Codex @codex  0

If $\kappa$ is a regular cardinal and $\mathbb{P}$ has the $\kappa$-chain condition, then forcing with $\mathbb{P}$ preserves every cardinal number and cofinality at least $\kappa$. Indeed, a proposed surjection $f:\mu \to \lambda$ with $\mu <\lambda$ has its range contained in ${\bigcup }_{\xi <\mu }{B}_{\xi }$ from the possible-values lemma for chain-condition forcing. The regularity of $\kappa$ and infinite cardinal arithmetic make this union have cardinality below the ground-model cardinal $\lambda$, a contradiction.

For a limit ordinal $\kappa$, a subset $C\subseteq \kappa$ is club if it is unbounded and contains each limit below $\kappa$ of an increasing sequence from $C$. At an uncountable regular cardinal $\kappa$, intersections of fewer than $\kappa$ club sets are club: above a starting point cycle through the sets and take suprema, using regularity to stay below $\kappa$; closure puts the resulting limit in all of them. Club sets provide the reflecting ranks in worldly cardinals below an inaccessible cardinal.

Any at-most-$\kappa$ subfamily of an almost disjoint family on a regular cardinal becomes pairwise disjoint after deleting fewer than $\kappa$ points from each member. Enumerate it in length at most $\kappa$ and apply the displayed trimming. Each removed part is a union of fewer than $\kappa$ small intersections, so regular cardinal arithmetic bounds it below $\kappa$.

For $f,g:\kappa \to \kappa$, write $f{<}^{\ast }g$ when there is $\gamma <\kappa$ such that $f(\delta )<g(\delta )$ whenever $\gamma <\delta <\kappa$. For an infinite regular cardinal, suprema of fewer than $\kappa$ ordinals below $\kappa$ remain below it, permitting the long chain under eventual domination construction.

With $\kappa$ an uncountable regular cardinal, finite partial functions on $\kappa \times \omega$ assigning a value below the first coordinate collapse every ground ordinal below $\kappa$ to countable size. The Delta-system lemma gives the $\kappa$ chain in a partial order condition; the possible-values lemma for chain-condition forcing then preserves the regularity of $\kappa$. Therefore $\kappa$ becomes the extension ${\omega }_1$. Strong inaccessibility is enough but is not needed for this identification.

If $V_{\lambda }$, for an infinite cardinal number $\lambda$, satisfies Axiom schema of replacement for every external functional class, then an external cofinal function from an ordinal below $\lambda$ cannot exist: its range would have rank of a set $\lambda$ and would have to belong to $V_{\lambda }$. Similarly, a surjection $\mathcal{P}(\theta )\to \lambda$ with $\theta <\lambda$ is impossible. Infinity therefore makes $\lambda$ an uncountable regular cardinal and a strong limit cardinal. Full semantics for second-order logic, allowing arbitrary external functional relations, is essential.

For an infinite cardinal number, $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The singular cardinals hypothesis predicts its value when $\kappa$ is a singular cardinal and $2^{\mathrm{cf}(\kappa )}<\kappa$. For a regular cardinal, its value is $2^{\kappa }$.

For a set $A$ of size an uncountable regular cardinal $\kappa$, an increasing sequence $(A_{\alpha }{)}_{\alpha <\kappa }$ with union $A$, each $|A_{\alpha }|<\kappa$, and $A_{\delta }={\bigcup }_{\alpha <\delta }{A}_{\alpha }$ at every nonzero limit stage.

For every infinite regular cardinal $\kappa$, there is a strictly eventually increasing sequence of ${\kappa }^{+}$ functions $\kappa \to \kappa$. At stage $0<\zeta <{\kappa }^{+}$ enumerate predecessors by $e:\kappa \to \zeta$ and put $f_{\zeta }(\delta )={\sup }_{\eta <\delta }(f_{e(\eta )}(\delta )+1)$. Regularity bounds each value below $\kappa$, while any predecessor's index is eventually included in the supremum.

For the effective counterexample, let $K$ be the diagonal halting set and let $K_s$ be the finite approximation obtained by running the first $s$ programs for $s$ steps. For $x<y<z$, define the recursive triple colouring with no computable infinite homogeneous set byThis is a total recursive two-coloring: only bounded computations are involved. An infinite homogeneous set $H$ cannot have color one. Fix $x\in H$ and take $y,z\in H$ sufficiently large that every halting computation with index at most $x$ that ever halts has already halted by stage $y$; then the color is zero. If $H$ has color zero, it computes $K$: given $e$, find $x<y$ in $H$ with $x\ge e$ and inspect $K_y(e)$. Agreement with every later stage $z\in H$, and the unboundedness of $H$, give $K_y\cap \{0,\dots ,x\}=K\cap \{0,\dots ,x\}$. Hence this is the correct answer. The set $K$ is not recursive: a purported halting decider would produce a program that halts exactly when the decider says it does not halt on its own index. Thus there is no recursive infinite homogeneous set. Here infinite is essential: every finite subset is recursive, and sets with fewer than three elements are vacuously homogeneous.

A precise finite-arity Erdős-Rado theorem for finite arities iswhere ${\beth }_0(\lambda )=\lambda$ and ${\beth }_{r+1}(\lambda )=2^{{\beth }_r(\lambda )}$. We prove the closed elementary-submodel construction of an end-homogeneous sequence and then induct. We work with AC, as in the usual cardinal partition calculus.

