Questions tagged [model-theory]
Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.
1,320 questions
6 votes
2 answers
693 views
Metamathematical/philosophical understanding of the smallest aleph
Forgive me for asking a perhaps low level question here but I suspect that the answer may be somewhat subtle and my confusions around it are profound. Working over ZFC, say (assuming the well-ordering ...
- 1,653
6 votes
0 answers
184 views
Dévissage in o-minimal structures
In his Esquisse d’un programme Grothendieck advocated a new foundations for topology making it more apt for geometry (…I'm far away from seeing completely through what Grothendieck meant exactly by &...
- 3,854
6 votes
1 answer
310 views
Set size comparison via non-existence of surjections
If $X, Y$ are sets, let us say that $X$ is strictly smaller than $Y$, in symbols $X \prec Y$, if $Y$ is non-empty and for every map $f:X\to Y$ we have $Y\setminus\text{im}(f) \neq \varnothing$. Our ...
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9 votes
1 answer
712 views
How random are Fraïssé limits really?
Up until now, I had just had the intuition that Fraïssé limits were in some sense "random" probabilistic objects in the same manner as the Rado graph. I was told recently that this intuition ...
- 563
11 votes
1 answer
491 views
Two definitions of regularity for ultrafilters
I am interested in the possible equivalence of two definitions of 'regular ultrafilter'. For context, an ultrafilter $\mathcal{D}$ is called $\lambda$-complete if, given any sequence $\langle X_{\...
- 333
6 votes
1 answer
355 views
Vopenka's principle with class-sized structures
Consider the following class-ized version of Vopenka's Principle: ($\mathsf{CVP}$) For every class $\mathcal{C}$ of graphs (or other structures of fixed set-sized similarity type), there are distinct ...
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11 votes
1 answer
608 views
Why aren't these functions o-minimal?
For brevity, say that $f:\mathbb{R}\rightarrow\mathbb{R}$ is o-minimal if the corresponding expansion of the reals $(\mathbb{R};+,\cdot,f)$ is o-minimal. I was surprised to learn that it is open ...
- 19.6k
8 votes
2 answers
905 views
Is an ultrapower essential for defining the hyperreals?
If AC holds, the hyperreals are typically defined using the ultraproduct construction. Without AC, such as in ...
- 1,391
3 votes
1 answer
162 views
Models of Second Order Arithmetic with non-standard length and all subsets of $\omega$?
Are there any non-standard models of RCA$_0$ such that every subset of $\omega$ appears as a restriction of a set in the second-order part to it's standard initial segment? In other words, does ...
- 3,967
12 votes
0 answers
320 views
Is there a best one-variable approximation to commutativity (either from above or below)?
Let $T_R$ be the theory of rings in the language $\{+,\cdot,-,0,1\}$. Let $A$ be the set of one-variable sentences which imply commutativity over $T_R$, and let $B$ be the set of one-variable ...
- 19.6k
22 votes
2 answers
2k views
Which fields satisfy first-order induction?
An amusing observation is that there are actually a fair number of familiar rings that satisfy the axioms of Peano arithmetic (in the language $\{+,\cdot,0,1\}$) except for the assertion that $0$ is ...
- 19k
5 votes
0 answers
283 views
When is "anomalously few substructures" possible?
Given a variety (in the sense of universal algebra) $\mathscr{V}$ axiomatized by a finite set of equations $E$, say that $\mathscr{V}$ is consistently gappy iff it is consistent with $\mathsf{ZF}$ ...
- 19.6k
6 votes
2 answers
524 views
One-variable approximations of commutativity
Say that a Jacobson sentence is a one-variable first-order sentence in the language of rings which modulo the ring axioms implies $\forall x,y(xy=yx)$. Jacobson's theorem provides many nontrivial ...
- 19.6k
8 votes
1 answer
462 views
What are the "sets of $\omega_2$-like ordinals"?
Given an uncountable cardinal $\kappa$, say that $A\subseteq\omega_1$ is a $\kappa$-pseudoclub iff there is some transitive structure $\mathcal{X}=(X; \in,\kappa,...)$ (i.e. $X$ is a transitive set ...
- 19.6k
4 votes
1 answer
238 views
Help in understanding proof that $S$ has this property
I am trying to read Jerome Keisler's book "Model Theory of Infinitary Logic". I got to the part about the undefinability of well-orders, and there I got stuck. For completeness of this post, ...
- 217
1 vote
0 answers
140 views
Non-algebraic proof of Robinson consistency for Infinite-valued Propositional Logic of Łukasiewicz
Recently I was looking at the proofs of Robinson Joint Consistency Theorem (henceforth RJCT) for first-order logic (and related systems) and, I was under the (mistaken) impression that a proof of this ...
- 135
1 vote
0 answers
141 views
Can there exist a Gitik model of ZF with the following nested property?
This is a continuation to a prior posting titled Can we have nested singular sets in a Gitik model? Is there a Gitik model $M$ of ZF, where $M$ satisfy the existence of some infinite set $A$ that has $...
- 13k
4 votes
2 answers
248 views
Is every external downshifting elementary embedding $j$ with $j(x)=j[x]$, an automorphism?
