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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2
votes
2answers
117 views

Recursion on a class instead of a set

According to Wikipedia, the recursion theorem states the following: Let $X$ be a set, and $f:X\to X$ a function. For any $x\in X$, there exists a unique function $g:\omega\to X$ such that $g(0)=x$ ...
2
votes
1answer
31 views

to show that there is no injection from a finite successor of finite ordinal to itself

im trying to prove this proposition let $\alpha \in \omega$ so i want to show that there is no injection (not necessarily keeps order) from $\alpha +1$ to $\alpha$ without using cardinality or the ...
2
votes
0answers
55 views

Is there any inconsistent large cardinal axiom which its inconsistency proof is essentially different from proof of Kunen inconsisteny theorem?

There is a long list of large cardinal axioms. Most of them deemed to be consistent with ZFC but there are also some axioms like existence of Reinhardt or $\omega$-huge cardinals which are natural ...
4
votes
1answer
53 views

Constructing a function whose domain is $\omega$ using successor operation recursively

Let $x$ be a set. Does there exist a functional relation $f:\omega\to \bf{V}$ which has the following property? \begin{eqnarray*} f(0)&=&x\\ f(1)&=&S(x)=x\cup\{x\},\\ ...
1
vote
2answers
54 views

Showing $\prod_{n < \omega} n = 2^{\aleph_0}$ [duplicate]

I have to show that $\prod_{n < \omega} n = 2^{\aleph_0}$. I'm having trouble getting started. I know that $2^{\aleph_0}$ is the set of binary sequences, or the space of functions from $\mathbb{N}$ ...
3
votes
1answer
73 views

$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
20
votes
4answers
1k views

Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?

I know the usual proof of the existence of an algebraic closure for any field using Zorn's Lemma. The answer to this previous question makes it clear that in general, some nonconstructive axiom (not ...
3
votes
1answer
424 views

How to prove that Cantor's normal form can produce all ordinal numbers

How do we prove Cantor's normal form can produce all ordinal numbers? Also, it's a bit difficult to picture $\omega_1$ as a format in Cantor's normal form. Can anyone show how to do this?
2
votes
1answer
40 views

For any $a\in S_\Omega$, either it has an immediate predecessor or there is an increasing sequence in $S_\Omega$ whose l.u.b. is $a$. [duplicate]

Specifically I would like to show: Let $S_\Omega$ be the minimal uncountable well-ordered set. For any $a\in S_\Omega$, either $a$ has an immediate predecessor in $S_\Omega$, or there exists an ...
2
votes
0answers
87 views

Formula Complexity of $\models_n$

I want to show $\models_0$ is $\Sigma_1$, and $\forall n \geq 1, \models_n$ is $\Sigma_n$. So for the base case, $\models_0 \ulcorner \phi \urcorner$ is true iff $\ulcorner \phi \urcorner \in ...
13
votes
1answer
959 views

What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
0
votes
1answer
69 views

Is there a set theory that handles collections of proper classes?

Is there a class of all classes? If not, can I define the “banana” so we have the “banana” of all classes? Is there a banana of all bananas? If I define an apple of all bananas, is there an apple of ...
3
votes
1answer
54 views

Topological form of Martin's Axiom

I'm currently studying consequences of Martin's Axiom: Martin's Axiom (MA): Suppose that $\left\langle P, \leq \right\rangle$ is a ccc partially ordered set and $\{D_\alpha\}_{\alpha < \lambda}$ ...
2
votes
1answer
66 views

Doubt about the proof of $V_\omega\models\mathsf{Separation}$

In Kunen's 'set theory', he introduce following theorem: Suppose that for each formula $\phi(x,z,\vec{w})$ with no variable besides the displayed ones free,$$\forall z, \vec{w}\in M:\{x\in z:\phi ...
-1
votes
1answer
46 views

Cardinality of the set of all monoids with countably many elements

How can I prove assuming the continuum hypothesis, that the cardinality of the set of all monoids with countably many elements has cardinality the same as that of the power set of the real numbers. ...
3
votes
1answer
264 views

A Dedekind infinite set has a countably infinite proper subset

Set A is Dedekind infinite, i.e. there is a bijective function from A onto some proper subset B of A. Please prove that A has a countably infinite proper subset.
13
votes
3answers
867 views

A nice introduction to forcing

I want to get acquainted with forcing, along with a few friends, and I'm looking for a text to introduce the basic notions (pardon the pun :) ). The point is to study a text (or texts, if they can be ...
6
votes
4answers
715 views

What are the prerequisites to Jech's Set theory text?

I'm looking for a book to self-study axiomatic set theory, and heard this was a classic. What are the main prerequisites for this text? My knowledge of set theory isn't too great. Probably the only ...
6
votes
1answer
174 views

Second reading on set theory? Any recommendations?