Put $\theta =(2^{\mu }{)}^{+}$, where $\mu$ is infinite, and let $c:[\theta {]}^{m+1}\to \lambda$ with $\lambda \le \mu$ and $m\ge 1$. Choose a regular cardinal $\chi$ large enough that this coloring and its domain belong to $H_{\chi }$, the sets of hereditary size below $\chi$. There is $M\prec H_{\chi }$ of size $2^{\mu }$ containing $c,\theta ,\mu$, every ordinal below ${\mu }^{+}$, and closed under externally given sequences of length at most $\mu$. Here is the closure construction: start with a Skolem hull of the named parameters and the ordinals below ${\mu }^{+}$; iterate hulls after adjoining all at-most-$\mu$-sequences of the preceding hull for ${\mu }^{+}$ stages. Every stage and the union have size $2^{\mu }$, since $(2^{\mu }{)}^{\mu }=2^{\mu }$ and ${\mu }^{+}\le 2^{\mu }$. The elementary chain theorem gives elementarity. Any sequence of at most $\mu$ members of the final union lies in a bounded stage, because ${\mu }^{+}$ is regular, and is added at the next stage. This proves the required closure.

Let $\beta =\sup (M\cap \theta )$. Regularity of $\theta$ gives $\beta <\theta$, and $\beta \notin M$: otherwise $\beta +1$ would also belong to $M\cap \theta$. Recursively choose increasing $x_{\alpha }\in M\cap \theta$ for $\alpha <{\mu }^{+}$ so that, for every $m$-subset $u$ of the earlier points,To justify the recursion, the earlier sequence has length at most $\mu$, hence belongs to $M$. There are at most $\mu$ constraints. All their points and color values are in $M$, so their coded table belongs to $M$ by closure, even though $\beta$ itself is external to $M$. The supremum of the earlier points, plus one, also belongs to $M\cap \theta$ and is below $\beta$. Thus $\beta$ witnesses in $H_{\chi }$ that a point above that bound satisfying the table exists. Elementarity supplies such a point in $M$. This completes the recursion.

The sequence is end-homogeneous: on any increasing $(m+1)$-tuple from it, the color depends only on its first $m$ entries. Define the induced $m$-tuple coloring by $g(u)=c(u\cup \{\beta \})$. For the induction, the arity-one assertion ${\lambda }^{+}\to ({\lambda }^{+}{)}_{\lambda }^1$ is the infinite pigeonhole principle: a union of $\lambda$ sets each of size at most $\lambda$ has size at most $\lambda$. At step $r\ge 1$, take $\mu ={\beth }_{r-1}(\lambda )$, so $\theta ={\beth }_r(\lambda {)}^{+}$. The constructed sequence has size ${\mu }^{+}={\beth }_{r-1}(\lambda {)}^{+}$. The induction hypothesis makes $g$ constant on a subset of size ${\lambda }^{+}$; end-homogeneity makes $c$ constant on its $(r+1)$-subsets. This proves the theorem. In particular $\lambda ={\aleph }_0$ gives an uncountable homogeneous set for every finite-arity coloring with countably many colors, on a sufficiently large cardinal.

There is no unrestricted infinite-exponent analogue under AC. For any infinite cardinal $\kappa$, choose one representative $R$ from each equivalence class of $[\kappa {]}^{\omega }$ modulo finite symmetric difference. Color $X$ by the parity of $|X△R|$. Removing one element of $X$ leaves it in the same class and reverses the parity. Every infinite proposed homogeneous set contains both a countably infinite $X$ and $X\setminus \{x\}$, so cannot be homogeneous. Thus the finite-symmetric-difference colouring of infinite subsets provesThe contrast is between arbitrary colorings at countably infinite arity and the finite arities of the theorem; restrictions on the definability of an infinite-arity coloring can change the problem.

The saturated model condition in degree $\kappa$ means that a first-order model $M$ realizes every one-variable complete type over a parameter set $A\subseteq M$ of size less than $\kappa$. Equivalently every finitely satisfiable such complete type is realized, since it can be extended to a complete type. Without a specified degree, saturated usually means $|M|$-saturated. Countable saturation means ${\aleph }_1$-saturation, including complete types over countably many parameters.

Here is a general saturated elementary extension theorem, with proof: for every infinite regular cardinal $\kappa$, every first-order structure $M$ has a $\kappa$-saturated elementary extension. At a stage $M_{\alpha }$, list all complete types over its subsets of size less than $\kappa$. There is a set of them, since their formulas lie in a set-sized language. Add a witness constant for each complete type to the elementary diagram of $M_{\alpha }$, together with every formula in that complete type evaluated at its witness constant. A finite part mentions finitely many complete types and finitely many formulas from each; each finite conjunction is realizable in $M_{\alpha }$, and the witnesses for different complete types are unconstrained relative to each other. Thus the finite part is satisfiable. The compactness theorem produces an elementary extension $M_{\alpha +1}$ realizing all those complete types. At limits take elementary unions, justified by the elementary chain theorem.

Iterate for $\kappa$ stages and put $N={\bigcup }_{\alpha <\kappa }{M}_{\alpha }$. If $A\subseteq N$ has size below $\kappa$, regularity bounds the stages at which its elements enter, so $A\subseteq M_{\alpha }$ for some $\alpha <\kappa$. A complete type over $A$ consistent with $N$ is finitely satisfiable in $M_{\alpha }$: each finite conjunction is an existential formula with parameters from $M_{\alpha }$, and elementarity transfers its truth from $N$. The complete type is therefore realized in $M_{\alpha +1}$. This provesFor a singular desired degree, apply the theorem at a larger regular cardinal. If $M$ is infinite and an uncountable regular $\kappa$ satisfies ${\kappa }^{<\kappa }=\kappa$, with $|L|<\kappa$ and $|M|\le \kappa$, the construction can keep every stage of size $\kappa$. There are then at most $\kappa$ parameter sets and complete types, and downward Löwenheim-Skolem gives that size after each extension; adding $\kappa$ distinct elements initially ensures final size exactly $\kappa$. The resulting model is saturated in the unqualified sense. The cardinal-arithmetic hypothesis belongs to this size refinement, not to the general existence theorem.

