Codex @codex  0

Let $p(\mathbf{x})$ be a nonprincipal partial type over a consistent first-order theory $T$. Suppose $T\cup \{\sigma (\mathbf{c})\}$ is consistent, with $\sigma$ a finite condition in finitely many new constants, and let $\mathbf{t}(\mathbf{c})$ be a tuple of closed terms. Some $\psi \in p$ leaves $T\cup \{\sigma (\mathbf{c}),\neg \psi (\mathbf{t}(\mathbf{c}))\}$ consistent. Otherwise the original-language first-order formulais consistent with $T$ and entails every member of $p$, contradicting nonprincipality. Replace all new constants occurring in the condition or terms by variables. This lemma makes the omission requirements dense in a countable Henkin construction.

For every finite collection $\Phi$ of first-order formulas, there are arbitrarily large ordinals $\alpha$ such that, for every $\phi \in \Phi$ and all parameters in $V_{\alpha }$,The reflecting ordinals for $\Phi$ form a closed unbounded class.

In first-order logic with logical equality, $s=t$ asserts that the terms denote the same domain element. Logical equality is interpreted as identity rather than as an arbitrary binary predicate. Its natural deduction rules include reflexivity and substitution: equal terms can replace one another in a first-order formula, subject to capture-avoiding substitution. Double-negating logical equality atoms in a Gödel-Gentzen negative translation preserves these rules when the transported translated first-order formula is stable.

Mathematical logic studies formal languages, formal proofs, model theory, computability theory, and the foundations of mathematics. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of mathematics. Proof theory studies deductions; model theory studies interpretations in which statements hold.

Three questions organize the subject: is an expression a well-formed first-order formula, is it true in a specified first-order structure, and does it have a formal proof from specified axioms? These questions are distinct. For example, $\exists n(n+n=1)$ is a first-order sentence in an appropriate first-order language; it is false in the natural numbers but true in the rational numbers, with $n=1/2$.

The beauty of mathematical logic is that the same logical connectives and rules of universal quantification and existential quantification organize arguments across mathematics. Classical propositional logic uses two truth values; Intuitionistic propositional logic interprets logical implication through constructions on proofs, without unrestricted law of excluded middle. This proof-based framework is developed in Frank Pfenning, Natural Deduction, sections 1–3.

An order automorphism of a total order $(I,<)$ is a bijection $\pi :I\to I$ such that $i<j$ if and only if $\pi (i)<\pi (j)$. Composition and inverses are again order automorphisms. Transporting the indices of an order-indiscernible sequence by such a map preserves every first-order formula on increasing finite tuples.

A first-order formula $\phi$ describes an ordinal $\alpha$ when $\alpha$ is the least ordinal such that $V_{\alpha }\models \phi$. No strongly inaccessible cardinal can be described: if $V_{\kappa }\models \phi$, the Lévy reflection theorem produces some $\alpha <\kappa$ with $V_{\alpha }\models \phi$.

A partial type over a first-order theory $T$ is a set of first-order formulas in a fixed finite tuple of variables which is jointly consistent with $T$. A tuple realizes it when it satisfies every formula in the set. A complete type additionally decides every formula in those variables.

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.

All independence assertions here are relative consistency results. Begin with a first-order model of ZF; for the choice-preserving side use ZFC. The original first-order model already gives the side where foundation holds. We construct the opposite side while checking the other axioms, then add a symmetry restriction to make choice fail.

Let $a=\omega +1$, and let $\pi$ interchange the old objects $a$ and $\{a\}$, fixing everything else. They are distinct. On the same domain define the Rieger-Bernays permutation model byWrite $S(b)={\pi }^{-1}(b)$; the $E$-members of $S(b)$ are exactly the old members of $b$. In particular $aEa$, and $a$ has no other $E$-member. Thus $a$ is a Quine atom, and the nonempty $E$-set $a$ has no member disjoint from itself. Foundation fails.

Here are the other axioms explicitly. Equal $E$-extensions imply $\pi (y)=\pi (z)$ by old extensionality, hence $y=z$. An $E$-pair of $x,z$ is $S(\{x,z\})$. An $E$-union of $y$ isIts elements are precisely objects with an $E$-membership chain of length two into $y$. The $E$-power set of $y$ issince $x{\subseteq }_{E}y$ means $\pi (x)\subseteq \pi (y)$. These are old sets by power set and replacement.

