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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5
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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 ...
4
votes
2answers
481 views

picking a witness requires the Axiom of Choice?

$\forall I, A:Set. I\subseteq \bigcup A\to \exists f:I\to A. \forall i\in I. i\in f(i)$ Does this theorem require the Axiom of Choice? To prove, I need to find $\forall i\in I$ a witness of that ...
3
votes
4answers
151 views

Why are partial orderings important?

I was reviewing my old Discrete Mathematics notes, and I came across a section describing how Partial Orderings are identified. I understand this, but I can't seem to recall/find information on why ...
12
votes
1answer
499 views

Do sets, whose power sets have the same cardinality, have the same cardinality?

Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$? Why? Thanks.
8
votes
1answer
551 views

The well ordering principle

Here is the statement of The Well Ordering Principle: If A is a nonempty set, then there exists a linear ordering of A such that the set is well ordered. In the book, it says that the chief advantage ...
5
votes
1answer
268 views

Is there a set with cardinality greater than N but less than R?

Is there a set with cardinality greater than the natural numbers but less than the real numbers? Is there a simple proof which shows this, if the answer is no?
3
votes
1answer
863 views

A transitive set of ordinals is an ordinal

This is Exercise III.2.20 of Bourbaki's Set Theory. (Von Neumann ordinals are actually called "pseudo-ordinals" by Bourbaki, but I simply call them ordinals here) Let $X$ be a transitive set, and ...
8
votes
3answers
533 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
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
290 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 ...
18
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
533 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
524 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
183 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
230 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
384 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
357 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
1k 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
203 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
307 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
86 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
248 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
320 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 ...
78
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
335 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 ...
8
votes
1answer
651 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
556 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 ...
32
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
371 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
243 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
478 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
570 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
281 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, ...
6
votes
2answers
951 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
356 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
310 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
183 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
118 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
460 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 ...