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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0
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2answers
409 views

How can I determine the cardinality of a set of polymorphic functions?

It seems obvious to me that the set of functions with the signature $\forall A. A \rightarrow A$ is "once-inhabited", i.e. there is only one such polymorphic function which "works" for any set $A$, ...
11
votes
2answers
497 views

When does $V=L$ becomes inconsistent?

In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$. The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ ...
3
votes
0answers
183 views

Can Egoroff's Theorem be Strengthened for A Sequence of Smooth Functions?

I have posted this question on MO, and I'll raise the question here in a more concrete way. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $I$ ...
0
votes
3answers
446 views

Infinite paths that connect two vertices?

This is a follow-up to another question concerning infinite paths which was admittedly ill-posed. I hope this question is posed better. The graph $N$ with vertex set $V(N) = \mathbb{N}$ and $(x,y) ...
6
votes
2answers
217 views

Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
5
votes
2answers
283 views

$\omega$-saturation of $(\mathbb{R},<)$

Could anyone of you explain me why $(\mathbb{R},<)$ is $\omega$-saturated? EDIT: do you know also why the theory of Boole algebras without atoms is $\omega$-categoric? Added: The added question ...
-2
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3answers
431 views

Infinite shortest paths in graphs

From Wikipedia: "If there is no path connecting two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite." This seems ...
9
votes
2answers
202 views

Easy Question about P-names

Notation(from Kunnen): $M$ is countable transitive model of $ZFC$, $P$ is a partial order that belongs to $M$, $G$ is a $P$-generic filter over $M$, $\hat{b}$ will be a $P$-name for $b\in M$, and ...
2
votes
1answer
197 views

Show that for any A, there exists { dom R | R in A }

I want to show that (a) $ \forall \alpha \; \exists \beta$ such that $\beta = \{ dom R \; | \; R \in \alpha \}$ I'm having difficulty proving this axiomatically. I'm not even certain that it can be ...
7
votes
1answer
321 views

powers of singular cardinals

I am trying to solve the following two problems: 1) if $\beta <\omega_1$, $2^{\aleph_1}<\aleph_{\omega_1}$, and $\aleph_\alpha^{\aleph_0} \leq \aleph_{\alpha +\beta}$ for a stationary set of ...
9
votes
2answers
268 views

How to show the existence of an infinite set of independent undecidable sentences?

How to show the existence of an infinite set of independent undecidable sentences? By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...
8
votes
1answer
219 views

Understanding an Easy Relative Consistency Proof

When proving that $(ZF-Reg)\vdash CON(ZF - Reg)\rightarrow CON(ZF)$ we start by defining the class $WF=\cup_{\alpha \in ORD} V_{\alpha}$. Then we prove with $ZF-Reg$ that actually $WF$ is a model for ...
5
votes
2answers
436 views

Why is cofinality important? And some help in proving absoluteness for $R(\kappa)$

I am starting to read the Kunen's book set theory. I could not understand the why we need cofinality. Why is it important? Also, I'm trying to solve some exercises (on page 146) about absoluteness ...
6
votes
1answer
103 views

Basic constructibility question

I'm currently reading J. D. Hamkins' paper "Unfoldable cardinals and the GCH," and I've run across a comment that I think I ought to find trivial, but I don't. On page 1187, he says that ...
11
votes
2answers
454 views

Any good decomposition theorems for total orders?

I'm learning about set theory and partial orders, well orders, total orders, etc. I'm curious to know of any good decomposition theorems that apply to all (or very general types of) total orders. ...
7
votes
1answer
546 views

No uncountable ordinals without the axiom of choice?

In Uncountable ordinals without power set axiom Francois Dorais explains that without the Power-set Axiom we cannot prove the existence of uncountable ordinals. I am guess that the power set of an ...
3
votes
1answer
567 views

Non-standard models of arithmetic for Dummies (2)

I've learned that there are (at least) three types of countable non-standard models of arithmetic, depending on the primitive operations: successor only → $\mathbb{N}$ followed by a copy of ...
5
votes
1answer
285 views

If you randomly choose a subset of the real line, what is the probability that it will be measurable?