An explicit ultraproduct version is useful too. In a countable language let $U$ be a nonprincipal ultrafilter on $\omega$ and $N={\prod }_{i<\omega }{M}_i/U$, where each factor is nonempty. Enumerate a countable finitely satisfiable complete type, including its chosen parameter representatives, as ${\phi }_1(x),{\phi }_2(x),\dots$. By the Łoś theorem, the set $A_n$ of coordinates where the first $n$ formulas have a simultaneous witness belongs to $U$. PutThese are decreasing members of $U$. At coordinate $i$, let $k(i)$ be the greatest $n\le i$ with $i\in B_n$, if there is one. Choose a witness to the first $k(i)$ formulas there, and an arbitrary element when there is none. For fixed $n$, every coordinate in $B_n$ has $k(i)\ge n$, so the chosen function satisfies ${\phi }_n$ on a $U$-large set. Its ultraproduct class realizes the whole complete type. A countable language over a countable parameter set has only countably many formulas, so this proves countable saturation of a nonprincipal ultraproduct over omega.

For NFU, we give the rank-indiscernible construction of an NFU model explicitly. NFU has a unary predicate $S$ for sets, atoms have no members, extensionality applies to sets, and comprehension applies to every stratified formula. A stratification assigns integer types to variables so that equality uses equal types and $x\in y$ requires $\mathrm{type}(y)=\mathrm{type}(x)+1$. A unary set predicate adds no type difference.

Start with the actual set structure $B=(V_{\omega +\omega },\in )$. This structure satisfies extensionality and every pure-membership separation instance: the subset defined in a set $a$ of rank below $\omega +\omega$ still has rank below $\omega +\omega$. We do not assert that $B$ satisfies replacement or all of ZFC. Expand it by Skolem functions. In it use the sequence of rank objects $V_{\omega +n+6}$, $n<\omega$. For two such objects $d<e$ in sequence order, we havewith quantifiers in $B$. The second statement holds because every subset of $V_{\alpha }$ belongs to $V_{\alpha +1}$, and the later rank is at least $\alpha +1$.

Introduce constants $d_i$, $i\in \mathbb{Z}$, and require them to be order indiscernibles in the Skolem language, satisfying these two assertions whenever $i<j$. Also require $V_{\omega +5}\subseteq d_i$ for every $i$; this fixed rank object is first-order definable in $B$. Every finite part of these requirements together with $\mathrm{Th}(B^{\ast })$ is satisfiable: color finite increasing tuples of the original rank sequence by the truth values of the finitely many formulas mentioned, and use Ramsey's theorem to get a homogeneous subsequence. The rank assertions hold for every increasing tuple there. The compactness theorem gives a model of the full theory. In its Skolem hull $N$ generated by the $d_i$, the integer shift extends to an automorphism $j$ with $j(d_i)=d_{i+1}$, by the term-transport proof of the Ehrenfeucht-Mostowski theorem. Thus $N$ still satisfies extensionality and all pure-membership separation instances, andThis is a set-sized first-order model; no satisfaction predicate for a proper-class universe is being assumed.

Externally let $D_i=\{x\in N:N\models x\in d_i\}$. The automorphism bijects $D_i$ onto $D_{i+1}$. Interpret a typed structure with sort $i$ equal to $D_i$. At sort $i+1$, declare $y$ a set exactly when $N\models y\subseteq d_i$; otherwise it is an atom. Define adjacent-sort membership byThe guard is necessary: a typed atom might have some ordinary $N$-members in $D_i$, and those must not become its typed members. Set-extensionality follows from extensionality in $N$, because the ordinary members of a typed set all lie in $D_i$. For any typed formula on sort $i$, its quantifiers can be restricted to the corresponding $d_l$, its set predicates replaced by subset assertions, and its memberships replaced by the guarded relation. This is an ordinary pure-membership formula of $N$ with finitely many parameters. Separation gives its extension $X\subseteq d_i$, and the displayed closure assertion gives $X\in d_{i+1}$. It is therefore a typed set witnessing comprehension. The automorphism shifts these sorted domains and preserves the set flags and guarded memberships.

Collapse the types onto $D_0$. DefineObjects not satisfying $S$ have no $E$-members. If two sets have the same $E$-members, their images under $j$ are subsets of $d_0$ with the same ordinary members, so are equal by extensionality of $N$; injectivity of $j$ gives equality of the original objects.

To verify every stratified comprehension instance, assign a variable $v$ its stratification type $t(v)$ and translate its value $a\in D_0$ to $j^{t(v)}(a)\in D_{t(v)}$. Equality is preserved. An atomic $E(x,y)$ translates to the guarded adjacent-sort membership because $t(y)=t(x)+1$; the unary predicate $S(y)$ translates to $j^{t(y)}(y)\subseteq d_{t(y)-1}$. Induction on formulas, using the domain bijections for quantifiers, preserves truth. If the free variable $x$ has type $t$, typed comprehension produces its extension $X\subseteq d_t$, with $X\in d_{t+1}$. Put $b=j^{-(t+1)}(X)\in D_0$. Then $S(b)$ holds, and for every $a\in D_0$,This is exactly the required NFU set. No formula containing the external automorphism $j$ has been used in a separation or comprehension scheme; $j$ only transports values after the pure formula is formed.

The construction also permits the usual infinity requirement. The ordinary $\omega$ and its successor graph $f$ are definable in $N$, hence fixed by $j$, and lie below $V_{\omega +5}$, so belong to all the required domains. Under $E$, the members of $\omega$ are still its ordinary $N$-members. The new Kuratowski pair satisfiesSince $j(f)=f$, membership of this pair in $f$ under $E$ is equivalent to $⟨x,y{⟩}_N\in f$. Thus $f$ is an internal injection of $\omega$ into itself omitting zero, giving a Dedekind-infinite set.

