All Questions
Tagged with set-theory model-theory
401 questions
60
votes
8
answers
6k
views
Is the ultraproduct concept fundamentally category-theoretic?
Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.
My ...
- 236k
43
votes
1
answer
2k
views
Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?
By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...
- 32.5k
39
votes
7
answers
6k
views
Is V, the Universe of Sets, a fixed object?
When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am ...
- 750
39
votes
3
answers
3k
views
Can one show that the real field is not interpretable in the complex field without the axiom of choice?
We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number ...
- 236k
34
votes
2
answers
3k
views
Ur-elemental surprises
For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ...
- 21.2k
33
votes
15
answers
7k
views
What's a magical theorem in logic?
Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less straightaway&...
33
votes
6
answers
5k
views
Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
- 19.6k
29
votes
4
answers
4k
views
How dangerous are set-size assumptions?
Many students' first introduction to the difference between classes and sets is in category theory, where we learn that some categories (such as the category of all sets) are class-sized but not set-...
- 18.7k
27
votes
4
answers
3k
views
What "metatheory" did early set theory/logic researchers use to prove semantic results?
Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" ...
- 4,897
27
votes
3
answers
4k
views
Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...
- 32.1k
27
votes
1
answer
2k
views
Are Conway's combinatorial games the "monster model" of any familiar theory?
This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE.
If I understand the answer to that question correctly, the surreal numbers have ...
- 4,897
26
votes
6
answers
3k
views
Is there a metamathematical $V$?
As with many of you, I've been following Peter Scholze's recent question about universes with great interest. In ring theory, we don't often have to deal with proper classes, but they occasionally ...
- 18.7k
26
votes
0
answers
1k
views
Where do uncountable models collapse to?
Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
- 21.2k
25
votes
2
answers
1k
views
Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary ...
- 21.2k
24
votes
4
answers
3k
views
A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
- 343
23
votes
1
answer
493
views
Can a non-standard model $M$ of $\mathsf{ZF}$ contain an internally infinite set that is externally smaller than $\aleph_0^M$?
Cardinalities in non-standard models of $\mathsf{ZF}$ are not generally reflected externally, but certain internal facts about cardinalities are always externally reflected (e.g., any internally ...
- 12.5k
23
votes
0
answers
682
views
CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
- 17.7k
22
votes
5
answers
1k
views
What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...
- 236k
22
votes
4
answers
2k
views
Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?
The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...
user21349
21
votes
3
answers
2k
views
Are there as many real-closed fields of a given cardinality as I think there are?
Let $\kappa$ be an infinite cardinal. Then there exists at least one real-closed field of cardinality $\kappa$ (e.g. Lowenheim-Skolem; or, start with a function field over $\mathbb{Q}$ in $\kappa$ ...
- 65.4k
21
votes
1
answer
864
views
Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
- 236k
20
votes
2
answers
2k
views
Is non-existence of the hyperreals consistent with ZF?
I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...
20
votes
5
answers
2k
views
Isomorphism types or structure theory for nonstandard analysis
My question is about nonstandard analysis, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of the nonstandard reals R*, there are of course many non-...
- 236k
19
votes
0
answers
563
views
What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?
Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
- 21.2k
18
votes
3
answers
1k
views
How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?
I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$.
Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
- 1,142
18
votes
1
answer
3k
views
Existence of a model of ZFC in which the natural numbers are really the natural numbers
I know that, from compactness theorem, one can prove that there are models of first order arithmetic in which there is some "number" which is not a successor of zero, in the sense that it is strictly ...
- 1,514
18
votes
3
answers
2k
views
Taking a proper class as a model for Set Theory
When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the ...
- 750
18
votes
2
answers
3k
views
Universe view vs. Multiverse view of Set Theory
Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...
- 1,465
18
votes
1
answer
1k
views
A topological version of the Lowenheim-Skolem number
This is a continuation of an MSE question which received a partial answer (see below).
Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on $...
- 21.2k
17
votes
3
answers
1k
views
Minimum transitive models and V=L
Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. ...
