Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc...). In slightly more detail, Woodin had established that the [](Axiom%20of%20determinacy.md) (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point lt;\lambda$. This axiom, a <a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">rank-into-rank</a> axiom, is known to be very strong and its use was first weakened to that of the existence of a [supercompact](Supercompact.md "Supercompact") cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the [](Projective.md#Regularity_properties) of sets of reals mentioned above. Shelah cardinals were the first cardinals to be devised by Woodin and Shelah. A cardinal $\delta$ is *Shelah* if for every function $f:\delta\to\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\delta$ such that $V_{j(f)(\delta)}\subseteq M$. Every Shelah is Woodin, but not every Woodin is Shelah: indeed, Shelah cardinals are always measurable and in fact [strong](Strong.md "Strong"), while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals. A related notion is *Shelah-for-supercompactness*, where the closure condition $V_{j(f)(\delta)}\subseteq M$ is replaced by $M^{j(f)(\delta)}\subseteq M$, a much stronger condition. The difference between Shelah and Shelah-for-supercompactness cardinals is essentially the same as the difference between strong and [supercompact](Supercompact.md "Supercompact") cardinals, or between [superstrong](Superstrong.md "Superstrong") and [huge](Huge.md "Huge") cardinals. Also, just like every Shelah is preceeded by a stationary set of strong cardinals, every Shelah-for-supercompactness cardinal is preceeded by a stationary set of supercompact cardinals. Much weaker, consistent with $V=L$ variant: A cardinal $κ$ is **virtually Shelah for supercompactness** iff for every function $f : κ → κ$ there are $λ > κ$ and $\bar{λ}< κ$ such that in a set-forcing extension there is an elementary embedding $j : V_{\bar{λ}}→ V_{λ}$ with $j(\mathrm{crit}(j)) = κ$, $\bar{λ} ≥ f(\mathrm{crit}(j))$ and $f ∈ \mathrm{ran}(j)$. If $κ$ is virtually Shelah for supercompactness, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-[extendible](Extendible.md "Extendible") cardinals for every $n < ω$ and if κ is 2-<a href="Iterable" class="mw-redirect" title="Iterable">iterable</a>, then $V_κ$ is a model of proper class many virtually Shelah for supercompactness cardinals.{% cite Gitmana %}