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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1answer
249 views

Proper classes and models of set theory

If I have a model of ZFC and a proper class in that model, is there always an extension to another bigger model where this proper class becomes a set? I know that this is possible in particular cases, ...
2
votes
1answer
295 views

Does a 'universal' group/ring/field/topology/etc. exist?

My question is inspired by the fact that there is no universal set (at least in ZF). There are many abstract objects such as group, ring, field, vector space, topology, etc. such that we can say about ...
3
votes
2answers
146 views

Existence of a subset of infinite model that's not definable

I'm working on the following problem: Show that any infinite model $\mathcal{M}$ has a subset of its domain that's not definable in $\mathcal{M}$ using parameters. I tried to follow a contradiction ...
7
votes
1answer
378 views

Grothendieck universe consistency

Is ZFC with existence of Grothendieck universe (variant: Grothendieck universe containing every given set) provable in ZFC to be equiconsistent with ZFC? If not, what else it may be equiconsistent ...
2
votes
1answer
401 views

Formalising real numbers in set theory

If I understand correctly, the real numbers can be formalised in a first-order, like in ZF. However, such a formalisation is not strong enough to ensure that all models of the reals are isomorphic. ...
17
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1answer
1k views

Proving “every set can be totally ordered” without using Axiom of Choice

It is known that the statement "every set can be totally ordered" is strictly weaker than Axiom of choice. How does one go about proving without using AC?
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2answers
140 views

Some questions concerning set-theoretic models of first-order theories

Consider a finitely axiomatized theory $T$ with axioms $\phi_1,...,\phi_n$ over a first-order language with relation symbols $R_1,...,R_k$ of arities $\alpha_1,...,\alpha_k$. Consider the atomic ...
6
votes
2answers
381 views

Fodor's lemma on singular cardinals

Fodor's lemma asserts that if $\kappa$ is a regular and uncountable cardinal, then if $f(\alpha)<\alpha$ for a stationary subset of $\kappa$, then it is constant on stationary subset. Suppose ...
11
votes
4answers
1k views

How can there be genuine models of set theory?

I know that this a beginner's question asked too many times, but I still didn't get an answer which lets me quit asking: Given that a model/interpretation of a theory (in the Tarskian sense) is a ...
19
votes
2answers
1k views

For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice

How to prove the following conclusion: [For any infinite set $S$,there exists a bijection $f:S\to S \times S$] implies the Axiom of choice. Can you give a proof without the theory of ordinal ...
5
votes
1answer
217 views

Generalizing $0^\#$

Background and motivation: The following theorem is due to Silver: If there exists a Ramsey cardinal then: For every $\aleph_0 < \kappa < \lambda$, $L_{\kappa}$ is an elementary ...
7
votes
0answers
372 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 ...
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0answers
212 views

Can the reduced product construction generate boolean-valued models?

In model theory, the reduced product construction contains a collection of structures or models, a set I that indexes the collection, and a filter U on I. Ultraproducts are a special case of reduced ...
10
votes
1answer
953 views

Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
15
votes
1answer
1k views

About a paper of Zermelo

This about the famous article Zermelo, E., Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (4), 514–516 (1904), available here. Edit: Springer link to the ...
5
votes
1answer
345 views

Consistency of $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$

It is known that if $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then $0^\#$ exists. I am familiar with the Feferman-Levy model in which $\omega_1$ is singular, which has the same ...
16
votes
1answer
815 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
2
votes
1answer
166 views

The Structure Of Weirdness in ZF + $\neg$ AC

Some questions and answers like this and this on Math Overflow suggest an ordering of large cardinals. I was wondering if there is any similar structure associated with $\text{ZF} + \neg \text{AC}$. ...
4
votes
1answer
131 views

If $V=L$, is every $V_\alpha$ an $L_\beta$?

I guess that amounts to if there is a continuous $f$ with $\mathbb{P}(\mathbb{L}_{f(\alpha)}) \cap \mathbb{L} = \mathbb{L}_{f(\alpha+1)}$ I seem to remember reading that it is, but I forget where or ...
6
votes
2answers
501 views

Zorn's lemma in abstract algebra?

It is well konwn that Zorn's lemma implies: Prop.1 Every commutative unital ring has a maximal ideal. Prop.2 Every proper ideal is contained in a maximal ideal in a unital ring. Question: Can we ...
5
votes
1answer
102 views

The real cofinality of singular cardinals in $L$ under $0^\#$

Suppose that $0^\#$ exists, is there a relatively simple way to show that for any ordinal $\lambda$, if $\lambda$ is a singular cardinal in $L$ then its real cofinality is $\omega$?
9
votes
1answer
231 views

$A^\#$ and inner models

For a set of ordinals $A$ we say that $A^\#$ exists if there exists a closed and unbounded class of indiscernibles, $I\subseteq\operatorname{Ord}$, for $L[A]$. Formally, if such class exists we define ...
4
votes
1answer
97 views

Constructibility of Skolem hulls of constructible sets

Suppose that $X$ is a constructible set, denote by $H^{L_{\alpha}}(X)$ the Skolem hull of $X$ in $L_{\alpha}$. Is $H^{L_{\alpha}}(X)$ constructible?
17
votes
3answers
1k views

Defining cardinality in the absence of choice

Under ZFC we can define cardinality $|A|$ for any set $A$ as $$ |A|=\min\{\alpha\in \operatorname{Ord}: \exists\text{ bijection } A \to \alpha\}. $$ This is because the axiom of choice allows any ...
11
votes
1answer
528 views

There's non-Aleph transfinite cardinals without the axiom of choice?

