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
230 views

About a ordinal-based definition of fast-growing functions

I try to understand the Löb-Wainer-hierarchy and one definition just doesn't open. I hope someone could clarify this to me. A fundamental sequence to limit ordinal $\alpha$ is $\omega$-sequence ...
10
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3answers
700 views

Difference between undecidable statements in set-theory and number theory?

Do all statements about the integers have a definite truth value? For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to ...
2
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1answer
367 views

Is there an axiom scheme exhausting all types of Mahlo cardinals?

Is there an axiom scheme exhausting all types of Mahlo cardinals? Mahlo cardinals may be considered as the first stage in the following construction : let $C_{0,0}$ be the class of all inacessible ...
2
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2answers
199 views

Can a 1-1 function be implied from a two-way onto pair of functions?

In DC Proof I've defined equal cardinality to mean there are two onto functions $g:A\to B$ and $h:B\to A$. I want to prove this means there exists a function $f:A\to B$ that is both 1-1 and onto ...
7
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2answers
635 views

Descriptions of sets and the Axiom of Choice

If we specifically do not assume the Axiom of Choice, are all the sets that we can prove to exist specified by some finite formula? (All the other Zermelo-Frankel set theory axioms seem constructive ...
15
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1answer
1k views

Infinite Set is Disjoint Union of Two Infinite Sets

A finite set is a set such that there exists a bijection from it to some finite ordinal. An infinite set is a set that is not finite. In ZF, can you prove that every infinite set is the union of two ...
8
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1answer
243 views

Reverse Mathematics of Well-Orderings

In Simpson's book, a well-ordered set $X$ is a linear ordering such that there are no functions $f : \mathbb{N} \rightarrow X$ which is decreasing. However, a familiar definition of well-ordering is ...
5
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1answer
255 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
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1answer
297 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
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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
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1answer
389 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
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1answer
404 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
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2answers
384 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 ...
12
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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 ...
20
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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
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1answer
219 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
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0answers
377 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 ...
1
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0answers
214 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
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1answer
955 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 ...
16
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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
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1answer
347 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
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1answer
821 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
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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
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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
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2answers
505 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
233 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
99 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
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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
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1answer
531 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
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1answer
210 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
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1answer
223 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
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1answer
412 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
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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
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1answer
217 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
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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
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1answer
417 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
332 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
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0answers
443 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
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1answer
279 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
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5answers
534 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
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1answer
174 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
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2answers
745 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
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2answers
274 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
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2answers
807 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
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1answer
113 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 ...