Suppose that you are working with the axiom of choice. If you randomly choose a subset of the real line, what is the probability that it will be measurable?
4
votes
3answers
224 views

Hausdorff ultrafilters

I know that Ramsey ultrafilters are Hausdorff ($\mathcal{U}$ is Hausdorff iff for every $f,g:\mathbb{N}\rightarrow\mathbb{N}$ $f(\mathcal{U})=g(\mathcal{U})$ then $f\cong_\mathcal{U} g$ $\;$). So if ...
3
votes
1answer
110 views

Small question on proof that any ordinal is the index of some initial ordinal

I'm reading a proof of a theorem, but an apparently trivial case has tripped me up. The theorem goes as For any ordinal $\alpha$, there is a initial ordinal $\phi$ such that $i(\phi)=\alpha$. Here ...
8
votes
1answer
288 views

Universal sets in metric spaces

Today I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is a polish space (metric, separable and complete) then there exists a $G\subseteq X\times Y$ which is open and ...
18
votes
4answers
857 views

Mathematical statement with simple independence proof from $\mathsf{ZF}$

Is it possible for someone with little set-theoretic knowledge (e.g., me) to understand the proofs that either $\mathsf{CH}$ or $\mathsf{AC}$ is independent of $\mathsf{ZF}$? I am looking for any ...
14
votes
3answers
458 views

Locally non-enumerable dense subsets of R

Today after lunch I was hungry for math problems so I started begging for some at the department and finally someone threw me this: Can $\mathbb{R}$ be partitioned into two non-countable dense ...
14
votes
7answers
1k views

Applications of ultrafilters

I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters? I'm ...
10
votes
3answers
983 views

Axiom of Choice: Can someone explain the fallacy in this reasoning?

Not a "set theory" guru (apologies if my terms are imprecise), but I have heard that it is an elementary result that the set of rational numbers has a measure of zero - intuitively meaning that the ...
3
votes
1answer
159 views

Rudin-Keisler order

I have to prove this: Fixed $\mathcal{U}$ a non-principal ultrafilter then (1) imples (2) where: 1) Every $f:\mathbb{N}\rightarrow\mathbb{N}$ is $\mathcal{U}$-eq. to a weakly increasing function. 2) ...
3
votes
1answer
151 views

For each extensive, monotone function there exists a closure operator that preserves the closed sets

To make the title a little bit more proecise: Let $X$ be a set and a map, $F:\mathcal{P} (X) \rightarrow \mathcal{P} (X)$ (where $\mathcal{P} (X)$ denotes the powerset of $X$), such that $A ...
22
votes
3answers
752 views

Are there uncountably many non homeomorphic ways to topologize a countably infinite set?

Today I was fooling around a bit trying to count the topologies on a finite set. I didn't make much progress, so I did some googling and quickly discovered it is an open problem to give a closed form ...
11
votes
1answer
451 views

Injective function and ultrafilters

An exercise left by my teacher let me think that the following statement is true: Let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$. Then every injective function ...
4
votes
1answer
111 views

Help on a definition

In many books I find these two definitions of Ramsey ultrafilters, extremely similar but different: 1)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists ...
3
votes
2answers
239 views

Problem on ultrafilters

I have to solve this problem: suppose to have an ultrafilter $\mathcal{U}$. Suppose that (1) holds, I want to prove (2): 1) for every partition $\mathbb{N}=\bigsqcup A_k$ $|A_k|=\aleph_0$ there ...
7
votes
2answers
331 views

Is $V$ under ZFC really a proper class?

Is $V$, the union of the von Neumann hierarchy, necessarily a proper class? Or is it only a proper class after you assume that it contains every set? (In that case, $V$ can't be a "set of all sets" ...
8
votes
1answer
432 views

Can any infinite ordinal be expressed as the sum of a limit ordinal and a finite ordinal?