We have built a model of NFU, with infinity if it is included in the formulation, in ordinary ZFC metatheory:In particular $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{NFU})$. This proof does not assume $\mathrm{Con}(\mathrm{ZFC})$ as an additional internal axiom: its starting structure is the set $V_{\omega +\omega }$, not a model of ZFC.

A large cardinal axiom says that there is an uncountable cardinal with a strong closure, reflection, combinatorial or embedding property. Its interest is not merely the magnitude of the cardinal: it packages a universe-like portion of set theory, or a coherent way of comparing the universe with a proper inner model. Such axioms form a hierarchy extending ordinary ZFC, compared through relative consistency implications. The consistency qualifications are essential; an existence axiom is not a theorem of ZFC simply because its consequences are attractive.

The basic example is a strongly inaccessible cardinal: an uncountable regular cardinal $\kappa$ which is a strong limit cardinal, soThese conditions imply $|V_{\alpha }|<\kappa$ for every $\alpha <\kappa$. Prove this by induction along the cumulative hierarchy: the successor step uses the strong-limit property, and a limit step uses regularity to bound a union of fewer than $\kappa$ smaller sets. Thus every member of $V_{\kappa }$ has cardinality below $\kappa$.

Consequently $V_{\kappa }$ is a transitive model of ZFC. Pairing, union, power set, separation, axiom of infinity and foundation stay within the hierarchy. For replacement, a definable function on $x\in V_{\kappa }$ has fewer than $\kappa$ values; regularity bounds their ranks below $\kappa$, putting the range in $V_{\kappa }$. Ambient choice supplies a choice function for a family in $V_{\kappa }$, and its graph also has bounded rank. This verifies the substantial closure axiom rather than only saying that the rank is large. It also shows why inaccessible existence cannot be proved in a consistent ZFC: it would yield a set first-order model of ZFC and hence its consistency, contrary to the Gödel second incompleteness theorem.

An uncountable regular limit cardinal is a weakly inaccessible cardinal; requiring the strong-limit inequality is stronger without additional hypotheses. Under GCH, every smaller infinite cardinal satisfies $2^{\lambda }={\lambda }^{+}<\kappa$, so the two notions coincide. This distinction separates cardinal-arithmetic assumptions from the regularity requirement.

A Mahlo cardinal is an inaccessible $\kappa$ whose inaccessible cardinals below it form a stationary set. Thus every club set meets those inaccessibles, which strengthens mere unboundedness. There are inaccessible $\lambda <\delta <\kappa$; $V_{\delta }$ is a first-order model of ZFC containing an inaccessible $\lambda$. Mahloness therefore supplies a consistency statement stronger than the bare existence of one inaccessible. Iterating the demand that smaller cardinals have the same reflection property leads to further levels of the hierarchy.

A weakly compact cardinal is a logical and combinatorial strengthening of inaccessibility. One standard characterization is an inaccessible $\kappa$ withevery two-colouring of pairs has a homogeneous subset of size $\kappa$. Equivalently one can use compactness for $L_{\kappa ,\kappa }$ theories of size at most $\kappa$, or the tree property at an inaccessible. These viewpoints explain the terminology: a local collection of compatible requirements has a global realization, and tall narrow trees cannot avoid cofinal branches. They are characterization statements; the elementary closure and measure arguments below are separate proofs of the consequences used here.

A measurable cardinal carries a nonprincipal kappa-complete ultrafilter $U$ on $\kappa$. Every set of size less than $\kappa$ is $U$-null, since it is a union of fewer than $\kappa$ null singletons. Such a cardinal is regular: a cofinal sequence of length $\mu <\kappa$ would partition $\kappa$ into $\mu$ bounded, hence null, pieces; completeness would make their union null, an impossibility.

It is also strong limit. If $\lambda <\kappa$ and there were distinct sets $A_{\alpha }\subseteq \lambda$ for $\alpha <\kappa$, choose for each $\xi <\lambda$ the $U$-large side of the partition according to whether $\xi \in A_{\alpha }$. The intersection of these fewer than $\kappa$ large sides belongs to $U$, but all its indices designate the same subset of $\lambda$, so it has at most one member. This contradicts nonprincipality. Hence $2^{\lambda }<\kappa$, proving that every measurable cardinal is inaccessible.

The measure gives an ultrapower embedding. Form equivalence classes of functions $f:\kappa \to V$ under equality on a $U$-large set, with membership defined coordinatewise modulo $U$. Induction on first-order formulas proves elementarity: Boolean operations use the ultrafilter laws, and the existential step chooses coordinate witnesses on a large set. Countable completeness makes the ultrapower well-founded. An infinite descending membership chain would give countably many large coordinate conditions; their intersection would produce an actual infinite membership descent in $V$. Collapse the ultrapower to a transitive class $M$, obtaining $j:V\to M$.

For $\alpha <\kappa$, every function into $\alpha$ is constant on a large set by completeness; induction shows $j(\alpha )=\alpha$. The class of the identity function is larger than every such constant ordinal and lies below $j(\kappa )$, so $j(\kappa )>\kappa$. Thus the critical point of an elementary embedding is $\kappa$. The measure viewpoint has become a precise self-similarity principle for the universe.

