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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10
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2answers
164 views

How to prove that Gödel's Incompleteness Theorems apply to ZFC?

Let us denote Robinson Arithmetic as Q and Primitive Recursive Arithmetic as PRA. Let $T$ be a formal theory formulated in the language of arithmetic. According to this page on the Stanford ...
2
votes
1answer
78 views

Modern algebra and set theory: ZFC vs. NBG

This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little: Is it not more natural consider NBG set theory as the foundation ...
0
votes
1answer
37 views

Function from an expression

How do you formalize, in ZFC set theory, the process of forming a function from an expression? Intuitively, I want to say something like this (I am guessing some vocabulary): There is an expression ...
4
votes
1answer
64 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
4
votes
1answer
104 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
0
votes
0answers
56 views

Choice-like axioms close to AC & rank-into-rank hypothesizes

Consider the following folklore theorems within ZF, Theorem 1: $V=L$ implies "$0^{\sharp}$ doesn't exist". Theorem 2: $V=L[U]$ implies "$0^{\dagger}$ doesn't exist". Theorem 3: $AC$ ...
8
votes
3answers
244 views

Is the “domain of discourse” in axiomatic set theory also a “set”?

The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of ...
30
votes
2answers
754 views

When should I be doing cohomology?

Background: I'm a logic student with very little background in cohomology etc., so this question is fairly naive. Although mathematical logic is generally perceived as sitting off on its own, there ...
0
votes
1answer
38 views

Can set containing not only all ordinal numbers exist?

I've always seen the same proof of non-existence of set of all ordianal numbers. It was based on a fact that such a set would fit into definition of ordinal number, and the set itself would be an ...
3
votes
1answer
104 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
2
votes
1answer
107 views

How much set theory and logic should typical algebraists/analysts/geometers know? (soft-question)

I know enough amount of set theory and logic to study grad-level math. However, I don't know more advanced set theory and logic, such as the ones on Kunen's or Shoenfield's texts. Although it's good ...
9
votes
2answers
173 views

Is it possible that every set can be specified?

Is it possible for there to be a model of ZFC with the property that, for every set $S$ in the model, there is a unary predicate in the language of ZFC such that $S$ is the is the only set satisfying ...
11
votes
3answers
391 views

Axiom of Choice: What exactly is a choice, and when and why is it needed?

I'm having trouble understanding the necessity of the Axiom of Choice. Given a set of non-empty subsets, what is the necessity of a function that picks out one element from each of those subsets? For ...
4
votes
0answers
101 views

A question regarding a paper of M. Magidor [closed]

I am interested in the following paper of M. Magidor: "On the role of supercompact and extendible cardinals in logic", Israel Journal of Mathematics, 05/1971; 10(2): 147-157. The abstract (which I ...
1
vote
1answer
66 views

Any known nontrivial undecidable/independent problems [closed]

Given ZF + standard model ℕ, are there any nontrivial, non-self-referential statements that are known to be independent of ZF + ℕ other than the Axiom of Choice? The halting problem isn't one. While ...
-1
votes
1answer
40 views

Set partitioning in ZFC

Does ZFC allow the partitioning of a set by claiming that a and b are in the same subset if f(a,b)? Cause I've once seen this tehnique being used in a proof but I can't see how it is consistent with ...
0
votes
1answer
53 views

Indiscernibility of indiscernibles in second order logic

It is not clear to me if the statements in $0^\#$ remain indiscernible when we move to second order logic. Or are there second logic formulas that can discriminate between first order indiscernibles?
4
votes
1answer
114 views

Statements comparable with Axiom of Choice in ZF

Let $AC$ denote any fixed statement of the Axiom of Choice in $ZF$. Consider the set of statements $\phi$ in the language of $ZF$ such that either $ZF+\phi$ proves $AC$ or $ZF+AC$ proves $\phi$. The ...
0
votes
0answers
48 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
8
votes
1answer
108 views

Examples of Forcing in Model Theory

My question is exactly my title: What are some examples of (set theoretic) forcing in model theory? I have been studying (combinatorial) set theory and model theory (independently of one another) for ...
1
vote
0answers
48 views

If full semantics higher order logic is set theory, which set theory is it?

I've been trying to get a handle on how higher order logic interacts with set theory. It's been stated convincingly that higher order logic with full semantics is set theory in sheep's clothing. For ...
1
vote
1answer
101 views

2 Questions regarding Relative Consistency Proofs

First Question: Let IC be the statement "There is an inaccessible cardinal." I have read that one cannot prove (in ZFC) the relative consistency of ZFC + IC w.r.t. ZFC. i.e. $ Con(ZFC) \rightarrow ...
1
vote
0answers
33 views

A question about HOD[A] and HOD(A)

HOD[A] (the class of sets hereditarily ordinal definable from $A$ where $A$ is some set) is known to be a transitive model of ZFC. HOD(A) (the class of sets hereditarily ordinal definable over $A$) is ...
4
votes
0answers
83 views

On the contradictory nature of large cardinals & choice-like axioms

Compare these two results: Theorem (Scott): $ZFC+V=L\vdash \nexists~\text{Measurable cardinal}$ Theorem (Kunen): $ZFC+AC\vdash \nexists~\text{Reinhardt cardinal}$ Now compare these two ...
15
votes
4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
5
votes
1answer
130 views

Examples of Extreme Anti-Choice Axioms

Axiom of Choice has many variants like the followings: There is a choice set for every family of non-empty sets. All sets are well-orderable. Of course in many cases one don't need AC to ...
0
votes
1answer
32 views

Tychonoff spaces with small weight

Let $\kappa$ be an infinite cardinal. Is there a Tychonoff space $(X,\tau)$ such that $|X| = 2^\kappa$ and $(X,\tau)$ has weight $\kappa$ (= a basis consisting of $\kappa$ elements)?
1
vote
0answers
47 views

Why is addition of powers of $\omega$ absorptive?