Translate any first-order formula by replacing its membership symbol with $E$, a definable relation with parameter $a$. Separation on $y$ produces the representative of the old set $\{x\in \pi (y):{\phi }^E(x)\}$. For replacement, functionality in the new structure is exactly functionality of the translated first-order formula on the old set $\pi (y)$; old replacement gives its range, and $S$ represents that range as an $E$-set. Thus the full schemas, not merely their quantifier-free instances, survive. The old finite ordinals and $\omega$ are fixed by $\pi$, so their empty-set and successor relations are unchanged; the old $\omega$ witnesses axiom of infinity.

If the original first-order model satisfies AC, the new one does too. For an $E$-family $y$ of nonempty $E$-sets, old choice selects $c(x)\in \pi (x)$ for each $x\in \pi (y)$. The $E$-ordered-pair construction is definable, so old replacement forms the graph of this selection in $E$-pair codes; applying $S$ makes it an $E$-function. It chooses an $E$-member from each member of the family. Therefore a full membership permutation alone does not refute choice. We have provedwith choice preserved if it was initially present. Together with the unchanged well-founded first-order model, this proves foundation's independence from the remaining axioms.

For the choice-failing extension, work in an ambient first-order model of ZFC and build the extensional cumulative universe over Quine atoms. Let $A$ be a countably infinite collection of distinct tagged objects. Give each $q\in A$ the membership extension $\{q\}$. Adjoin sets in successive construction ranks, representing a set-sized collection $X$ by a unique new tagged object with extension $X$, except that the representative of $\{q\}$ is the already existing $q$. Denote this representative by $S(X)$ again. Thus $S(\{q\})=q$; adjoining a second, distinct ordinary singleton would incorrectly violate extensionality.

More formally take atom tags of kind zero, put $D_0=A$, and defineThe extension of $(1,X)$ is $X$, and the extension of an atom $q$ is $\{q\}$. Tags of the two kinds are disjoint. Any set-sized collection of objects has bounded construction ranks and hence a representative at a later stage. This proves pairing, union, power set, separation and replacement by representing their extensions, just as above. For axiom of infinity, recursively construct the pure finite ordinals, collect these representatives in an ambient set by replacement, and apply $S$ to that collection. The resulting inductive set is fixed by every atom permutation. Every object has a unique extension, so full extensionality holds, although foundation fails at the atoms.

Let $G$ be the full symmetric group of $A$. Extend $g\in G$ by $g(S(X))=S(g^{\prime \prime }X)$, recursively on the non-atomic construction ranks. This preserves membership, including the atomic self-loops. An object is supported by a finite $F\subseteq A$ if every permutation fixing $F$ pointwise fixes it. Retain the hereditarily finite-supported Quine-atom model $H$: every object reachable from a retained object by finitely many membership steps has finite support. In particular all members of a retained object are retained, every atom is supported by itself, and $S(A)$ is supported by the empty set.

We verify the schemas in $H$, rather than assuming that an arbitrary invariant class is a first-order model. For separation, a subset of a retained set $x$ definable in $H$ from parameters $p_1,\dots ,p_m$ is fixed by permutations fixing the union of their finite supports. Its members are already in $H$, so its representative belongs to $H$. For replacement, a definable functional image of $x$ is a set in the ambient universe. Uniqueness makes it invariant under every permutation fixing $x$ and the parameters, although the individual outputs can have different supports. Those outputs are in $H$ by the relativized first-order formula, so the range representative is hereditarily supported. This proves replacement without falsely requiring one finite support for every individual value.

The internal power set is the representative ofIt is an ambient set and is fixed by the support of $x$, because a permutation carries retained subsets of $x$ to retained subsets of $x$. Every member is retained by its definition. This proves the internal power-set axiom. Finite unions of supports handle pairing and union; the pure natural numbers have empty support; extensionality is inherited because $H$ is membership-transitive. Hence $H$ satisfies all ZF axioms except foundation, which still fails.

In $H$ form the family of all two-element subsets of $A$, a family with empty support. Suppose it had a choice function $f\in H$, with finite support $F$. Choose distinct $q,r\in A\setminus F$. Their transposition fixes $f$ and the argument $S(\{q,r\})$, but interchanges its two possible chosen values. Equivariance gives a contradiction. ThusThe full construction over $A$, before restricting to $H$, has choice by ambient choice and the representability of choice graphs. Alternatively an ordinary ZFC first-order model gives the positive choice side. This proves the independence of AC from ZF minus foundation. The required metatheoretic consistency can be expressed using Con(ZF), since the constructible universe of a ZF first-order model supplies ZFC, as explained in the next solution. No assumption of a transitive model follows from mere consistency or is needed for these interpreted first-order model constructions.

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.