I have in past six-ish months studied through the Herbert Enderton's Elements of set theory book. Up to the point the book is great,I loved most parts of it and learned almost everything up to the ...
1
vote
1answer
64 views

A Question Regarding the Origin of the Axiom of Symmetry

It is my understanding that Chris Freiling's "Axiom of Symmetry" is based on a counterexample to CH given by Sierpinski in his book "Hypothese de continu". Since I neither read nor speak French, I ...
6
votes
1answer
103 views

Regarding functions on $\omega_1$

I've been trying to prove a property that apparently all functions $g: \omega_1 \rightarrow \omega_1$ have, where $\omega_1$ is the least uncountable ordinal. For $\alpha \in \omega_1$, define ...
3
votes
0answers
47 views

Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
2
votes
1answer
119 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
1
vote
1answer
68 views

A question from Kunen's book: chapter VII (H9), about diamond principle

Assume $(\mathbb{P}$ is c.c.c.$)^M$ and $\Diamond$ holds in $M[G]$. Show that $\Diamond$ holds in $M$. Hint: It is sufficient to verify $\Diamond^-$ in $M$. Should I try to create a ...
18
votes
4answers
593 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
7
votes
3answers
223 views

Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
1
vote
1answer
33 views

Initial segments of well-ordered sets are isomorphic

I want to prove that if $(X,\prec)$ and $(Y,<)$ are well-ordered sets that X must be isomorphic to an initial segment of $Y$ or vice versa. I am trying to do this by defining the function: ...
0
votes
1answer
39 views

Disjoint union of sets $A_i$ indexed by a set $I$ is still a set?

I have a question in set theory considering a disjoint union of sets. My question is: if we form the following disjoint union $$\bigsqcup_{i \in I} A_i$$ where the $A_i$ are are sets and $I$ is a set ...
7
votes
1answer
117 views

Consistency strength of the “club ultrafilter”

What are the consistency strengths of $$ZF+``\text{The club filter on $\omega_1$ is an ultrafilter}"$$ and $$ZF + DC + ``\text{The club filter on $\omega_1$ is an ultrafilter}"?$$ I know that the ...
0
votes
1answer
133 views

Platonist research on the cardinality of the reals

Apologies to any formalist! Here's the basic thought: $\mathbb{R}$ is a well-defined concept with unambiguous meaning in reality. Everyone can imagine an infinite series of digits (signifying the ...
4
votes
2answers
75 views

Does every set have a derangement? [duplicate]

A derangement of a set $A$ is a bijection from $A$ to itself with no fixed points. Is it the case that every infinite set has a derangement.
4
votes
0answers
75 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
5
votes
2answers
301 views

Equivalent characterizations of ordinals of the form $\omega^\delta$

Let $\alpha$ be a limit ordinal. Show the following are equivalent: $\forall \beta, \gamma<\alpha (\beta+\gamma<\alpha)$ $\forall \beta<\alpha(\beta+\alpha=\alpha)$ $\forall X\subset ...
17
votes
1answer
475 views

On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
5
votes
1answer
140 views

What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
4
votes
1answer
100 views

Does it make sense to form a set whose elements are proper classes?

In chapter 4 of Handbook of Categorical Algebra, vol 1, the author defines a "subobject of $A$" as "an equivalence class of monomorphisms with codomain $A$" (for a suitable notion of equivalence). He ...
2
votes
1answer
76 views

Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
3
votes
0answers
59 views

Equivalent forms of Jensen's diamond principle [duplicate]

I try to prove that these four statements are equivalent: $\Diamond$ There are $A_\alpha\subseteq \alpha\times\alpha$ for $\alpha<\omega_1$ s.t. for all $A\subset \omega_1\times\omega_1$, ...
2
votes
1answer
66 views

Axiom of Choice Equivalent

I'm trying to prove the following statement is equivalent to the Axiom of Choice: "For any set $A$, there exists a function $F$ with dom $F = ⋃A$ and for each $x ∈ ⋃A$, $x ∈ F(x) ∈ A$." (1) The ...
4
votes
2answers
230 views

Is there any category theoretic proof for independence of Continuum Hypothesis?

Both of set theory and category theory could be a foundation for mathematics. Many set theoretic arguments could be translated to a category theoretic argument and vice versa. Question: Is there ...
4
votes
0answers
62 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
3
votes
1answer
87 views

Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
2
votes
0answers
40 views

Show that if A is a finite set of non empty sets, then A has a choice function.

I'm trying to prove that if A is a finite set of non empty sets, then A has a choice function. My approach is to construct such function. Since it is a finite set we can say that $A=\left \{ ...
0
votes
1answer
83 views

Induction in a first order system with ZF

Suppose I have some characterization of the natural numbers $N$ in a first-order system under ZF. To be precise, I have $N = \lbrace n: \forall w:( w\space is\space inductive) \rightarrow n \in w ...
4
votes
1answer
95 views

What goes wrong when you try to reflect infinitely many formulas?

The reflection principle in ZFC shows that you can construct a set that reflects finitely many formulas. Suppose we wanted to reflect {$\phi_n$} and we construct a set $M_n$ to reflect $\phi_1, ... , ...
6
votes
1answer
120 views

Does iterating the consistency of ZFC answer any natural questions?

The following is a natural question that occurred to me, but I'm not sure if it's even well-defined since I haven't read the literature on iterating consistency statements. Let $Con_0(ZFC)=Con(ZFC)$ ...
2
votes
0answers
56 views

Monomorphisms in a concrete category

Let $\mathcal{C}$ be a concrete category, i.e., a category which admits a faithful functor $C:\mathcal{C}\rightarrow \mathsf{Set}$. It is certainly not the case that $f$ a mono in $\mathcal{C}$ ...
11
votes
2answers
921 views

How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
6
votes
1answer
325 views

Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.

Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
0
votes
0answers
52 views

Books and articles on model theory for set theory

I'm interested in books and/or articles which explore a little more in depth the model theory of set theory. I'm aware that most books on set theory have a section or two on models (e.g. Jech, Kunen), ...