I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum ...
2
votes
1answer
98 views

Help on two exercises

I need some help on two exercise: Here is the first: Let $\mathcal{U}$ be a non-principal ultrafilter. Suppose that every $f:\mathbb{N}\rightarrow\mathbb{N}$ is $\mathcal{U}$-equivalent to a ...
4
votes
2answers
507 views

Help on a proof about Ramsey ultrafilters

I was studying in this proof the implication $1\Rightarrow3$. I don't understand why $\{x_n,x_m\}\in[X]^2$ and $x_m>x_n$ implies $x_m\in X_{x_n}$. I believe it is not true, I think, first of all, ...
2
votes
1answer
118 views

Exercise on ultrafilters

I have to prove this: if $\mathcal{U}$ is a p-point and $f:\mathbb{N}\rightarrow\mathbb{N}$, if $f$ is not $\mathcal{U}$-equivalent to a costant, then it is $\mathcal{U}$-equivalent to a function ...
4
votes
1answer
376 views

ZFC + $\exists$ Standard model $\rightarrow$ Con(ZFC + $\exists \omega$-model)

$ZFC + \exists V_\alpha$ model of $ZFC \vdash Con(ZFC + \exists$ transitive standard model of $ZFC)$ and then $ZFC + \exists$ transitive standard model of $ZFC \vdash Con(ZFC + \exists \omega-model$ ...
2
votes
3answers
137 views

Identity of indiscernibles

It follows from the axioms of identity alone that $x = y \Rightarrow \big((\forall z) x \in z \equiv y \in z\big)$ and $x = y \Rightarrow \big((\forall z) z \in x \equiv z \in y\big)$. One of the ...
15
votes
2answers
694 views

Intersection of Algebraic Topology/Geometry and Model Theory/Set Theory

Is there any intersection between the ideas of Algebraic Topology/Geometry (I know that there is most certainly a non-trivial intersection between Algebraic Geometry, Algebraic Topology, Arithmetic ...
1
vote
1answer
139 views

convex sets and intersecting lines

Let's say that A is a convex set in $R^2$. Now assume that L is a line in $R^2$. $L=\{x: p\cdot x = t\}$ where p and x are both contained in $R^2$, $p\cdot x$ is the inner product of p and x, and t ...
23
votes
6answers
2k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
3
votes
1answer
475 views

Undecidability in ZFC

I would like to know whether the following statement is true or not: there is some first-order formula $\varphi$ such that "$\varphi$ is valid in all finite structures" is true although not provable ...
8
votes
1answer
234 views

How to prove an extension of ZFC is conservative

Working in ZFC. I've defined a function-like binary predicate $R$ on a proper class. It has to be recursive; i.e. $R(a,b)$ must usually depend on one or more $R(c,d)$ for some $c$s and $d$s ...
8
votes
4answers
492 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
6
votes
1answer
322 views

Finite subsets of a set A in the definable power set of A

I'm working through Kunen's famous book on set theory and I'm puzzled by exercise 19 of chapter VI. Background for the exercise: In chapter V (Definition 1.1) the author defines certain function of ...
5
votes
2answers
356 views

Sum and product of ultrafilters

can anyone tell me, please, two ultrafilters such that $\mathcal{U}\otimes\mathcal{V}\neq\mathcal{V}\otimes\mathcal{U}$ and others two such that ...
4
votes
1answer
219 views

p-point in $\beta\mathbb{N}$

my definition of p-point in $\beta\mathbb{N}$ is: $\mathcal{U}$ is a p-point if and only if every $\{A_n:n\in\mathbb{N}\}\subset\mathcal{U}$ has a pseudo-intersection, i.e. a $B\in\mathcal{U}$ such ...
1
vote
1answer
659 views

Prove something is a partial order

A relation $\mathrm{R}$ is defined on the set of all positive integers by: $x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$. Prove that $\mathrm{R}$ is a partial ...
4
votes
1answer
283 views

First-order Indistinguishibility of “the continuum”

Let us consider two different models of the continuum $\mathbb{R}$ (that is, we take two arbitrary ZF-models, and we look at the continuum in each one of these models). Let us now suppose that we ...
1
vote
1answer
47 views

I have two sets, each increasing at a certain constant rate. I need to find x given y

I have the following sets: x | 6 | 8 | 10 | 12 | 14 ... y | 4 | 5 | 6 | 7 | 8 ... I need to find the value of y given any positive value of x. I ...