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Let the first order formulas $p(x)$ and $wi(x)$ assert "$x$ is a large cardinal of type $p$" and "$x$ is weakly inaccessible" respectively.

The large cardinal type $p$ is upward reflecting if $ZFC\vdash \forall \kappa~~(p(\kappa)\longrightarrow\exists \lambda>\kappa~~wi(\lambda))$.

In the other words existence of a large cardinal of type $p$ like $\kappa$ implies existence of a large cardinal (at least weakly inaccessible) above $\kappa$.

Which large cardinals are upward reflecting?

In particular, are Shelah cardinals upward reflecting?

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Huge cardinals. –  Andres Caicedo Feb 7 at 1:34
    
Anyway, reflection means something other than the existence of large cardinals, so the name is not appropriate. What is the precise difficulty that is not letting you answer the question on your own in the case of Shelah cardinals? –  Andres Caicedo Feb 7 at 1:35

1 Answer 1

up vote 9 down vote accepted

There are numerous large cardinal notions that are upward reflecting in the sense you have described. On the one hand, some of the very largest notions have much stronger versions of the upward reflecting property, since indeed the large cardinal property $p$ itself reflects upward, in the sense that every instance of a $p$ cardinal has largest instances of $p$ cardinals.

  • For example, if $\kappa$ is a rank-into-rank cardinal, meaning that it is the critical point $\kappa$ of an elementary embedding $j:V_\lambda\to V_\lambda$, then $j(\kappa)$ is also a rank-into-rank cardinal, witnessed by the application $j(j)$, the union $\bigcup_{\alpha<\lambda}j(j\upharpoonright V_\alpha)$, which is an elementary embedding $V_\lambda\to V_\lambda$ with critical point $j(\kappa)$.

  • A similar argument works with $j:V_{\lambda+1}\to V_{\lambda+1}$ and $j:L(V_\lambda)\to L(V_\lambda)$, which are amongst the very strongest large cardinal properties.

  • If $\kappa$ is Reinhardt (considered in GB without AC), witnessed by $j:V\to V$, then $j(\kappa)$ is similarly Reinhardt, witnessed by $j(j)$.

A bit lower, we have further examples with the huge cardinals, which are the critical points of embeddings $j:V\to M$ where we have $M^{j(\kappa)}\subset M$. This implies $V_{j(\kappa)}\subset M$ and since $j(\kappa)$ is a large cardinal in $M$, it is a limit of many inaccessible and much larger large cardinals in $M$ which must therefore have these properties also in $V$. Similar reasoning applies to the almost huge and superhuge cardinals, as well as to the superstrong cardinals.

In particular, this reasoning applies to Shelah cardinals. A cardinal $\kappa$ is Shelah if for every $f:\kappa\to\kappa$ there is $j:V\to M$ with critical point $\kappa$ and $V_{j(f)(\kappa)}\subset M$. If you use the function $f(\gamma)=$next inaccessible after $\gamma$ plus $1$, then the witnessing $j:V\to M$ will have $V_{j(f)(\kappa)}=V_{\delta+1}\subset M$, where $\delta$ is the next inaccessible cardinal after $\kappa$ in $M$. In particular, this implies that $\delta$ really is inaccessible in $V$, and so Shelah cardinals are upward reflecting in your sense.

Meanwhile, Woodin cardinals are not upwardly reflecting, and indeed, there are numerous large cardinals that do not necessarily exhibit your upward reflecting property, including measurable cardinals, strong cardinals, strongly compact cardinals, supercompact cardinals, and many others. The reason is that if $\kappa$ exhibits any of these large cardinal properties, and $\lambda\gt\kappa$ is the next inaccessible cardinal above $\kappa$, then $\kappa$ will continue to exhibit the large cardinal inside $V_\lambda$, where there will be no inaccessible cardinals above $\kappa$. (Note, we may work in a model of GCH, where the distinction between strongly inaccessible and weakly inaccessible evaporates.) Basically, any large cardinal property that is witnessed sufficiently locally, so that it remains true whenever one cuts off the universe at an inaccessible level, will fail to be upwardly reflecting in your sense.

Lastly, let me mention that one shouldn't think that it is only the very large large cardinals that are upward reflecting. Consider the uplifting cardinals, which are rather weak in consistency strength, below Mahlo cardinals. The uplifting cardinals and even the pseudo uplifting cardinals are upward reflecting, since they are inaccessible cardinals $\kappa$ for which there is an elementary extension $V_\kappa\prec V_\lambda$, and this implies that $\lambda$ must be a limit of inaccessible cardinals above $\kappa$.

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Prof. Hamkins, Thank you very much for your useful answer. I have another question here. By the answer, Woodin and Shelah cardinals are so similar in definition and so different in behavior (at least in the case of being upward reflecting). Can this difference be problematic in forcings for these large cardinals? Precisely, are forcings which can preserve Shelah cardinals in generic extensions essentially different from forcings which preserve Woodin cardinals? –  user46647 Feb 7 at 12:56
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The difference between Woodin and Shelah cardinals arising because with a Woodin cardinal, it is much smaller cardinals rather than the cardinal itself that witnesses the property as a critical point of an embedding, and these smaller cardinals do have inaccessible cardinals above them. For forcing, for example, Woodin cardinals are easily seen to be indestructible by forcing that does not add subsets to $\kappa$, but Shelah cardinals are not necessarily indestructible that way. –  Joel David Hamkins Feb 7 at 13:23

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