REVIEW 4 minor 1 cited by
On Tameness, Measurability and the Independence Property
T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read A subfield of the real numbers of cardinality continuum defines a non-Borel set of squares in the language of rings, and has the independence property.
desk verdict A correct and genuinely new construction of a subfield of R that is not NIP and ∅-defines a non-Borel set of squares; the main proof is sound, with only two terse spots that are fillable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing construction is Proposition 2.3, a transfinite induction that produces a family $(A_\alpha)_{\alpha<\mathfrak c}$ of pairwise disjoint subsets of $\mathbb R$, algebraically independent over the base field $K$, such that each $A_\alpha$ has inner measure $0$ and the complement of $A_\alpha$ also has inner measure $0$; such sets cannot be Borel. At each induction step, a classical result on Minkowski differences of positive-measure sets is used to place the next real number outside a prescribed Borel set while preserving algebraic independence. The definable non-Borel set is $D=\{y^2:y\in K\}$, defined by the formula $\exists y\, (x=y^2)$. The independence property follows from a classical result that $\mathbb Z$ is definable without parameters in $F(t)$ whenever $F$ is a real field with an archimedean ordering and $t$ is transcendental over $F$.
What would settle it
Check the asserted “without loss of generality” in Proposition 2.3: for $K=\mathbb Q$ and $C=[0,1]$, the Minkowski difference is a neighborhood of 0; one would need to confirm that every such neighborhood contains a $\mathbb Q$-algebraically independent set of cardinality $\mathfrak c$. If some small $\delta>0$ yields only smaller algebraically independent sets, the induction step and hence the main theorem fail.
Extended reading notes
Core claim
The paper's main theorem (Theorem 3.3) asserts the existence of a subfield $K\subseteq \mathbb R$ of cardinality $\mathfrak c$ such that $K$, viewed as a structure in the language of rings $\{+,-,\cdot,0,1\}$, has the independence property and $\emptyset$-defines a set $D\subseteq K$ that is not a Borel set with respect to the order topology on $K$. The set $D$ is the set of squares in $K$, defined by the formula $\exists y\, (x=y^2)$. The field is built as $K=\mathbb Q(\sqrt{A_{\ge 0}}\cup A')$ for two disjoint algebraically independent sets $A,A'\subseteq \mathbb R$ that are non-Borel in a strong sense: each has inner measure $0$ and its complement also has inner measure $0$. The proof shows that if $D$ were Borel, it would have to be approximated by Borel sets in $[0,1]$ with measure both $1$ and $0$ simultaneously, a contradiction. Because $\mathbb Z$ is $\emptyset$-definable in $K$, the independence property follows; because $\mathbb Z$ is definable, $K$ is undecidable.
Load-bearing premise
The construction's transfinite induction assumes, without proof, that at every stage a continuum-sized algebraically independent set can be placed inside the Minkowski difference of the complement of any positive-measure Borel set; if that “without loss of generality” step fails, the non-Borel sets $A_\alpha$ cannot be produced.
Editorial extensions
If this is right
- The set of squares in $K$ is not only non-Borel but also not of the form $M\cap K$ for any Lebesgue measurable $M\subseteq \mathbb R$, since the proof's measure argument goes through with Lebesgue measure.
- $K$ cannot be o-minimal, since every definable set in an o-minimal ordered field is Borel in the order topology; in particular $K$ is not real closed and not almost real closed.
- $\mathbb Z$ is $\emptyset$-definable in $K$, so $K$ is undecidable and has the independence property; hence $K$ is a wild field from the tame-geometry perspective.
- $K$ admits $2^{\mathfrak c}$ pairwise non-isomorphic archimedean orderings and $2^{\mathfrak c}$ pairwise non-isomorphic non-archimedean orderings, so the failure of tameness does not depend on choosing one ordering.
- The motivating question of whether every NIP ordered field has Borel definable sets remains open: the constructed $K$ has the independence property, so it does not settle that question.
Reading between the lines
- The same method would produce a non-Borel set of squares in any subfield $K=\mathbb Q(S)$ where $S$ is a set of continuum many algebraically independent reals arranged so that some of their square roots lie in $K$ and some do not; the transfinite construction is one way to get such $S$, but the definability-to-measurability failure may be much more common.
- If the unproved “without loss of generality” step in Proposition 2.3 is repaired, the construction yields a whole family of $\mathfrak c$ many pairwise disjoint non-Borel algebraically independent sets; this could be used to build many non-isomorphic wild subfields, possibly including some that avoid the independence property if the $\mathbb Z$-definability step were replaced.
- A natural next test is whether an NIP subfield of $\mathbb R$ can define a non-Borel set; the present construction suggests that the obstacle is not the orderings or the Borel $\sigma$-algebra but the algebraic independence of the defining parameters, so an NIP example would need a different definability mechanism.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the relation between model-theoretic tameness and Borel measurability for subfields of R in the language of rings. Its main result (Theorem 3.3) constructs a subfield K of R of cardinality c that has the independence property and ∅-defines the set D of its squares, which is not Borel with respect to the order topology on K. The proof combines a transfinite construction (Proposition 2.3) of continuum many pairwise disjoint, measure-theoretically wild, algebraically independent subsets of R with R. Robinson's theorem that Z is definable in purely transcendental extensions of real fields, and a measure argument showing that D cannot equal B∩K for any Borel B⊆R. The paper also records that K is not o-minimal, is undecidable, and admits maximal families of archimedean and non-archimedean orderings, and it situates the result relative to Shelah's Conjecture on NIP fields.
