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

7
votes
2answers
250 views

Cardinality of the collection of all compact metric spaces

I was wondering today, that if $\mathscr{M}$ is the collection of all sets that admit a metric generating a compact topology, then Is $\mathscr{M}$ a set in ZFC? If it is, what is the cardinality ...
3
votes
2answers
118 views

Splitting the Real Line

By definition a $\mathfrak c$-dense subset of $\mathbb{R}$ has $\mathfrak c$-sized intersection with every non-empty open set. Using transfinite recursion it is quite easy to prove that every ...
18
votes
6answers
2k views

Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this ...
5
votes
3answers
1k views

What's wrong with this proof of the inconsistency of the axiom of choice?

Let $\mathscr{T}$ be the (countable) collection of all theorems provable in ZFC. Define an equivalence relation on $\mathscr{T}$ by $\phi\sim\psi$ iff $(\phi \iff \psi)$. In other words, two theorems ...
3
votes
1answer
81 views
4
votes
1answer
67 views

The Diamond Principle implies the Club Principle.

So the Diamond and the Club principles are both combinatorial principles in set theory. They are defined as follows (there are thinner definitions but I stick to this ones is $\omega_1$, as I am sure ...
6
votes
0answers
79 views

Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
0
votes
1answer
39 views

Ordinal arithmetic and limit ordinals

Suppose $1\leq\xi<\omega_1$ is a countable ordinal and that $1\leq\zeta<\omega^\xi$. Is it always true that $\zeta+\omega^\xi=\omega^\xi$? If so, why?
3
votes
1answer
40 views

Splitting Stationary Sets

So by a theorem of Solovay we know that a stationary subset of a regular (uncountable) cardinal $\kappa$ can always be split into $\kappa$-many stationary subsets. Is the regularity assumption ...
0
votes
1answer
80 views

Hanf Numbers and Decidability

Currently reading J.L. Bell's Models and Ultraproducts and at the end of Chapter 4 section 4 the authors comment that "In spite of the fact that most languages can easily be shown to possess Hanf ...
4
votes
1answer
62 views

Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
5
votes
3answers
166 views

Lists of sets as objects of ZF axiomatics

I have a naive question about foundations of mathematics. A common opinion of most mathematicians is that the essential part of mathematics can be reduced to ZF(C) axioms. I do not quite understand ...
1
vote
0answers
44 views

If $M\prec L_{\omega_1}$ then $M = L_\alpha$ for some $\alpha$ - we need a condition to prove it?

I try to prove the exercise 13.17 in Jech: If $M\prec (L_{\omega_1},\in)$, then $M=L_\alpha$ for some $\alpha.$ [Show that $M$ is transitive. Let $X\in M$. Let $f$ be the $<_L$-least ...
4
votes
1answer
80 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 ...
1
vote
3answers
72 views

Why do we have Axiom of Pairing but we don't have its generalisation, i.e a collection exists instead of pairing axiom?

For unions we have the generalised axiom, not just union for pairs. But for pairing we don't have generalisation, that a collection exists for any number of sets. If we have some axiom that says ...
1
vote
1answer
37 views

Can a sum of a nonprincipal ultrafilter and a principal ultrafilter be equal to the nonprincipal ultrafilter?

If $ \mathcal U$ is a nonprincipal ultrafilter and $\mathcal V$ is a principal ultrafilter, can $ \mathcal U \oplus \mathcal V$ be equal to $\mathcal U$ ?
1
vote
0answers
72 views

is axiom of powers required?

we can form any subset of a set using axiom of specification. then we can form collection of subsets using axiom of pairing repeatedly. we have for every condition S , (A={x belongs to X & S(x)} ...
10
votes
0answers
181 views

The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need ...
7
votes
1answer
107 views

For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$

I'm trying to do a past paper question which states: $$ \text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}. $$ I'm supposed to be able to do this without the axiom of ...
9
votes
1answer
314 views

Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?