Why is it the case that if $\alpha<\beta$, $\omega^\alpha+\omega^\beta=\omega^\beta$? Is it because ...
2
votes
1answer
117 views

Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?

A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), ...
2
votes
0answers
40 views

Do large cardinal properties tend to be semiabsolute?

I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question. Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ ...
0
votes
1answer
49 views

Well-foundedness is not a first order property.

In the book 'Logic, Induction and Sets' by Thomas Foster I read the following in page 100 (Section 'The language of predicate logic'): "We can show that well-foundedness is not a first-order property ...
2
votes
1answer
44 views

Hereditary cardinality and products of sets

To start, suppose that $\lambda$ is an infinite cardinal and suppose that $\alpha, \beta \in \mathbf{H}_\lambda$, where $\mathbf{H}_\lambda = \{x : \left|\operatorname{trcl}(x)\right| < \lambda\}$. ...
0
votes
1answer
107 views

A question regarding non-(Lebesgue)-measurable sets in models of ZFC+$2^{\aleph_0}$=$\aleph_2$

Let $\mathscr V$ represent a set of Vitali's type. It is known that $|\mathscr V|=2^{\aleph_0}$. Does $\mathscr V$ have any measure-theoretic properties in models of, say, ...
6
votes
1answer
162 views

ZF and The Cardinality of The Set of Finite Subsets

In a comment on one of my answers, I claimed that the abelian group generated by a set of $S$ generators, each of order two, could take on any infinite cardinality; this is equivalent to saying that, ...
0
votes
0answers
34 views

is there a known set in ZF, such that we can't find a well order on? [duplicate]

is there a known set in ZF, such that we cant find a well order on? since the axiom of choice $(AC)$ and it's negation is consistent with ZF, i wonder if we have a concrete example of a set $A$ that ...
0
votes
0answers
27 views

About a function ranging filters

Let $U$ be an (infinite) set and $N$ be an (infinite) index set. I denote $\mathfrak{A}$ the set of filters on $U$ (including the improper filter). Let $f$ be an $N$-ary relation that is a set of ...
1
vote
0answers
50 views

Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
1
vote
0answers
81 views

About models of ZFC and models in general.

So I've been attending lectures in Set Theory lately and been struggling with the following. When studying the universe of sets V our approach is: let ZFC be consistent, then a model V of the theory ...
3
votes
2answers
149 views

A question regarding the Power Set Axiom in ZFC

It is known that the Axioms of ZFC are not necessarily independent of each other. For example, it can be shown that one can derive Separation from Replacement even though both are listed as axioms of ...
-1
votes
1answer
28 views

How to prove that a union of cardinals is a cardinal [duplicate]

I have this question: Let $\omega_1$ the least uncountable cardinal, and for all $n \in \omega$, $n \geq 1$. Let $\omega_{n+1}$ the least cardianal greater than $\omega_n$. Show that $$\bigcup_{n \in ...
2
votes
1answer
45 views

Notation for the class of all cardinals

I have seen the notation for the class of all ordinals to be $\rm Ord$ or $\rm On$, is there an analogous notation for the class of all cardinals?
0
votes
1answer
38 views

If $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$

I would like to show that if $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$ where $\kappa$ is an infinite cardinal. I'm certain it ...
2
votes
0answers
75 views

“constructible but not definable”

T. Jech in his monograph Ch. 18, p. 301 writes "... the set T is constructible, but not definable in L." My guess is that this means that T (a subset of, say, some ordinal) is definable in some ...
1
vote
1answer
51 views

Unordered pair of proper classes

The usual Kuratowski ordered pair function does not work on proper classes, because if $A,B$ are proper classes and $\langle A,B\rangle=\{\{A\},\{A,B\}\}$, then since $A\notin\{A\}$ and so on you get ...
2
votes
0answers
47 views

Weight of topology related to sizes of open sets?

I'm wondering if there is a notion relating the weight of a topological space (=minimal base cardinality) to the sizes of its open sets. In particular I'm looking for properties of spaces, whose ...
6
votes
1answer
149 views

Turing invariance on large sets

Definition: A function $f: 2^{\omega} \rightarrow 2^{\omega}$ is Turing invariant if $x \equiv_T y \rightarrow f(x)\equiv_T f(y)$. Question I (under $ZFC$): Let $f: 2^{\omega} \rightarrow 2^{\omega}$ ...
6
votes
1answer
91 views

Separating disjoint sets of size $\aleph_1$ with Borel sets

Question: Is it consistent with ZFC that every pair of disjoint sets $A,B\subseteq\mathbb{R}$, both of size $\aleph_1$, can be separated by a Borel set? This statement is clearly false under CH; take ...
1
vote
0answers
38 views

Cardinal exponentiation ambiguity

This is a following passage from Jech on cardinal exponentiation. Am I correct in that $\kappa^\lambda$ means two different things here? Namely, that the first $\kappa^\lambda$ is the set of all ...
4
votes
1answer
85 views

$H(\kappa)$ a model of all the axioms of ZFC for $\kappa$ not inaccessible

Is this possible? For each cardinal $\kappa$, we can define $H(\kappa) = \{ x \mid |trclx| < \kappa\}$, where trcl is the transitive closure, and $|A|$ is the cardinality of $A$. For every regular ...
2
votes
1answer
53 views

Can somebody explain (and ideally reference) this strange use/version of the Pressing Down Lemma?

In Stevo Todorcevic's "A dichotomy for P-ideals of countable sets" (link, page 261 at the bottom [page 11 in the pdf]), the following confusing situation comes up: (Context: $\mathcal I$ is a P-Ideal ...