This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...

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13
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315 views

Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
13
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0answers
326 views

Is every model of modular arithmetic either even or odd?

Modular Arithmetic (MA) has the same axioms as first order Peano Axioms (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
11
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0answers
362 views

Formulations of Singular Cardinals Hypothesis

I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are: SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
9
votes
0answers
158 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
8
votes
0answers
150 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
7
votes
0answers
45 views

Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
7
votes
0answers
79 views

Carlson's model and Sierpinski sets

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by ...
7
votes
0answers
250 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
6
votes
0answers
111 views

Skolem Hulls in $H_{\omega_2}$

Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \epsilon, \prec, f_0, f_1, ...)$, some expansion of $H_{\omega_2}$ in a countable language, with $\prec$ giving a well-order. Does there ...
6
votes
0answers
112 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
6
votes
0answers
148 views

Large Cardinal Inequalities

Solovay showed that the existence of $0^\dagger$ follows from the existence of two measurable cardinals. We know existence of a measurable cardinals is weaker than existence of $0^\dagger$ so we ...
6
votes
0answers
75 views

On locally small category

Maybe this is a very trivial question for those who are familiar with Set theory. Let $C$ be a category, and $\hat{C}$ be its presheaves. I hear that if both $C$ and $\hat{C}$ are locally small, ...
6
votes
0answers
207 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
6
votes
0answers
107 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
6
votes
0answers
219 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
6
votes
0answers
136 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
6
votes
0answers
147 views

Half of a Cohen real

I recently heard from a friend that Zapletal gave a talk at Toronoto where he constructed a proper forcing which adds a real which infinitely often equals every ground model real but doesn't add a ...
6
votes
0answers
94 views

Why can countable recursive constructions not be done using countable choice?

Why can countable recursive constructions not be done using countable choice? For example, replace $Z$ by $\omega$ in the following theorem Why can't one prove it using countable choice? (The proof ...
6
votes
0answers
298 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
5
votes
0answers
164 views

Largest provably existing ordinal in ZF without power set

If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ...
5
votes
0answers
112 views

Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ...
5
votes
0answers
156 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
5
votes
0answers
318 views

Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
4
votes
0answers
64 views
+50

What happens if we replace “regularity” in GCH with other conditions?

We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is regular. GCH1. If $\kappa$ is an infinite cardinal ...
4
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0answers
52 views

Do I run into set-theoretic problems if I allow Homs in my category to be proper classes?

I speak very little set theory, so please , have patience with potential misconceptions below. Let $C$ be some category. This is, as we all know, given by a class of objects $Ob(C)$ and for each $a,b ...
4
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0answers
48 views

Full Theory of Structure of Set Theory and Generic Extensions

Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of ...
4
votes
0answers
89 views

What axioms are necessary to prove Mostowski Collapse?

I've been reading Schimmerling's A Course on Set Theory, and have been enjoying it a lot so far. However, some times he's less clear than I think he could be. For example, during a proof of Mostowski ...
4
votes
0answers
46 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
4
votes
0answers
144 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
4
votes
0answers
80 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
4
votes
0answers
76 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
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0answers
80 views

Infinite “String” of Implication Statements

This question is inspired by the conversations at Does this require transfinite induction? First of all, does an infinite string of implication statements have a conclusion? I don't think so, but I ...
4
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0answers
76 views

Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
4
votes
0answers
76 views

What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
4
votes
0answers
62 views

What is the name of the function that indexes Grothendieck universes?

Assume Tarski-Grothendieck set theory. Then Grothendieck universes form a well-ordered proper class, so we can let $U_\alpha$ denote the $\alpha$'th Grothendieck universe, where $\alpha$ is an ...
4
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0answers
1k views

Lists of open problems in set theory

Are there any publicly available lists of open problems in set theory besides the following ones? (And if so, what are they?) http://www.math.wisc.edu/~miller/res/problem.pdf ...
4
votes
0answers
84 views

Width and height of partial ordered sets

The width $w$ of a partial ordered set(poset) is defined as the cardinality of the maximum antichain. By Dilworth Theorem, we know it is equivalent to the minimum number of chains in any partition. ...
4
votes
0answers
126 views

Set theoretic arguments to prove the existence of a certain null set

Let me recall the well-known Carleson's theorem. Theorem (Carleson). Let $f$ be any periodic $L^2[0, 2\pi]$ function. Let $\hat{f}(n)$ be its Fourier coefficients. Then we have $$\lim_{N \to ...
4
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0answers
172 views

Property of Galvin-Hajnal rank

In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$. Lemma 2.2.5 of Introduction to Cardinal ...
3
votes
0answers
52 views

Is it known if “Homotopy type theory” (HTT) can consistently model objects beyond V?

I read the free book on HTT but could not find an answer to this question. If that were the case HTT would be stronger than ZFC+LC (for any choice of the Large Cardinal axiom). That would be ...
3
votes
0answers
48 views

Absoluteness of Satisfaction Relation for Models of the type $J_\alpha$

Is the satisfaction relation absolute between $J_\alpha \subseteq J_\beta$? That is, given a language $L$, a $L$-structures $M$, a formula $\varphi$, and $x$ which are all in $J_\alpha$, is it true ...
3
votes
0answers
118 views

Levy collapse gone bad

Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following? (1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, ...
3
votes
0answers
123 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
3
votes
0answers
55 views

A Question Regarding the Relation Between Second-Order Arithmetic and the Constructible Universe

Colin McLarty, in this MathOverflow question, writes, Simpson's book Subsystems of Second Order Arithmetic shows $Z_2$ can interpret some fragments of ZF strong enough to give good theories of ...
3
votes
0answers
55 views

Does high closure of a forcing poset imply triviality?

The general intuition for forcing (at least for me) is that forcing with $P$ doesn't dramatically affect objects of size $> |P|$. Of course new large things appear, but these are usually "echoes" ...
3
votes
0answers
57 views

If we're interested in adjoining more stuff to a model of set theory, what is the appropriate notion of “embedding”?

I sometimes like to imagine that I have a model $V_0$ of set theory, not necessarily well-founded, to which we adjoin more stuff, like an inaccessible cardinal (together with the least number of sets ...
3
votes
0answers
77 views

Weakly Normal Ultrafilters

A filter $F$ on $\lambda$ is weakly normal if and only if for all $f : \lambda \rightarrow \lambda$ such that $\{\xi < \lambda : f(\xi) < \xi\} \in F^+$, then there is some $\beta < \lambda$ ...
3
votes
0answers
113 views

Has anyone published a “wish-list” of sentences that are independent of ZFC, but which are intuitively plausible?

I've substantially rewritten this question, since the original was rather badly written. Based on theorems about finite sets and/or finite cardinal numbers, we can make conjectures about infinite ...
3
votes
0answers
46 views

Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
3
votes
0answers
170 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...