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

Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
3
votes
1answer
119 views

If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
2
votes
2answers
148 views

Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
3
votes
1answer
108 views

Extending Banach-Tarski paradox?

I've learned the Banach-Tarski paradox as following: The points on the sphere (but not the fixed points) are drawn as a square grid, form each point there are three new directions plus the direction ...
7
votes
3answers
684 views

Banach-Tarski theorem without axiom of choice

Is it possible to prove the infamous Banach-Tarski theorem without using the Axiom of Choice? I have never seen a proof which refutes this claim.
6
votes
1answer
633 views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...
3
votes
3answers
629 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
1
vote
1answer
35 views

The set of ordinals $< \alpha$ of a given cofinality $\kappa< \text{cf}(\alpha)$ is stationary

I don't understand the last line of the following proof : I understand that we climbed up to $g(\kappa)$ in $\kappa$ steps (i.e. cf$\big(g(\kappa)\big) \leq \kappa$) but can we be sure that we did ...
2
votes
1answer
39 views

Clarifying some doubts on the definition of “extensional classes”

On page 68 of Jech's "Set Theory" 4th Edition there is the following definition : A class $\mathcal{M}$ is extensional if the relation $\in$ on $\mathcal{M}$ is extensional, i.e., if for any ...
1
vote
1answer
195 views

Reference request, Descriptive set theory

I was wondering what a good text would be to learn descriptive set theory out of? Hopefully something more in the spirit of Kunen's text on the introduction to independence proofs.
2
votes
4answers
256 views

Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
1
vote
4answers
308 views

What is the future of Set Theory if it is NOT the foundation of Mathematics?

Homotopy Type Theory has been receiving a lot of hype during the last couple of years and been hailed as the most effective foundation of all Mathematics compared with Set Theory. My question: If ...
2
votes
1answer
66 views

What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples: The set of all finite ...
1
vote
1answer
88 views

Is this behavior of the continuum function consistent with ZFC.

I have an axiom, based on how finite cardinalities work. As we know, every ordinal can be written as the sum of a unique limit ordinal $L$ (where $0$ is a limit ordinal axiom for the purposes of this ...
5
votes
1answer
279 views

$\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the ...
11
votes
1answer
231 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
1
vote
1answer
33 views

Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right ) $$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
2
votes
0answers
54 views

Supplementing FOL with the Härtig quantifier

"The Härtig quantifier captures a certain fairly large fragment of second-order logic and contributes to understanding higher-order logic." (p. 1154 ...
27
votes
5answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
1
vote
1answer
48 views

Indiscernibles and colorations.

Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of ...
1
vote
0answers
34 views

Condition for an Ultrafilter to be Ramsey.

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 ...
2
votes
1answer
72 views

Non analytic ideals on $\omega$

I would like to gather examples of NON analytic ideals on $\omega$. However, I have found nothing in the books and papers I have consulted. Could anyone tell me some reference/s?
1
vote
1answer
48 views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
1
vote
1answer
49 views

How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
1
vote
1answer
45 views

P-names are comprehensible in the ground model

If my understanding of $\mathbb{P}$-names $\tau$ is correct, they must be comprehensible by someone living in the ground model $\mathcal{M}$. However, the value $\tau[G]$ of $\tau$ need not be ...
3
votes
1answer
124 views

Must $\mathfrak{sd} = \mathfrak{d}$?

This should be fun. Let $\mathfrak{sd}$ be the least cardinal $\kappa$ such that there exists $\langle s_i : i < \kappa \rangle$ satisfying the following. (1) Each $s_i: \omega^{\omega} \to ...
6
votes
1answer
154 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
4
votes
1answer
95 views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
3
votes
0answers
49 views

Does “$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$” implies the axiom of choice? [duplicate]

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that ...
0
votes
0answers
35 views

Intersection with the empty set in Principia Mathematica

In Whitehead and Russell's Principia Mathematica, "arithmetical class-sum" is defined as I have tried my best to decipher the dated notation in the book. It seems that the sum is defined as the ...