at the level ofsecond-order arithmetic,thereverse mathematicsprogram has much to say. Consistency strength issues are a usual part ofset theory,since this is acomputabletheory that can certainly model most of mathematics. The most widely used set of axioms of set theory is calledZFC.When a set-theoretic statementAis said to be equiconsistent to anotherB,what is really being claimed is that in the metatheory (Peano arithmeticin this case) it can be proven that the theories ZFC+Aand ZFC+Bare equiconsistent. Usually,primitive recursive arithmeticcan be adopted as the metatheory in question, but even if the metatheory is ZFC or an extension of it, the notion is meaningful. The method offorcingallows one to show that the theories ZFC, ZFC+CH and ZFC+¬CH are all equiconsistent (where CH denotes thecontinuum hypothesis).

When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, or ZF+AD, set theory with theaxiom 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 bylarge cardinals.For example:

• the negation ofKurepa's hypothesisis equiconsistent with the existence of aninaccessible cardinal,
• the non-existence of special$\ Omega _{2}$-Aronszajn treesis equiconsistent with the existence of aMahlo cardinal,
• the non-existence of$\ Omega _{2}$-Aronszajn treesis equiconsistent with the existence of aweakly compact cardinal.

See also

• Large cardinal property

References

• Akihiro Kanamori(2003).The Higher Infinite.Springer.ISBN3-540-00384-3