I can't find anything on this anywhere. The book I'm largely using at the moment is based around ZFC, so it makes no mention of anything other than the Aleph numbers, but according to Wikipedia on the ...
7
votes
1answer
209 views

Fixed point: sets and measures

Let $X$ be a Borel space with a Borel measure $\mu$. Suppose $\xi: X\times X\to\mathbb R_{\geq 0}$ is a continuous function and put $s(x) = \{y\in X:\xi(x,y) = 0\}$. For any set $b\in\mathcal B(X)$ we ...
7
votes
1answer
216 views

Equivalent ways to describe the Mitchell order

For a measurable cardinal $\kappa$, we define an ordering over $\kappa$-complete ultrafilters as follows: Suppose $W,U$ are both $\kappa$-complete free ultrafilters over $\kappa$, we say that $U\lhd ...
3
votes
1answer
410 views

Short open problems or undecidable statements in ZFC

I guess Goldbach's conjecture is a good example of a short open problem in number theory, and "Goodstein sequences reach 1" is a good example of a statement undecidable from first-order Peano ...
80
votes
4answers
6k views

What are the Axiom of Choice and Axiom of Determinacy?

Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple ...
6
votes
1answer
212 views

Silver indiscernibles and definable injections

This is a follow up to my previous question on Silver indiscernibles. Background: Suppose that $0^\#$ exists, $\alpha<\lambda$ are limit ordinals, $i_\alpha$ is the $\alpha$th Silver ...
5
votes
1answer
207 views

Silver indiscernibles and constructibility

We know that if $0^\#$ exists then it's not in $L$. For an infinite ordinal $\alpha$, denote by $I_\alpha$ the initial segment of length $\alpha$ of Silver indiscernibles. Question: For which ...
4
votes
1answer
414 views

Can it be shown that ZFC has statements which cannot be proven to be independent, but are?

I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would ...
2
votes
2answers
324 views

Can logic be defined in terms of sets? Can sets be defined using logic?

Can logic be defined in terms of sets? Can sets be defined using logic? If both answers are positive, is one reduction preferable to the other? In what sense?
12
votes
0answers
436 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 ...
2
votes
1answer
284 views

Separable metric space with 0-dimensional kernel

Suppose $X$ is a separable metric space, let $D(X)$ denote the Cantor-Bendixson derivative of $X$, and $D_\alpha(X)$ the $\alpha$-th derivative of $X$. We denote $\operatorname{Ker}(X)$ the kernel of ...
5
votes
1answer
276 views

$\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the ...
10
votes
5answers
529 views

Natural isomorphisms and the axiom of choice

The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the intuitive notion of "an isomorphism ...
4
votes
1answer
172 views

Cohen forcing and bounds on GCH violation

I recall (though it might be my faulty memory) an exercise in some book (perhaps Jech?) that was along the lines "Suppose $\mathrm{GCH}$ holds in $V$ and $P$ is a Cohen forcing that adds $\kappa$ ...
6
votes
2answers
738 views

Cardinality of sets of subsets of $\mathbb{N}$

If we dont assume CH, is there a procedure to construct or define a set of subsets of $\mathbb{N}$ such that we cannot prove it to be of cardinality $\aleph_0$ or $\aleph_1$? Or if we assume not CH, ...
5
votes
1answer
118 views

The cardinality of $L_\alpha$ in $L$

It's true (in $V$) that for any infinite ordinal, $|L_\alpha|=|\alpha|$. My question: Is it also true in $L$? i.e., does $L$ itself satisfies $|L_\alpha|=|\alpha|$ for any infinite ordinal $\alpha$?
5
votes
2answers
273 views

Number of Ramsey ultrafilters

Hereafter, "ultrafilter" is intended to mean "nonprinicpal ultrafilter on the set, $\omega$, of natural numbers." An ultrafilter $U$ is Ramsey if for every partition $P$ of $\omega$ into sets not in ...
6
votes
2answers
796 views

Diamond Principle and its variants - two questions

The standard formulation of the Diamond Principle $\Diamond$ is as follows: There exists a sequence $\langle f_\alpha:\ \alpha<\omega_1\rangle$ of functions $f_\alpha:\alpha\to2$ such that for ...
5
votes
1answer
112 views

Exercise about Galvin-Hajnal rank

I'm stuck at exercise 9a of chapter 2.2 of Introduction to cardinal arithmetic by Holz, Steffens and Weitz. It is as follows: Assume that $\kappa > \omega$ is a regular cardinal, $I$ is the ...
65
votes
1answer
4k views

How do we know an $ \aleph_1 $ exists at all?

I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have ...
4
votes
1answer
220 views

Are these sets stationary?

Assume $\kappa > \omega$ is a cardinal of uncountable cofinality and $S$ is a stationary set in $\kappa$. If $\alpha < \kappa$ is a successor ordinal then the set $S+\alpha=\{\sigma+\alpha : ...
2
votes
2answers
175 views

Accessible formal specification and explanation of First Order Logic?

I am trying to get good at proofs by working through How To Prove It. Unfortunately I am very bothered by the fact that I do not understand all the formalities in First Order Logic + set theory ...
5
votes
2answers
2k views

Cantor-Bernstein-like theorem: If $f\colon A\to B$ is injection and $g\colon A\to B$ is surjective, can we prove there is a bijection as well?

I've been trying to find this proof: If there exists $f \colon A\to B$ injective and $g \colon A \to B$ surjective, prove there exists $h \colon A \to B$ bijective. I thought of using ...
4
votes
2answers
164 views

Independence of the comprehension axiom

What's wrong with the following line of reasoning, if so? The comprehension axiom of Zermelo's set theory would be provable by the other axioms, if the following was provable: ($*$) All ...
11
votes
1answer
648 views

The categories Set and Ens

I've recently started reading Categories for the Working Mathematician and I'm a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like ...
4
votes
0answers
178 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 ...