Consistency strength,constructibility or high cardinals?

Question

When do we say that an assertion has a higher consistency strength: when it follows from high cardinal assumptions or from $V=L$? Both are quite strong requirements, but going to opposite directions.

Answer 1

Consistency strength is ultimately a number-theoretic measurement of a theory. How strong it is.

More specifically, assuming a theory $T$ is consistent, can we prove that a theory $T'$ is consistent as well?

For example, assuming $\sf ZF$ is consistent, we can prove that $\sf ZFC$ and $\sf ZFC\rm+V=L$ and $\sf ZFC\rm+V\neq L$ are all consistent. Therefore all these theories have the same consistency strength.

On the other hand, assuming that $\sf ZFC+\rm\kappa\text{ is measurable is consistent}$ we can prove that $\sf ZFC+\operatorname{Con}(ZFC)$ is consistent, and therefore the former is strictly stronger in terms of consistency strength.

Similarly, $\sf ZF+DC+\rm\text{Every set of reals is Lebesgue measurable}$ is equiconsistent with $\sf ZFC\rm+\text{Every }\Sigma^1_3\text{ set of reals is Lebesgue measurable}$ and with $\sf ZFC\rm+\exists\kappa\text{ inaccessible}$. All of which are strictly stronger than the consistency of just plain $\sf ZFC$.

But you're question is fundamentally backwards. We say that $T$ has a higher consistency strength when it implies the consistency—or even existence—of large cardinals. Not when it is implied by large cardinals or $V=L$.