If $M$ is a model of $\sf ZF$, and $j:M \to M$ is an external elementary embedding that moves an $M$-ordinal $\alpha$ downwardly, i.e. $j(\alpha) <^M \alpha$. Suppose, we add that $j(x)=j[x]$ for ...
- 13k
2 votes
0 answers
116 views
Equivalence among $\tau$-theory, elementary topos and Mitchell-Bénabou language
In Johnstone's book "Sketches of an Elephant: A topos theory compendium, volume 2" (referred as Elephant), he defined a higher-order typed (intuitionistic) signature (simplified as $\tau$-...
- 29
9 votes
0 answers
248 views
Which sets of sentences can be "continuously" decided in an ultraproduct?
Motivated by Łos' theorem, given a countable sequence of structures $\mathscr{A}=(A_i)_{i\in\mathbb{N}}$ each in the same language say that a second-order sentence $\varphi$ in that language is Ł-...
- 19.6k
3 votes
0 answers
176 views
Can every elementary embedding be extended to an isomorphism? [closed]
Let $L$ be a first order language. Given an elementary embedding $h:M\to N'$ between two $L$-structures, the range $h[M]$ is an elementary substructure of $N'$, and is isomorphic to $M$. Can this ...
8 votes
0 answers
188 views
Lower bounds on positive $\sin$-polynomials
I am aware that the structure $(\mathbb{R},+,\cdot,<,\sin)$ is extremely wild. Indeed, since the natural numbers are definable in such structure, one can define, via first-order formulas without ...
- 457
2 votes
0 answers
157 views
Is there a general way to translate well-founded and non-well founded models of theories?
In the Boffa construction for proving NFU, we have a model $M$ of MacLane set theory with an external downshifting automorphism on it. Now, clearly $\in^M$ is not $\in \restriction \! M$, where $\in$ ...
- 13k
3 votes
2 answers
268 views
Indiscernible sequences as orbits of automorphisms
I just started reading Simon's A Guide to NIP theories, and in the proof of Lemma 2.7 Simon uses the fact that any monotonic injection $\tau:I\to I$ defines a partial automorphism such that $a_i\...
- 563
5 votes
1 answer
332 views
Can we have nested singular sets in a Gitik model?
Working in a Gitik model of $\sf ZF$, where every set has a countable cofinal. Can we have some non-well-orderable set $A$ that has $\mathcal P(A)$ being a countable union of nested sets each of ...
- 13k
1 vote
0 answers
110 views
Clone isomorphism and definitional equivalence
Let $T$ and $T'$ be two equational theories, such that there is a isomorphism between the clones $\mathrm{Cl}(T)$ and $\mathrm{Cl}(T')$ associated with these theories. Are these theories $T$ and $T'$ ...
5 votes
0 answers
193 views
Rationality and stable rationality in first-order logic
A very important problem in the intersection of (birational) algebraic geometry (function fields), algebra (field theory) and logic is the Elementary Equivalence vs Isomorphism Problem of Fields. ...
- 3,758
9 votes
1 answer
367 views
Self-referential theories (in a Lindenbaumian sense)
Say that a first-order theory $T$ in the language of Boolean algebras is LSR ("Lindenbaum self-referential") iff the Lindenbaum algebra of sentences of $T$, construed as a Boolean algebra in ...
- 19.6k
6 votes
1 answer
319 views
Löb's Theorem and Large Cardinal Analogs for Second Order Arithmetic
Löb's Theorem tells us we can't consistently supplement a sufficently strong theory with the schema $$\operatorname{Prov}(\phi) \implies \phi$$ at least when $\operatorname{Prov}$ captures provability ...
- 3,967
6 votes
1 answer
428 views
Logic with an ambient structure: downward Löwenheim–Skolem
Given an uncountable first-order structure $\mathfrak{M}$ in a countable language, say that an $\mathfrak{M}$-structure is a countable-language expansion of a substructure of $\mathfrak{M}$ with ...
- 19.6k
4 votes
0 answers
321 views
On an invariant of $\aleph_0$-categorical theories
Let $T$ be an $\aleph_0$-categorical theory (complete, and in a countable language) and fix $\mathscr{M}=(M;...)\models T$. For each tuple $\overline{\varphi}$ of formulas such that $(M;\overline{\...
- 19.6k
7 votes
1 answer
263 views
Non-conservative subrings of $\mathbb{C}$
This is a follow-up to this answer of mine. Say that a small ring is a subring $A$ of $\mathbb{C}$ such that $\mathbb{C}$ has infinite transcendence degree over the algebraic closure of $A$. Is there ...
- 19.6k
3 votes
1 answer
120 views
Does definable compactness imply that a definable open cover parameterized by a definable set has a finite subcover?
Let $\mathcal M$ be a sufficiently saturated o-minimal structure. In the paper Generic Sets in Definably Compact Groups, Y. Peterzil and A. Pillay introduced the following: Let $X\subseteq M^n$ be a ...