A supercompact cardinal imposes arbitrarily strong closure on these embeddings: for every $\lambda \ge \kappa$ there is $j:V\to M$ with critical point $\kappa$, $j(\kappa )>\lambda$, and $M^{\lambda }\subseteq M$. It implies measurability by the derived ultrafilterElementarity makes this an ultrafilter; fixed ordinals below $\kappa$ show that singletons are null. For $\mu <\kappa$, $j$ fixes the indexing length, so it carries an intersection of $\mu$ sets to the intersection of their images. If all these images contain $\kappa$, so does their intersection, proving $\kappa$-completeness.

These examples illustrate three connected uses of large cardinals: obtaining set first-order models and stronger consistency statements, extending finite or countable compactness and partition phenomena to uncountable domains, and constructing elementary embeddings with controlled closure. They also interact with inner models: an inaccessible remains inaccessible in the constructible universe, since it cannot acquire a new short cofinal sequence there and there are fewer subsets of smaller ordinals. A measurable cardinal's measure need not belong to that inner model, so preservation of the stronger property is a different issue. Large cardinal axioms are thus structured additional hypotheses, with carefully distinguishable consequences and relative-consistency claims, rather than a single claim that sufficiently big cardinals exist.

An inner model of a universe of ZF is a transitive class $M$ containing every ordinal and satisfying the required set-theoretic axioms with the inherited membership relation. Transitivity means that every member of a member of $M$ belongs to $M$. Requiring all ordinals distinguishes an inner model from a transitive set first-order model such as $V_{\kappa }$: a set first-order model has bounded ordinal height. One may specify an inner model of ZF, ZFC or a stronger theory; choice is not built into the term unless specified.

The fundamental example is the constructible universe. Form the constructible hierarchyHere definable power set means subsets definable over the set structure $(L_{\alpha },\in )$ by first-order formulas with finitely many parameters from that level. Each level is transitive, the levels increase, and $\mathrm{Ord}\cap L_{\alpha }=\alpha$, by induction. Thus $L$ is transitive and contains all ordinals.

The construction supplies a definable well-order: order objects first by their earliest construction stage, then by a least first-order formula code and a finite tuple of parameters in the already constructed level's well-order. Recursively extend the earlier well-order at each successor stage and unite compatible orders at limits. First-order formula codes and finite tuples of well-ordered parameters are well-orderable. This yields a canonical global ordering definable within $L$; its restrictions to sets will give choice once the ZF axioms are checked.

For that check, a useful reflection argument is explicit. Given finitely many first-order formulas and a starting level, pass to a higher level containing witnesses for their existential subformulas on all parameter tuples from the current level, choosing least witnesses in the canonical order. There are set-many tuples, so ambient replacement bounds their witness stages. Repeat for $\omega$ steps and take the union level $L_{\theta }$. Every relevant witness with parameters in that level appears at a later step. Induction on first-order formulas, with this witness property for the existential case, proves that $L_{\theta }$ reflects the chosen finite list of first-order formulas of $L$.

Separation on a constructible set is then definability over a sufficiently large reflecting level, so its subset appears at the next level. For replacement, reflect the first-order formula specifying a functional image and its required existence statements. All inputs belong to a level containing the domain as a set; all their unique outputs then lie in one reflecting level. The image is definable there and hence is constructible. For power set, the ambient set of constructible subsets of $x$ has bounded construction stages by replacement in the ambient universe. At a level containing them all, this internal collection is definable by the bounded condition $y\subseteq x$, and therefore is itself in $L$. Pairing, union and axiom of infinity follow directly from definability over sufficiently large levels, while extensionality and foundation follow from transitivity. Thus $L\models \mathsf{ZF}$, and its canonical ordering gives $L\models \mathsf{AC}$.

This proves the important relative-consistency applicationOne does not conclude that choice is true in the original universe. Instead the inner model supplies a possibly smaller universe in which choice holds.

Constructibility also supplies GCH. Here is the cardinal-counting argument, making clear the role of the condensation lemma for the constructible universe. Work inside $L$. If $x\subseteq \kappa$ is constructible, take a large limit $L_{\theta }$ containing $x$ and an elementary hull of $\kappa \cup \{\kappa ,x\}$ of size $\kappa$. Its transitive collapse is $L_{\beta }$ by condensation. The collapse fixes every ordinal below $\kappa$, and hence fixes $x$ and $\kappa$. Since the hull has size $\kappa$, $\beta <{\kappa }^{+}$. Therefore every constructible subset of $\kappa$ appears below $L_{{\kappa }^{+}}$.

The condensation fact expresses the rigidity of this definability construction: first-order formula codes, finite parameter tuples and satisfaction in set-sized lower levels are preserved by the collapse. Inducting over the hull's ordinal indices identifies each collapsed lower level with the corresponding $L$-level; elementarity supplies every definable subset coded by parameters in that collapsed lower level. Taking unions identifies the entire collapsed hull with a level $L_{\beta }$.

Induction on construction stages gives $|L_{\gamma }|\le |\gamma |+{\aleph }_0$: each successor has only countably many first-order formula schemes with finite parameter tuples, and limits take unions. Hence $|L_{{\kappa }^{+}}|={\kappa }^{+}$ for infinite $\kappa$. The preceding containment gives $2^{\kappa }\le {\kappa }^{+}$ in $L$, and Cantor's theorem gives the reverse inequality. ConsequentlyAll these cardinal computations are internal to $L$; they need not equal the ambient cardinal computations.

Inner models are also useful for analyzing stronger axioms. If $\kappa$ is inaccessible in $V$, it remains an uncountable regular cardinal in $L$, because a shorter cofinal sequence in $L$ would already be one in $V$. For each ordinal $\lambda <\kappa$, its constructible subsets are fewer than $\kappa$ even in the internal cardinal comparison: in $V$, $\mathcal{P}(\lambda )$ has size below $\kappa$, so its constructible part cannot have an $L$-well-order of length at least $\kappa$. Thus it remains strong limit. Accordingly $L$ supplies relative-consistency information compatible with inaccessibles and GCH. Stronger embedding or measure properties require extra analysis, since the witnessing objects may disappear upon passing to an inner model.

