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New Foundations uses ordinary extensionality and comprehension restricted to stratified formulas. Its atom-permitting weakening is New Foundations with urelements.

We give a relative consistency proof: a model of ZFC yields a model of NFU. This proves the requested consistency relative to the usual background set theory; it does not claim the optimal, much weaker consistency bound for bare NFU.

Recall that New Foundations with urelements has a unary set predicate $S$, atoms without members, extensionality restricted to sets, and comprehension for every stratified formula. Stratification assigns integer types to variables, with equal types at equality and adjacent types at membership. We construct a model of simple typed set theory with atoms with a type-shifting automorphism, then identify its types using that automorphism.

Start with a model of ZFC and a Skolem expansion of it. By the compactness theorem there is a model $N$ of the same expanded theory with an increasing integer-indexed order-indiscernible sequence of ordinals $⟨c_i:i\in \mathbb{Z}⟩$, all above $\omega$. Here is the finite-satisfiability argument. A finite fragment names only finitely many $c_i$ and asks for equal truth values of finitely many formulas on increasing tuples. In the starting model choose an infinite increasing sequence of ordinals above $\omega$. Colour its finite subsets by these truth vectors, successively treating each tuple length. The Ramsey's theorem supplies an infinite homogeneous subsequence for the finitely many colours and arities. Its first required number of elements interprets all named $c_i$ and satisfies that fragment. Compactness gives the entire sequence. This argument works for the countable expanded language because each fragment uses only finitely many formulas.

Replace $N$ by the Skolem hull of the $c_i$. The mapis well defined by indiscernibility: equal terms remain equal after shifting their increasing lists of indices. The same argument preserves every formula. Shifting indices backwards gives its inverse, so $j$ is an automorphism of the hull, and $j(c_i)=c_{i+1}>c_i$. The Skolem hull is elementary by its witness functions. This is a concrete Ehrenfeucht-Mostowski model; it is not an elementary class self-embedding of the actual well-founded universe.

Set $D_i=V_{c_i}^N$, interpreted externally as the objects belonging to that internal rank segment, for every integer $i$. At type $i$ declareand define adjacent-type membership byThe members of a type-$i+1$ set are exactly a subset of $D_i$; a type-$i+1$ atom has no ${\epsilon }_i$-members. Every internal subset of $D_i$ is an object of $D_{i+1}$, since $c_{i+1}>c_i$. Extensionality for sets is inherited from $N$. For a typed formula involving finitely many types, restrict its quantifiers to the corresponding internal $D_k$ and express each $S_k$ and ${\epsilon }_k$ by the displayed definitions. Separation in $N$ gives the required subset of $D_i$, which represents its comprehension at type $i+1$. This verifies the typed theory even though many higher-type objects are atoms. The automorphism $j$ carries $D_i,S_i,{\epsilon }_i$ to the next type.

On domain $D_0$, define $S(x)=S_0(x)$ andIf $S(y)$ is false, $S_1(j(y))$ is false, so $y$ has no $E$-members. For sets $y,z$, equality of their $E$-extensions gives equality of the typed sets $j(y),j(z)$ and hence $y=z$. To check stratified comprehension, translate a variable of type $k$ into $j^k(x)$. The automorphism laws make membership and equality translate exactly. A stratified predicate on a variable of type zero defines a typed subset $B$ of $D_0$, represented by a set at type one. Then $b=j^{-1}(B)\in D_0$ has $E$-extension exactly that predicate. A uniform type shift handles any other type assigned to the comprehended variable. The resulting untyped structure is a model of NFU.

Use a relative consistency construction: a model of ZF gives one of ZFC through $L$, and we construct a model of New Foundations with urelements from it. In the language with membership and a unary predicate $S$, NFU requires that atoms have no members, extensionality holds between sets, and every stratified formula has a set extension. A type assignment gives equal variables equal types and raises type by one across membership. We prove the actual comprehension scheme, rather than invoke its consistency by name.

Start with a model $M\models \mathsf{ZFC}$ and Skolemize it. Apply the Ramsey construction from Question 1 to obtain increasing ordinal indiscernibles $(c_i{)}_{i\in \mathbb{Z}}$, all greater than $\omega +\omega$. The finite satisfiability step uses the infinite increasing list $\omega \cdot 3,\omega \cdot 4,\dots$ in the original model. Include the ordinal and ordering requirements in the finite truth colorings. Ramsey and compactness produce the indiscernibles; their Skolem hull $N$ still models ZFC. The index shift gives an external structure automorphism $j$ of $N$ with $j(c_i)=c_{i+1}$. Such a model is necessarily externally ill-founded: $(c_0,c_{-1},c_{-2},\dots )$ is an external descending ordinal sequence. No well-founded-model hypothesis is being smuggled in.

For every integer $i$, let $D_i$ be the external domain consisting of the objects of $N$ belonging internally to $V_{c_i}^N$. At type $i+1$, declare $S_{i+1}(y)$ iff $N$ says $y\subseteq V_{c_i}$. Use membership of $N$ only for flagged targets; an unflagged object has no typed members. A flagged object is a set of the preceding type. Set-extensionality holds since all members of a flagged set lie in $D_i$.

Every typed comprehension instance holds. Translate its finitely many typed quantifiers into quantifiers bounded by the appropriate sets $V_{c_r}^N$. Separation in $N$ gives its extension $A\subseteq V_{c_i}^N$ for the distinguished free variable of type $i$. Since $c_i+1\le c_{i+1}$, this subset is an object of $D_{i+1}$ and satisfies its set predicate. Also $j:D_i\to D_{i+1}$ is bijective and preserves the typed relations and flags. This is the required type-shifting automorphism.

Collapse to the single domain $D_0$ by definingIf $S(y)$, its image $j(y)$ is a subset of $V_{c_0}$, so its entire extension is visible in $D_0$. The flag is necessary: an unflagged object might otherwise have a partial visible extension. Flagged coextensive objects have equal images under $j$, by extensionality in $N$, and hence are equal; atoms have no $E$-members.

Now take a stratified untyped formula $\phi (x,\bar{a})$. Shift its type assignment so that $x$ has type zero, allowing negative integer types for other variables. Send a variable of assigned type $r$ to $j^r$ of its untyped value. Equality is preserved, and membership $uEv$ translates to flagged ordinary membership from type $r$ to type $r+1$. A set predicate translates to its flag at that type. Quantifiers correspond bijectively under $j^r:D_0\to D_r$. Thus induction on the formula gives exact agreement between its untyped truth and this typed translation.

Typed comprehension supplies an object $A\in D_1$, flagged at type one, containing exactly the type-zero solutions. Let $b=j^{-1}(A)\in D_0$. It is flagged at type zero, and for every $x\in D_0$, $xEb$ iff $x\in A$ iff $\phi (x,\bar{a})$. This proves all stratified comprehension, including an empty set and the universal set. We have constructedIf infinity is included in the convention for NFU, the same construction supports it: $\omega$ and its successor graph lie far below $c_{-1}$ and are fixed by $j$, since they are definable without parameters in $N$. The domain of the new set $\omega$ is the old $\omega$, since $j(\omega )=\omega$. An ordered pair in the new membership relation is $j^{-2}$ of the old Kuratowski pair. The old successor graph is fixed as a whole by $j$, so its elements are exactly these recoded pairs $(n,n+1)$ as $n$ ranges over $\omega$. It therefore supplies an actual new successor injection with proper range. This does not assume that $j$ fixes each nonstandard natural number. This is the rank-shifting automorphism construction of NFU; its logical ingredients are precisely the proved indiscernible-model construction and ordinary set separation in the typed domains.