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In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In...

question is actually inconsistent. In case 1, we say that A1 and A2 are equiconsistent. In case 2, we say that A1 is consistency-wise stronger than A2 (vice...

showed that NF is equiconsistent with TST plus the axiom scheme of "typical ambiguity".[citation needed] NF is also equiconsistent with TST augmented...

non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with...

strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another...

Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his...

choice (this is equivalent since these two theories have been proved equiconsistent; that is, if one is consistent, the same is true for the other). This...

model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry. This model is conformal which means that the...

does not suffer from this defect, and indeed was subsequently proved equiconsistent with NF by Hao Wang. Absolute infinite Burali-Forti, Cesare (1897),...

theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until...

used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if...

is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. William Mitchell proved, with the help of the Dodd-Jensen core model...

candidate proofs are current in the NF community suggesting that NF is equiconsistent with Zermelo set theory without Choice. A modification of NF, NFU, due...

function s : N → N satisfies the Peano axioms. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with...

cardinals. Δ 2 1 {\displaystyle \Delta _{2}^{1}} (lightface) determinacy is equiconsistent with a Woodin cardinal. If Δ 2 1 {\displaystyle \Delta _{2}^{1}} determinacy...

critical point κ such that j(κ) > θ and Mκ ⊆ M. Tall cardinals are equiconsistent with strong cardinals. Hamkins, Joel David (2009), "Tall cardinals"...

and the Kurepa hypothesis. The failure of the Kurepa hypothesis is equiconsistent with the existence of a strongly inaccessible cardinal. A variation...

first-order Peano arithmetic. The two systems are provably (in a weak system) equiconsistent. ACA0 can be thought of as a framework of predicative mathematics, although...

One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explosion: from a contradiction...

{\displaystyle {\mathsf {PA}}} . However, H A {\displaystyle {\mathsf {HA}}} is equiconsistent with both P A {\displaystyle {\mathsf {PA}}} as well as with the theory...