- 7,545
17
votes
1
answer
720
views
Would an oracle for Rayo's function let you compute a model of $(V, \in)$?
Working in Kelly-morse set theory, let $R$ be an oracle that can compute Rayo's function. Can $R$ compute a countable model $M = (\mathbb N,\in_M)$ that is elementary equivalent to $(V, \in)$?
- 6,353
17
votes
1
answer
434
views
End-extension which Mostowski collapses a fake well ordering
Assume $\alpha$ is admissible, $R\in L_\alpha$ is a linear order that don't have any infinite descending chain in $L_\alpha$. Is there always an end extension $M$ of $L_\alpha$, such that $M\vDash KP$ ...
- 1,490
17
votes
0
answers
1k
views
Non-rigid ultrapowers in $\mathsf{ZFC}$?
Originally asked and bountied at MSE:
Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\...
- 21.2k
16
votes
6
answers
2k
views
Application of Fraïssé construction in set theory
As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...
- 1,872
16
votes
4
answers
1k
views
Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?
I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of (...
- 2,111
16
votes
2
answers
1k
views
When does Vopěnka's principle hold?
Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\...
- 21.2k
16
votes
2
answers
1k
views
How special is first-order $\mathsf{PA}$?
This is a modified version of a question which was asked and bountied at MSE without success.
Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic.
There are various "...
- 21.2k
16
votes
3
answers
1k
views
Vopěnka's Principle for non-first-order logics
(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.)
Vopěnka's Principle ($VP$) states that, given any ...
- 21.2k
16
votes
1
answer
746
views
Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?
Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
- 12.5k
15
votes
4
answers
2k
views
Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?
The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
- 21.2k
15
votes
5
answers
2k
views
Intended interpretations of set theories
In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...
- 1,465
15
votes
2
answers
2k
views
Do the real numbers "know" that they are countable in a larger model?
(This was first posted to math.stackexchange but had no answers there after several days):
Let ${\mathbb R}$ be the set of real numbers in whatever is your favorite model of $ZFC$. Then (by Levy ...
- 23k
15
votes
2
answers
2k
views
Is "There exists an unbounded non-measurable set but no bounded non-measurable set" consistent with $\mathsf{ZF}$?
This is a follow-up to this question. We say that a set $A \subseteq \mathbb{R}$ is bounded if there exists a finite interval $(a,b)$ such that $A \subseteq (a,b)$.
Working in $\mathsf{ZFC}$, the ...
- 1,412
15
votes
1
answer
1k
views
Pointwise algebraic models of set theory
Let $\mathfrak{M} = \langle M, E \rangle$ be a structure for the language of set theory, and take some $B \subseteq M$ and $m \in M$. Say that $m$ is definable over $B$ iff there is a formula $\phi(x,\...
- 1,081
15
votes
4
answers
2k
views
Where is the end of universe?
In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
user36136
14
votes
2
answers
1k
views
If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set ...
- 236k
14
votes
2
answers
810
views
Is it possible to define an internal model of ZFC which is not set-like and which is not elementary equivalent to any definable set-like model?
(1) Are there formulae $\varphi_D(x)$ and $\varphi_\in(x,y)$ defining an internal model $\mathcal{N}$ of $ZFC$ where $\mathcal{N}$ is not set-like and no definable, set-like, internal model $\mathcal{...
- 1,142
14
votes
3
answers
2k
views
Tarski's truth theorem — semantic or syntactic?
I was reading the sketch of the proof of Tarski's theorem in Jech's "Set Theory", which appears as Theorem 12.7, thinking that it would be an interesting result to really understand. As stated in the ...
- 18.7k
14
votes
1
answer
839
views
Is there a model of ZFC that can define a "longer" model of ZFC to which it is isomorphic?
Suppose $ZFC$ is consistent. Is there a model $\mathcal{M}$ of $ZFC$ and formulae $\varphi_D(x)$ and $\varphi_\in(x,y)$ that define (in $\mathcal{M}$) the domain and membership relation of a model $\...
- 1,142
13
votes
1
answer
742
views
Is forcing computable?
By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...
- 28.2k