1 vote
0 answers
291 views
Set-theoretic geology on ZFC subsystems
What is known now about the set-theoretic geology of $\sf ZFC$ subsystems? It is now known that $\sf ZFC^-$ is violate ground model definability in many cases. Is "$\sf ZFC^-+\neg Pow$+All ...
- 1,391
2 votes
0 answers
161 views
Searching for Literature on NFU
I am searching for an introductary text on NFU, that investigates NFU as a set of Axioms formulated in a formal language. I found a text from Randall Holmes, but by skimming trough it, it appeared to ...
- 21
3 votes
2 answers
390 views
The existence of a partial order with "bounded self-saturation"
I want to know whether a partial order with bounded suborders of suitably many isomorphism types exists. More precisely, let $\mathbb{P} = (P, \leq_{\mathbb{P}})$ be a partial order and $\kappa$ be a ...
- 1,360
7 votes
1 answer
397 views
When do the "Gaifman metrics" not vary much?
Given a structure $\mathfrak{A}=(A;...)$ in a finite language $\Sigma$, let $\Phi_\mathfrak{A}$ be the set of all finite tuples of $\Sigma$-formulas $\overline{\varphi}$ such that $\mathfrak{A}_{\...
- 19.6k
3 votes
0 answers
252 views
Can this extension of ZC evade having distinct bi-interpretable extensions?
What property should an extension of $\sf ZC$ have in order for it to evade having distinct yet bi-interpretable extensions. Which might be seen as a merit by some, foundationally speaking. Is ...
- 13k
12 votes
3 answers
717 views
History of invariant types in model theory
For a first-order structure $M$, $p\in S_x(M)$ is an invariant type if there is a 'small' set $A\subseteq M$ (usually 'small' means $\lvert A\rvert<\lvert M\rvert$) such that, for all $g\in\...
- 2,512
8 votes
1 answer
317 views
On varieties of lattices admitting "large" free complete members
Let $E$ be an equational theory (in the sense of universal algebra) in the language of lattices. Given a cardinal $\kappa$, say that $E$ is $\kappa$-cheap iff there is a set-sized complete lattice $L$ ...
- 19.6k
5 votes
1 answer
418 views
Reverse Chang's conjecture
The two-cardinal transfer property $(\kappa,\lambda)\rightarrow(\kappa',\lambda')$ says "any structure of type $(\kappa,\lambda)$ has an elementarily equivalent structure of type $(\kappa',\...
- 541
5 votes
0 answers
196 views
Failures of $\square$ from Chang-type reflection
For infinite cardinals $\nu \leq \lambda, \mu \leq \kappa$, let $\langle \kappa, \mu \rangle \twoheadrightarrow \langle \lambda, \nu \rangle$ be the assertion that, whenever $\langle f_i: i < \...
- 2,033
6 votes
2 answers
346 views
Must a non-elementary chain model have an elementary submodel with the induced chain being elementary
Let $L$ be a countable relational language. Let $\kappa$ be an uncountable regular cardinal. Suppose that $\langle M_\alpha:\alpha\leq\kappa\rangle$ is a strictly increasing and continuous chain of $...
- 2,665
3 votes
1 answer
198 views
Is a closed, normal subgroup of a definably compact definable group also definable?
Fix a saturated, o-minimal structure $\mathcal M$ with definable Skolem functions/definable choice (the last two being equivalent by [1]). Let $G \subseteq M^n$ be a definable group that is definably ...
11 votes
2 answers
865 views
Axiomatization of Euclidean geometry
The motivation of this question is finding an axiomatization of Euclidean geometry. I consider Tarski's axioms a satisfactory axiomatization of those parts of Euclidean geometry that do not include ...
- 113
4 votes
1 answer
259 views
Is $\{\langle\omega^M,(2^\omega)^M;\in^M\rangle\mid M\models\mathsf{ZFC}\}$ an elementary class?
We work in $\mathsf{ZFC}+\operatorname{Con}(\mathsf{ZFC})$. If $M$ is a model of $\mathsf{ZFC}$, then denote by $R_M$ the two-sorted first-order structure $\langle\omega^M,\mathcal{P}(\omega)^M;{\in^M}...
- 2,512
2 votes
1 answer
332 views
Does the field of rational numbers have NIP?
Does $(\mathbb{Q},0,1,–,+,×)$ have NIP (Not the Independence Property)? I cannot find an answer let alone a proof on the Internet.
- 123
5 votes
0 answers
210 views
Theories ambivalent to formula complexity
A common aspect of mathematical reasoning that formal proof environments can struggle to capture is the way mathematicians often work without paying any attention to the underlying formal language -- ...
- 3,967
4 votes
1 answer
370 views
Morley degree of variety
Let K be an expansion of an algebraically closed field. Let V be an irreducible subvariety of K^n. Assume that K is strongly minimal. Is the Morley degree of V equal to 1? What happens if we moreover ...
2 votes
1 answer
167 views
On "unary-representable" relation algebras
This question is belatedly inspired by an answer of Keith Kearnes. Say that a relation algebra $\mathcal{A}=(A;0,1,\check{\Box}, \overline{\Box},I, \circ, \wedge,\vee)$ is unary-representable if ...
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