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 ...

learn more… | top users | synonyms

1
vote
1answer
115 views

Relative consistency : “T proves F” VS “arithemic proves that T proves F”

This is related to my previous question. The thing was to show : $ZF \vdash \neg Con(ZF + AF) \longrightarrow \neg Con(ZF)$ If $ZF + AF$ is inconsistant then there is a finite number of $ZF + AF$ ...
2
votes
1answer
288 views

How can ZF prove relative consistency for itself?

This is related to my first question. In order to get what I don't get, I ll go with something much more specific here. It is well known that $ZF \vdash Con(ZF) \longrightarrow Con(ZF + AF)$. The way ...
17
votes
4answers
1k views

Why does Cantor's diagonal argument yield uncomputable numbers?

As everyone knows, the set of real numbers is uncountable. The most ubiquitous proof of this fact uses Cantor's diagonal argument. However, I was surprised to learn about a gap in my perception of the ...
10
votes
1answer
526 views

Logic, set theory, independence proofs, etc

I have some big troubles trying to understand specific set theory stuff. Especially when we demonstrate something about set theory we always have to keep our demonstration in set theory, typically ...
2
votes
2answers
523 views

Find a countable set of functions from an uncountable set?

I'm a uni student doing a real analysis course and am finding it very interesting, but at the same time very confusing. One question that has me stumped is how to get a countable set of functions ...
1
vote
1answer
182 views

Alternative proof of the cardinality of the set of all mappings

Can anyone please tell me if there is any other proof for the cardinality of all mappings, that is not by induction, i.e., not this one ...
4
votes
1answer
227 views

Does this make sense $\aleph_0+\aleph_1+\aleph_2$?

Let $\mathcal{A}$ denote the collection of all subsets A of an uncountable set $\Omega$ for which either A or $A^c$ are countable. Let $\mu(A)$ denote the cardinality of A. Define $\phi(A)$ equal to ...
4
votes
3answers
1k views

Proof: Cartesian Product of Two Sets is a Set ZF

Hi I'm having trouble with this proof. I'm not sure if I did the first part right and how I should use the second part properly. I am following the hint in the book to use the Axiom of Replacement ...
11
votes
1answer
380 views

Why is the real line not used in Descriptive Set Theory?

In most Descriptive Set Theory books, the rationale for working with the Baire space ($\mathbb{N}^{\mathbb{N}}$) as opposed to the real line ($\mathbb{R}$) is that the connectedness of the latter ...
6
votes
1answer
228 views

Strong cardinals and reflection

I'm new to all this large cardinal thing and I have trouble in proving the following: If $\kappa$ is a $\gamma$-strong cardinal, for some large enough $\gamma$, then $\kappa$ is ...
6
votes
1answer
348 views

Some questions regarding (relative) constructibility and the condensation lemma

I've got a question regarding the constructible universe and I'm a bit confused about the Condensation Lemma for the universe constructible from some set $A$. Help will be greatly appreciated: Let's ...
12
votes
5answers
2k views

What is a good text in intermediate set theory?

I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to ...
6
votes
1answer
201 views

If P is k-c.c. and C is club in k in M[G] then C contains a club in M

I've seen this written several places without proof, so I assume it's not difficult, but I am not getting it. Let $\mathbb P$ be a $\kappa$-c.c. notion of forcing, and let $C\in M[G]$ be club in ...
3
votes
2answers
192 views

What is the significance of “classes”?

In the introduction of Hungerford's Algebra (p. 2), he gives a rather trivial example of a class that is not a set, but what is the purpose of even having this term defined? Is it useful, other than ...
5
votes
1answer
304 views

Cardinality of sets of functions with well-ordered domain and codomain

I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated. If $X$ and $Y$ are ...
1
vote
1answer
85 views

How do I explicitly see that the Ultrapower map is the identity below its critical point?

I apologize in advance for how basic this question is... Let $j:V\rightarrow V/U$ be the ultrapower map where U is an ultrafilter on a set S, and $j(x)=[c_x]$. Now, let $f\in j(0)$. Then $f$ is ...
2
votes
1answer
99 views

Show that there is some $p$ in $\omega$ for which $m+p^+=n$.

This is an old exercise I wrote up, but I'm unhappy with my solution. I assume only the basic properties of addition and multiplication for natural numbers as sets. Assume that $m$ and $n$ are ...
4
votes
2answers
247 views

Equivalence of two definitions of Rudin–Keisler equivalence

Let $U$ is an ultrafilter on a set $X$, and $V$ an ultrafilter on a set $Y$. Wikipedia says: Ultrafilters $U$ and $V$ are Rudin–Keisler equivalent, $U\equiv_{RK}V$, if there exist sets $A\in U$, ...
64
votes
5answers
3k views

What are the issues in modern set theory?

This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed ...
3
votes
4answers
319 views

Is it possible to prove that the metric space is an open set without choice?