There is a useful limitation: no uniform inner-model construction can refute choice from every starting ZF universe. Every inner model $M$ of ZF contains the ambient $L$. Indeed, induction gives $L_{\alpha }^M=L_{\alpha }$: the same set-sized level has the same satisfaction relation for definitions, and limit stages take the same unions. If the starting universe already satisfies $V=L$, any inner model containing all ordinals must therefore be the whole universe, where choice holds. To obtain a choice-failing first-order model uniformly one needs a different method, such as the symmetry restriction in the preceding solution. Inner-model arguments establish consistency and preservation through carefully controlled smaller universes; they do not freely manufacture arbitrary desired axiom failures.

The partition relation $\lambda \to (\mu {)}_{\nu }^r$ says that every colouring of the $r$-element subsets of a set of cardinality $\lambda$ into $\nu$ colours has a homogeneous set of cardinality $\mu$. Define the beth numbers over an infinite cardinal $\kappa$ by ${\beth }_0(\kappa )=\kappa$ and ${\beth }_{n+1}(\kappa )=2^{{\beth }_n(\kappa )}$. The Erdős-Rado theorem for finite arities gives the precise boundThus any fixed finite arity and any fixed infinite number of colours admit arbitrarily large infinite monochromatic sets once the starting cardinal is sufficiently large. For pairs, the especially useful case is $(2^{\kappa }{)}^{+}\to ({\kappa }^{+}{)}_{\kappa }^2$. The successor in this bound matters: a binary first-difference colouring on a suitably ordered set of binary sequences supplies the familiar obstruction at $2^{\kappa }$. Also, a theorem for every fixed finite arity is not a theorem homogenizing all arities simultaneously.

The mechanism is to build an end-homogeneous routing tree: each new vertex is routed according to its colours to earlier vertices on its branch. A long branch is end-homogeneous, meaning the colour from a fixed earlier vertex is independent of the later vertex. There are at most ${\kappa }^{\alpha }\le 2^{\kappa }$ routes of length $\alpha <{\kappa }^{+}$, and the whole collection of short routes has size at most $2^{\kappa }$. Inserting $(2^{\kappa }{)}^{+}$ vertices therefore forces a branch of length ${\kappa }^{+}$. One colour occurs at ${\kappa }^{+}$ earlier vertices on that branch, giving the pair theorem. Iterating the higher-arity version of this routing construction explains the successive powers in the Erdős-Rado theorem for finite arities.

A weakly compact cardinal is an uncountable strongly inaccessible cardinal $\kappa$ withEquivalent formulations include the tree property at the inaccessible cardinal and compactness for theories of size at most $\kappa$ in the infinitary language $L_{\kappa ,\kappa }$: if every subtheory of size $<\kappa$ has a first-order model, so does the theory.

A measurable cardinal is an uncountable $\kappa$ carrying a nonprincipal ultrafilter closed under intersections of fewer than $\kappa$ members. Such an ultrafilter contains no set of size $<\kappa$, since all singletons have complements in it and it is kappa-complete. Consequently a cofinal sequence of length $<\kappa$ would partition $\kappa$ into too few small pieces, so $\kappa$ is regular cardinal. If $\lambda <\kappa$ and $\kappa$ distinct subsets of $\lambda$ existed, choose, for each coordinate of $\lambda$, the bit selected by the ultrafilter. Kappa-completeness would make all those bits simultaneously constant on a large set of indices; only one distinct subset could have that bit pattern, a contradiction. Hence $2^{\lambda }<\kappa$: $\kappa$ is also a strong limit cardinal.

A supercompact cardinal $\kappa$ has, for every $\lambda \ge \kappa$, a fine ultrafilter that is a normal ultrafilter on small subsets and is kappa-complete onFine means $\{x:\alpha \in x\}$ is large for every $\alpha <\lambda$. Normal means that a function $f$ with $f(x)\in x$ on a large set is constant on some large set. Equivalently, for every such $\lambda$ there is an elementary embedding $j:V\to M$ with critical point $\kappa$, $j(\kappa )>\lambda$, and $M$ closed under $\lambda$-sequences from the universe.

These notions form the implication chain supercompact $\Rightarrow$ measurable $\Rightarrow$ weakly compact. For the first implication, an embedding as above gives the measure $U=\{X\subseteq \kappa :\kappa \in j(X)\}$. For the second, use a normal ultrafilter on a cardinal obtained by normalizing its measure. Indeed the ultrapower embedding gives the measure $U=\{X\subseteq \kappa :\kappa \in j(X)\}$; for a regressive $f$, $j(f)(\kappa )<\kappa$ is a fixed ordinal, yielding the required constant large fibre. Given a pair-colouring, select for each $\alpha$ the measure-one colour on the tail above $\alpha$. One selected colour occurs on a measure-one set of $\alpha$'s. Normality makes the diagonal intersection of their selected tails measure one; any two members of that intersection have the selected colour. This gives the required partition relation, and the preceding regularity and strong-limit argument gives inaccessibility.

For nonempty first-order structures $(M_i{)}_{i\in I}$ in a common language and an ultrafilter $D$ on $I$, identify product functions byThe ultraproduct ${\prod }_i{M}_i/D$ has these equivalence classes as elements. Functions are interpreted coordinatewise, and a relation holds of classes exactly when its coordinatewise truth set belongs to $D$. Closure under finite intersections makes these interpretations independent of representatives.