Significance. If correct, the result provides a striking example that first-order definability in the pure ring language on a subfield of R does not imply Borel measurability, and it clarifies that the independence property can coexist with non-Borel definable sets. The construction is elementary and largely self-contained, using only standard facts such as Steinhaus's Theorem and R. Robinson's theorem, and the measure-theoretic contradiction in Theorem 3.3 is elegant. The paper does not answer Question 1.1 for NIP fields, but it supplies a natural test case and connects the question to Shelah's Conjecture. The main proof is mathematically sound modulo two local justifications discussed below, neither of which affects the central claim.
minor comments (4)
- [§2.1, Proposition 2.3] In the choice of a_{σ,α}, the inference 'Since |D_{σ,α}|<|T_{σ,α}|=c, this yields |U_{σ,α}|=c' is only justified when the field generated by K, D_{σ,α}, and U_{σ,α} has cardinality <c, which holds for |K|<c but not in general for the stated hypotheses. The conclusion is nevertheless correct, because for every t∈T_{σ,α} the witnesses u,v with t=u−v lie in U_{σ,α}, so T_{σ,α}⊆U_{σ,α}−U_{σ,α}, and |U_{σ,α}−U_{σ,α}|≤|U_{σ,α}| for infinite U_{σ,α}; please either replace the argument with this observation or restrict the proposition to |K|<c, the case actually used in Theorem 3.3.
- [§2.1, Proposition 2.3] The 'without loss of generality' step that scales T_{σ,α} into the Steinhaus difference (R\C_σ)−(R\C_σ) should be spelled out: multiplying each t by a sufficiently small nonzero rational q∈Q⊆K preserves algebraic independence over K and cardinality, and Steinhaus's Theorem supplies an interval around 0 contained in the difference.
- [§3, Lemma 3.2] Lemma 3.2 is stated without proof and is used essentially in Theorem 3.3; since it is described as straightforward, please include a short proof (for instance, via the rational function field Q(C∪C′) and unique factorization) or a precise reference.
- [§3, Theorem 3.3] The notation A_{≥0} is used without definition; please define A_{≥0}=A∩[0,∞), and note explicitly that the equalities µ_*(A∩[0,1])=µ_*(A′∩[0,1])=1 follow from Lemma 2.5 together with µ_*(R\A)=µ_*(R\A′)=0.
Circularity Check
No circularity found: the construction of K and D is self-contained, and the only self-citations are motivational.
full rationale
The derivation chain is not circular. Proposition 2.3 constructs the sets A_alpha using Steinhaus's Theorem (external), cardinality facts about Borel sets, and transfinite induction; the non-Borel conclusion is not embedded in the construction. Theorem 3.3 applies Proposition 2.3 (implicitly with base field Q, as allowed by Remark 2.4(a)), builds K = Q(sqrt(A_{>=0}) union A'), and proves D = {y^2 : y in K} is non-Borel by an inner/outer measure contradiction using Lemma 2.1, Lemma 2.5, and Lemma 3.1. The set D is defined as squares, not as 'the non-Borel set', and the measure argument derives D notin B(tau_K) from A_{>=0} subset of D and A' intersect D = empty set; neither inclusion is a fitted parameter. The independence property is imported from R. Robinson's external theorem [22] together with the standard fact that Z has IP, not from a self-citation. The only self-citations ([17], [15], [16]) are motivational or in the Further Work section and are not used as proof inputs. The reader's flagged 'without loss of generality' scaling step in Proposition 2.3 is a potential proof gap about cardinality or algebraic independence, not a circularity: the proposition's conclusion is not assumed in its hypothesis. Hence no circular step is present.
Assumptions & free parameters
assumptions (7)
- standard math ZFC set theory
- standard math Borel measure on R exists with µ((a,b))=b−a
- standard math Steinhaus's Theorem (Fact 2.2)
- standard math R. Robinson's Theorem (Fact 2.7)
- standard math Lemma 3.2: algebraic independence of square roots and non-square property
- standard math Cardinal arithmetic facts: |B(R)|=c, |σ|^2<c for σ<c
- standard math Hölder's Theorem: archimedean ordered subfields of R embed uniquely into R
Cite this review
Pith. "Pith review of On Tameness, Measurability and the Independence Property." pith.science (2026). https://pith.science/paper/RSMDOHZR
@misc{pith2026250608733,
author = {Pith},
title = {Pith review of: On Tameness, Measurability and the Independence Property},
year = {2026},
howpublished = {\url{https://pith.science/paper/RSMDOHZR}},
note = {Machine review of arXiv:2506.08733}
}
abstract
In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield $K$ of the real numbers that lacks several of such tameness properties. As our main result, we present a first-order formula in the language of rings that defines a non-Borel set in $K$. Moreover, $K$ has the independence property and admits both archimedean and non-archimedean orderings.
Forward citations
Cited by 1 Pith paper
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Lipschitzian SLLNs for random functions
Empirical averages of random locally Lipschitz functions converge to their expectation in the Lipschitz pseudometric under separability or NIP-definability conditions, giving uniform convergence of subdifferentials.
Reference graph
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