Suppose that $(X, \rho)$ is a metric space, $|X| > 1$. Is it possible to prove that $X$ is an open set without assuming the axiom of choice? As I understand it, the challenge is to find a way to ...
77
votes
16answers
7k views

Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
6
votes
2answers
331 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
3
votes
1answer
86 views

Given a set of 2D points (x,y) (cloud of points), find the points that, when connected, will contain all other points

Given a set of 2D points I have to find the points that when connected will form a polygon that contains all the points in the set. A quick example: imagine you have a set ...
7
votes
1answer
641 views

Equivalence relation on a proper class

We define cardinality as an equivalence relation on sets. But the class of all sets is not a set, so how do we do that? In particular, I'm interested in the proposition that equivalence classes form a ...
9
votes
4answers
554 views

Explicit well-ordering of $\mathbb{N}^{\mathbb{N}}$

Is there an explicit well-ordering of $\mathbb{N}^{\mathbb{N}}:=\{g:\mathbb{N}\rightarrow \mathbb{N}\}$? I've been thinking about that for awhile but nothing is coming to my mind. My best idea is ...
30
votes
3answers
7k views

First-Order Logic vs. Second-Order Logic

Wikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic ...
10
votes
1answer
368 views

Is the sentence “$(A,\in)\models ZFC$” absolute?

I know that we can assume that formulas are objects in $V_\omega$, and that notions such as formula and satisfiability for a standard model (when the universe is a set) are definable and absolute. ...
8
votes
1answer
241 views

On the existence of closed form solutions to finite combinatorial problems

Is it possible that a finite combinatorial problem may admit a closed form solution, and for it to be impossible in practice to prove the validity of this solution? I'm not sure if a rigorous ...
6
votes
2answers
472 views

Constructing Infinite Cartesian Products without AC

I recently stumbled across the wikipedia page on equivalents to the Axiom of Choice. I noticed that every infinite Cartesian product of a non-empty family of non-empty sets being non-empty was ...
21
votes
2answers
567 views

Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?

Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
3
votes
1answer
276 views

Class models in set theory and category theory

Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, ...
5
votes
2answers
914 views

Some questions concerning the size of proper classes in ZFC

For some formulae $\phi(x)$ it can be proved from the axioms of ZFC, that there is no set $X$ with $(x)x\in X \equiv \phi(x)$. Thus the collection $\lbrace x\ |\ \phi(x)\rbrace$ is a proper class. ...
0
votes
2answers
355 views

Equality of abstract structures

Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing ...
6
votes
2answers
307 views

Does the principle of mathematical induction extend to higher cardinalities?

Does the principle of mathematical induction extend to a cardinality larger than that of the countably infinite?
3
votes
1answer
182 views

Describing all elements in the set algebra generated by given sets in an infinite product

Let $X_i$ be a sequence of probability spaces and define $\displaystyle X=\prod_{i=1}^\infty X_i$ Let $A$ be the algebra on $X$ generated by the sets of the form $$\displaystyle \prod_{i=1}^{n-1} X_i ...
1
vote
1answer
117 views

What is the cardinality of a subset of the hyperbolic upper half plane?

Given a subset of the hyperbolic upper half plane, say an ideal triangle (so with vertices on the boundary), what is the cardinality of all points contained in the interior?
16
votes
1answer
456 views

Forcing Classes Into Sets

I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions. I know from Jech's Set Theory that you can force that the class of ordinals in the world ...
21
votes
2answers
1k views

How can there be alternatives for the foundations of mathematics?

How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of ...
11
votes
1answer
2k views

What does it mean to say a map “factors through” a set?

Consider the following diagram: commutative diagram (Sorry, It wouldn't let me directly post the image.) What does it mean precisely to say "$f$ factors through $G/\text{ker}(f)$"? Does it mean $f ...
7
votes
1answer
319 views

Every member of an ordinal is an ordinal

How to prove that, if a is an ordinal and b is in a, then b is an ordinal? Here are the definitions I'm using. A set is an ordinal number if it is transitive and well-ordered by ∈. A set T is ...
8
votes
2answers
384 views

What does Martin's Maximum imply for $P(\mathbb{R})$?

Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in ...
4
votes
1answer
189 views

Closed subset of stationary set and AC questions

Hope I'm not spamming too much by asking questions on separate threads. I have 2 more questions, not connected one to another, in any way: 1. Show that every stationary set in $\aleph_1$, contains, ...
8
votes
1answer
294 views

Cardinal equality question

The question contains 2 stages: 1. Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$ This one is pretty clear by induction and by applying Hausdorf's formula. 2. Prove ...
5
votes
0answers
226 views

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of ...
5
votes
2answers
1k views

Some questions in Set Theory

I have some exam questions that were left unanswered for me: Suppose that for every $\alpha<\kappa$ there is a subset $A_\alpha$ of $\kappa$ of cardinality $\kappa$. Show that there is a subset X ...
12
votes
1answer
765 views

bound on the cardinality of the continuum? I hope not

Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for ...
1
vote
4answers
298 views

Does the reflexivity mean maximum element in partial ordering $x \leq x$ for every $x \in S$

The notation of my teacher confuses me in the title. If x is the same on the both sides, it seems trivial. So I suspect the x is ...
19
votes
1answer
861 views

Bijection between $2^{\mathbb R}$ and $\mathbb{ R ^ R}$

I'm well aware of the standard proof based on cardinality arithmetic to show that these two sets have the same cardinality and the question of defining a bijection between the two sets came up. I ...
8
votes
1answer
887 views

Polish Spaces and the Hilbert Cube

I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : ...
13
votes
4answers
2k views

Is there such a thing as a countable set with an uncountable subset?

Is there such a thing as a countable set with an uncountable subset? Actually I know the answer. Well, I believe I know the answer, which is NO. Unfortunately, the professor in a Theory of ...