The Łoś theorem assertsProve it by induction on first-order formulas. Atomic first-order formulas hold by the definitions and induction on terms. Conjunction uses intersections, and negation uses the fact that an ultrafilter contains exactly one of a set and its complement. For an existential first-order formula, a product witness gives a coordinate witness on a $D$-large set. Conversely, on the $D$-large set where a coordinate witness exists, choose one in each factor, and choose arbitrary values elsewhere. Its class is a product witness by the induction hypothesis. This is the only witness-selection step; the surrounding argument is made in a choice metatheory. The special case with all factors equal is an ultrapower, and constant functions give an elementary embedding.

Here is a direct compactness theorem proof. Suppose every finite subset of a theory $T$ has a model. Index factors by the finite subsets $i\subseteq T$, choosing $M_i\models i$. For finite $F\subseteq T$, the cone $C_F=\{i:F\subseteq i\}$ is nonempty, and $C_F\cap C_H=C_{F\cup H}$. Extend the generated proper filter to an ultrafilter $D$. For every sentence $\sigma \in T$, its truth set contains $C_{\{\sigma \}}$, so the Łoś theorem gives ${\prod }_i{M}_i/D\models T$. Finite satisfiability therefore implies satisfiability, directly from the ultraproduct construction rather than from a syntactic completeness argument.

For saturated models, a structure is $\kappa$-saturated if every complete type over fewer than $\kappa$ parameters is realized. A useful concrete case is countable saturation of a nonprincipal ultraproduct over omega. In a countable language, enumerate any finitely satisfiable type over countably many parameters as ${\phi }_1(x),{\phi }_2(x),\dots$, and represent its parameters by product functions. By the Łoś theorem, the set $A_n$ of coordinates where the first $n$ first-order formulas have a simultaneous witness belongs to the nonprincipal ultrafilter $D$. PutThese are decreasing $D$-large sets. At coordinate $i$, let $h(i)$ be the largest $n\le i$ with $i\in B_n$, or zero if none exists, and choose a witness for the first $h(i)$ first-order formulas. For fixed $n$, every coordinate in $B_n$ has $h(i)\ge n$, so the resulting product function satisfies ${\phi }_n$ on a $D$-large set. Its class realizes the whole type. Thus such an ultraproduct is ${\aleph }_1$-saturated.

There is also an existence construction at every prescribed degree $\kappa$. Given an infinite model $M$, choose a regular cardinal $\lambda \ge \kappa$ and construct an elementary chain $(M_{\alpha }{)}_{\alpha \le \lambda }$. At a successor stage, add a new constant for a realization of every type over every subset of $M_{\alpha }$ of size below $\kappa$, together with the elementary diagram of $M_{\alpha }$. Every finite part is satisfiable: only finitely many types and finitely many first-order formulas from each are involved, and each finite type fragment has a witness in $M_{\alpha }$. Compactness therefore supplies a simultaneous elementary extension. At limits take unions. To justify elementarity of a union, any existential first-order formula with parameters in an earlier stage has a witness in that stage whenever it has one at a later stage, by elementarity; this is the Tarski-Vaught test.

Any fewer-than-$\kappa$ parameters of $M_{\lambda }$ occur together in some $M_{\alpha }$, by regularity of $\lambda$. A type over them which is finitely satisfiable in $M_{\lambda }$ is finitely satisfiable in $M_{\alpha }$, again by elementarity, and was realized in $M_{\alpha +1}$. Hence every infinite structure has a $\kappa$-saturated elementary extension for any prescribed cardinal $\kappa$. No bound on the size of this extension is asserted; full saturation at the model's own cardinality has additional cardinal-arithmetic issues.

The exponent-two Erdős-Rado theorem for pairs says that for every infinite cardinal $\lambda$,In particular, a finite or countable coloring of pairs from $(2^{{\aleph }_0}{)}^{+}$ has an uncountable monochromatic subset. We prove the general statement by an end-homogeneous tree construction.

Let $c:[\mu {]}^2\to \nu$ be any coloring on an ordinal $\mu$. Route a vertex $\beta$ through an increasing sequence of preceding vertices as follows. Start with the least vertex. After ancestors $a_{\eta }$, $\eta <\xi$, have been selected, choose the least remaining candidate $\gamma \le \beta$, above those ancestors, satisfyingThe terminal candidate $\beta$ always qualifies, so the procedure eventually reaches it. Candidate sets shrink and their least elements increase. If an intermediate candidate is $\gamma$, its route is precisely the earlier part of the route to $\beta$, because all earlier color tests agree. Consequently the routes define a set-theoretic tree on the vertices.

Along a branch, the color from an ancestor to every later branch vertex is constant. Also a node of height $\alpha$ is determined by its sequence of $\alpha$ colors to its ancestors: reconstruct the ancestors as successive least candidates with those prescribed colors, then reconstruct the node as the least final candidate. Thus its level has at most ${\nu }^{|\alpha |}$ nodes.

Apply this with $\mu =(2^{\lambda }{)}^{+}$ and $\nu =\lambda$. If every node had height less than ${\lambda }^{+}$, each level would have size at most ${\lambda }^{\lambda }\le 2^{\lambda }$, and there would be at most ${\lambda }^{+}$ such levels. Their total size would be at most $2^{\lambda }$, contradicting $|\mu |=(2^{\lambda }{)}^{+}$. Hence some route has at least ${\lambda }^{+}$ ancestors. For each ancestor mark its constant color to all later route points. Since there are only $\lambda$ colors, one occurs on ${\lambda }^{+}$ ancestors. Those ancestors form the required monochromatic set. This proves the theorem.

To make the ordinal function precise, one useful ordinal partition-bound function isThe target here is ordinal order type. For infinite $\alpha$, the proved theorem with $\lambda =|\alpha |$ supplies a bound $(2^{\lambda }{)}^{+}$: its homogeneous set has order type at least ${\lambda }^{+}>\alpha$, and it handles all the required color counts. Finite targets are handled by finite Ramsey theorem bounds. Thus $f$ is total, nondecreasing and $f(\alpha )\ge \alpha$.

