Reflecting cardinals
Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak Kripke-Platek set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-separation after reflection down to a $V_\alpha$ containing the range of the defined function. Several philosophers have advanced philosophical justifications of large cardinals based on ideas arising from reflection.
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Reflection theorem
The Reflection theorem is one of the most important theorems in Set Theory, being the basis for several large cardinals. The Reflection theorem is in fact a "meta-theorem," a theorem about proving theorems. The Reflection theorem intuitively encapsulates the idea that we can find sets resembling the class $V$ of all sets.
Theorem (Reflection): For every set $M$ and formula $\phi(x_0...x_n,p)$ ($p$ is a parameter) there exists some limit ordinal $\alpha$ such that $V_\alpha\supseteq M$ such that $\phi^{V_\alpha}(x_0...x_n,p)\leftrightarrow \phi(x_0...x_n,p)$ (We say $V_\alpha$ reflects $\phi$). Assuming the Axiom of Choice, we can find some countable $M_0\supseteq M$ that reflects $\phi(x_0...x_n,p)$.
Note that by conjunction, for any finite family of formulas $\phi_0...\phi_n$, as $V_\alpha$ reflects $\phi_0...\phi_n$ if and only if $V_\alpha$ reflects $\phi_0\land...\land\phi_n$. Another important fact is that the truth predicate for $\Sigma_n$ formulas is $\Sigma_{n+1}$, and so we can find a (Club class of) ordinals $\alpha$ such that $(V_\alpha,\in)\prec_{{T_{\Sigma_n}}\restriction{V_\alpha}} (V,\in)$, where $T_{\Sigma_n}$ is the truth predicate for $\Sigma_n$ and so $ZFC\vdash Con(ZFC(\Sigma_n))$ for every $n$, where $ZFC(\Sigma_n)$ is $ZFC$ with Replacement and Separation restricted to $\Sigma_n$.
Lemma: If $W_\alpha$ is a cumulative hierarchy, there are arbitrarily large limit ordinals $\alpha$ such that $\phi^{W_\alpha}(x_0...x_n,p)\leftrightarrow \phi^W(x_0...x_n,p)$.
Reflection and correctness
For any class $\Gamma$ of formulas, an inaccessible cardinal $\kappa$ is $\Gamma$-reflecting if and only if $H_\kappa\prec_\Gamma V$, meaning that for any $\varphi\in\Gamma$ and $a\in H_\kappa$ we have $V\models\varphi[a]\iff H_\kappa\models\varphi[a]$. For example, an inaccessible cardinal is $\Sigma_n$-reflecting if and only if $H_\kappa\prec_{\Sigma_n} V$. In the case that $\kappa$ is not necessarily inaccessible, we say that $\kappa$ is $\Gamma$-correct if and only if $H_\kappa\prec_\Gamma V$.
- A simple Löwenheim-Skolem argument shows that every infinite cardinal $\kappa$ is $\Sigma_1$-correct.
- For each natural number $n$, the $\Sigma_n$-correct cardinals form a closed unbounded proper class of cardinals, as a consequence of the reflection theorem. This class is sometimes denoted by $C^{(n)}$ and the $\Sigma_n$-correct cardinals are also sometimes referred to as the $C^{(n)}$-cardinals.
- Every $\Sigma_2$-correct cardinal is a $\beth$-fixed point and a limit of such $\beth$-fixed points, as well as an $\aleph$-fixed point and a limit of such. Consequently, we may equivalently define for $n\geq 2$ that $\kappa$ is $\Sigma_n$-correct if and only if $V_\kappa\prec_{\Sigma_n} V$.
A cardinal $\kappa$ is correct, written $V_\kappa\prec V$, if it is $\Sigma_n$-correct for each $n$. This is not expressible by a single assertion in the language of set theory (since if it were, the least such $\kappa$ would have to have a smaller one inside $V_\kappa$ by elementarity). Nevertheless, $V_\kappa\prec V$ is expressible as a scheme in the language of set theory with a parameter (or constant symbol) for $\kappa$.
Although it may be surprising, the existence of a correct cardinal is equiconsistent with ZFC. This can be seen by a simple compactness argument, using the fact that the theory ZFC+"$\kappa$ is correct" is finitely consistent, if ZFC is consistent, precisely by the observation about $\Sigma_n$-correct cardinals above.
$C^{(n)}$ are the classes of $\Sigma_n$-ordinals. These classes are clubs (closed unbounded). $C^{(0)}$ is the class of all ordinals. $C^{(1)}$ is precisely the class of all uncountable cardinals $α$ such that $V_α = H(α)$. References to the $C^{(n)}$ classes (different from just the requirement that the cardinal belongs to $C^{(n)}$) can sometimes make larger cardinal properties stronger (for example $C^{(n)}$-superstrong, $C^{(n)}$-extendible, $C^{(n)}$-huge and $C^{(n)}$-I3 and $C^{(n)}$-I1 cardinals). On the other hand, every measurable cardinal is $C^{(n)}$-measurable for all $n$ and every ($λ$-)strong cardinal is ($λ$-)$C^{(n)}$-strong for all $n$.[1]
A cardinal $\kappa$ is reflecting if it is inaccessible and correct. Just as with the notion of correctness, this is not first-order expressible as a single assertion in the language of set theory, but it is expressible as a scheme (Lévy scheme). The existence of such a cardinal is equiconsistent to the assertion ORD is Mahlo.
If there is a pseudo uplifting cardinal, or indeed, merely a pseudo $0$-uplifting cardinal $\kappa$, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus Ord is Mahlo. You can get this by taking some $\lambda\gt\kappa$ such that $V_\kappa\prec V_\lambda$.
$\Sigma_2$ correct cardinals
The $\Sigma_2$-correct cardinals are a particularly useful and robust class of cardinals, because of the following characterization: $\kappa$ is $\Sigma_2$-correct if and only if for any $x\in V_\kappa$ and any formula $\varphi$ of any complexity, whenever there is an ordinal $\alpha$ such that $V_\alpha\models\varphi[x]$, then there is $\alpha\lt\kappa$ with $V_\alpha\models\varphi[x]$. The reason this is equivalent to $\Sigma_2$-correctness is that assertions of the form $\exists \alpha\ V_\alpha\models\varphi(x)$ have complexity $\Sigma_2(x)$, and conversely all $\Sigma_2(x)$ assertions can be made in that form.
It follows, for example, that if $\kappa$ is $\Sigma_2$-correct, then any feature of $\kappa$ or any larger cardinal than $\kappa$ that can be verified in a large $V_\alpha$ will reflect below $\kappa$. So if $\kappa$ is $\Sigma_2$-reflecting, for example, then there must be unboundedly many inaccessible cardinals below $\kappa$. Similarly, if $\kappa$ is $\Sigma_2$-reflecting and measurable, then there must be unboundedly many measurable cardinals below $\kappa$.
The Feferman theory
This is the theory, expressed in the language of set theory augmented with a new unary class predicate symbol $C$, asserting that $C$ is a closed unbounded class of cardinals, and every $\gamma\in C$ has $V_\gamma\prec V$. In other words, the theory consists of the following scheme of assertions: $$\forall\gamma\in C\ \forall x\in V_\gamma\ \bigl[\varphi(x)\iff\varphi^{V_\gamma}(x)\bigr]$$ as $\varphi$ ranges over all formulas. Thus, the Feferman theory asserts that the universe $V$ is the union of a chain of elementary substructures $$V_{\gamma_0}\prec V_{\gamma_1}\prec\cdots\prec V_{\gamma_\alpha}\prec\cdots \prec V$$ Although this may appear at first to be a rather strong theory, since it seems to imply at the very least that each $V_\gamma$ for $\gamma\in C$ is a model of ZFC, this conclusion would be incorrect. In fact, the theory does not imply that any $V_\gamma$ is a model of ZFC, and does not prove $\text{Con}(\text{ZFC})$; rather, the theory implies for each axiom of ZFC separately that each $V_\gamma$ for $\gamma\in C$ satisfies it. Since the theory is a scheme, there is no way to prove from that theory that any particular $\gamma\in C$ has $V_\gamma$ satisfying more than finitely many axioms of ZFC. In particular, a simple compactness argument shows that the Feferman theory is consistent provided only that ZFC itself is consistent, since any finite subtheory of the Feferman theory is true by the reflection theorem in any model of ZFC. It follows that the Feferman theory is actually conservative over ZFC, and proves with ZFC no new facts about sets that is not already provable in ZFC alone.
The Feferman theory was proposed as a natural theory in which to undertake the category-theoretic uses of Grothendieck universes, but without the large cardinal penalty of a proper class of inaccessible cardinals. Indeed, the Feferman theory offers the advantage that the universes are each elementary substructures of one another, which is a feature not generally true under the universe axiom.
Maximality Principle
The existence of an inaccessible reflecting cardinal is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if $\kappa$ is an inaccessible reflecting cardinal, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model with an inaccessible reflecting cardinal.
References
- Bagaria, Joan. $C^{(n)}$-cardinals. Archive for Mathematical Logic 51(3--4):213--240, 2012. www DOI bibtex