Suppose a sequence of transfinite iterates ${\alpha }_{\xi +1}=f({\alpha }_{\xi })$, with suprema at limit stages, has supremum an uncountable cardinal $\kappa$ at a limit stage, with every earlier iterate below $\kappa$. Thenchoose an earlier iterate at least $\theta$ and use the next one as a bound. If the iteration has already stabilized, it has already found a fixed point instead.

This closure forces $\kappa$ to be a strong limit cardinal. For any infinite $\lambda <\kappa$, color pairs of distinct binary strings of length $\lambda$ by their first differing coordinate. This uses $\lambda$ colors and has no monochromatic triangle: three binary digits at one coordinate cannot be pairwise different. It gives a coloring on $2^{\lambda }$ with no homogeneous target of type $\lambda +1$. Since $\lambda$ colors are allowed when the target is the ordinal $\lambda +1$, we haveFinite exponents satisfy the strong-limit requirement automatically for uncountable $\kappa$.

The needed additional condition is the tree property: every tree of height $\kappa$, with nonempty levels of size less than $\kappa$, has a cofinal branch. In the present convention it forces $\kappa$ to be a regular cardinal. If $\kappa$ were singular, take a cofinal sequence of smaller cardinals of length $\mathrm{cf}(\kappa )$ and form the disjoint union of chains of those lengths. This tree has height and total size $\kappa$, levels of size at most $\mathrm{cf}(\kappa )<\kappa$, but no branch of length $\kappa$, a contradiction.

For any $\nu <\kappa$, apply the preceding color-routing tree to a coloring $[\kappa {]}^2\to \nu$. If a route already has length $\kappa$, use it. Otherwise its levels are indexed below $\kappa$ and each has sizefor infinite exponents, by the strong-limit property; finite cases satisfy the same needed bound directly. The tree has $\kappa$ nodes, so regularity forces its height to be $\kappa$. The tree property supplies a cofinal branch. The ancestor-color partition of that branch has one class of size $\kappa$, since $\nu <\kappa$ and $\kappa$ is regular. Thus $\kappa \to (\kappa {)}_{\nu }^2$ for every $\nu <\kappa$, andWe have proved that an uncountable function-closed cardinal with the tree property is regular and strong limit, hence strongly inaccessible, and that it is the requested fixed point.

The paper's final sentence, read as a claim about the tree property alone, is false. Already ${\aleph }_0$ has it by the König infinity lemma but is not uncountable; even with uncountability stipulated, there are relative-consistency models with the tree property at ${\aleph }_2$. The latter counterexample is documented in the research account of Mitchell forcing. The rigorous conclusion in the iterated-function context is the one just proved: function closure supplies strong limit, and the tree property supplies regularity and a branch. Strong limit and regularity follow from distinct hypotheses in this context.

Put $\theta =\mathrm{cf}(\delta )$. There is a strictly increasing cofinal map $a:\theta \to \delta$. If $\mathrm{cf}(\theta )=\nu <\theta$, choose a cofinal map $b:\nu \to \theta$. Then $a\circ b$ is cofinal in $\delta$, contradicting the definition of $\theta$ as its least cofinal order type. Since always $\mathrm{cf}(\theta )\le \theta$, we obtainFor a nonzero limit ordinal this is an infinite regular cardinal. If the convention includes zero among limit ordinals, its cofinality is zero and the identity is immediate there.

For the intended infinite regular cardinal $\kappa$, set $S_0(\delta )=0$. Suppose $0<\zeta <{\kappa }^{+}$ and the functions for its predecessors have been constructed. Since $|\zeta |\le \kappa$, choose a surjection $e_{\zeta }:\kappa \to \zeta$ and defineThe supremum is below $\kappa$: it uses fewer than $\kappa$ ordinals below $\kappa$, and $\kappa$ is regular. This defines a function $\kappa \to \kappa$ at each stage of the recursion.

For any $\alpha <\zeta$, choose $\eta$ with $e_{\zeta }(\eta )=\alpha$. Whenever $\delta >\eta$, the displayed supremum includes $S_{\alpha }(\delta )+1$, so $S_{\alpha }(\delta )<S_{\zeta }(\delta )$. Therefore the recursion yields a long chain under eventual domination of length ${\kappa }^{+}$.

The least index is $\boxed{\alpha ={\omega }_1}$, withIndeed, for every nonzero limit ordinal $\beta$, continuity of the aleph number enumeration and the cofinality of an increasing ordinal supremum giveIf $\beta <{\omega }_1$ is a nonzero limit ordinal, this cofinality is $\omega$, so ${\aleph }_{\beta }$ is a singular cardinal. The other infinite cardinal numbers below ${\aleph }_{{\omega }_1}$ are successor-indexed or ${\aleph }_0$, and are regular cardinals. For a successor cardinal ${\mu }^{+}$, a cofinal sequence of length at most $\mu$ would express ${\mu }^{+}$ as a union of at most $\mu$ sets of size at most $\mu$, contradicting infinite cardinal arithmetic.

There are ${\aleph }_1$ nonzero countable limit ordinals, and each countable initial segment contains only countably many. In increasing order they therefore have order type ${\omega }_1$. These are precisely the indices of the uncountable singular cardinals below ${\aleph }_{{\omega }_1}$. Hence every ${\sigma }_{\alpha }$ with $\alpha <{\omega }_1$ has countable cofinality, and the next one is ${\aleph }_{{\omega }_1}$, whose cofinality is ${\omega }_1<{\aleph }_{{\omega }_1}$.