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
1answer
45 views

Independent families of sets

I'm having a difficulty understanding some exercises related to independent families of sets. Recall that $ \mathcal{A} $ is $\lambda$-independent if for any disjoint $ P, Q \in \mathcal{A} : |P|, |Q| ...
2
votes
1answer
25 views

Existence of a particular element of an ultrafilter

I'm getting to know some ultrafilter theory. I'm stuck on the following exercise: Suppose $ \mathcal{U} $ is an ultrafilter on $ \omega $. Prove that there exists $ A \in \mathcal{U} $ such that ...
4
votes
1answer
41 views

Consistency of ZFC + “for every function there exists a class inaccessible to it”

Is ZFC + the following statement consistent (and if so, is it equiconsistent to some known large cardinal): For every function $f:ORD \rightarrow ORD$ such that: $f(\alpha)\geq \alpha$, $\alpha ...
0
votes
0answers
34 views

Maximal antichains in a forcing which adds surjections

Let $P$ be a separative partial order such that $\left| P \right| \leq \left| \alpha \right|$ and $$\Vdash_P\exists f(f\colon\omega\to\alpha\text{ is surjective}\land f\notin\check V).$$ I want to ...
1
vote
1answer
53 views

The generalized continuum hypothesis can't fail first at $\omega_{\omega_{1}}$

I am willing to prove that the GCH cannot first fail at a singular cardinal. For this purpouse I am following the strategy outlined by Kunen in his 2013 book (see Exercises III.6.16-6.17-6.18-6.19). I ...
2
votes
1answer
66 views

Why wasn't Bertrand Russell surprised by the set of all sets that contain themselves?

Russell's paradox deals with the question: "Does the set of all sets that do not contain themselves, contain itself?" What about the question: "Does the set of all sets that contain themselves, ...
12
votes
2answers
194 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
2
votes
1answer
27 views

If $a\sqcup b$ and $a\times b$ biject, then $b$ either injects or surjects in-/onto $a$

Let $a$ and $b$ be sets such that there is a bijection $a\sqcup b\to a\times b$. Show, without assuming the Axiom of Choice, that there is either a surjection $b\to a$ or an injection $b\to a$. ...
3
votes
1answer
56 views

Well-pruned $\kappa$-Suslin trees have the $\kappa$-Baire property

I'm having troubles seeing that a well-pruned $\kappa$-Suslin tree, with $\kappa$ uncountable and regular, has the $\kappa$-Baire property. I saw in Kunen's Set Theory that it can be seen by proving ...
0
votes
1answer
37 views

Does the MK axiom of infinity and the Power set axiom hold in the cumulative hierarchy?

Question: Define the cumulative hierarchy inductively as: $$V_0 = \emptyset$$ For each ordinal $\alpha$: $$V_{\alpha+1} = \mathcal{P}(V_\alpha)$$ ie the power set of $\alpha$ and for any nonzero ...
31
votes
6answers
2k views

Do the axioms of set theory actually define the notion of a set?

In Henning Makholm's answer to the question, When does the set enter set theory?, he states: In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is ...
1
vote
1answer
44 views

Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in L. Since ...
0
votes
0answers
49 views

$\Pi_2$ sentence $\sigma$ such that $L_{\omega_1}\vDash \sigma$ but false in L [closed]

Working on an excersise asking me to (in part) find such a sentence. Having a hard time. Any hints?
1
vote
1answer
34 views

A proof of maximality of an antichain in a complete Boolean algebra

Suppose $(\mathbf{B},\leqslant)$ is a complete Boolean algebra and let $|\mathbf{B}|=|\gamma|$. Let $C=\langle c_\alpha\mid \alpha<\gamma\rangle$ be a maximal descending chain without a lower ...
1
vote
1answer
35 views

Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
1
vote
1answer
27 views

Order-preserving maps on a directed-complete poset

Definitons. A poset $P$ is directed if every finite subset has an upper bound in $P$. It is directed-complete if every directed subset has a least upper bound in $P$. For any poset $P$ let $[P\to P]$ ...
3
votes
1answer
106 views

Justification of ZFC without using Con(ZFC)?

I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in ...
1
vote
0answers
49 views

A classification of the types of poset extensions that raise local-global principle is equivelent to the axiom of choice?

EDIT: Completely overhauled the question, remove category theory from it (stated in plain set theory): The original question Adding relations to a partial order Let $X$ be a set and let $P: Pow(X) ...
6
votes
1answer
103 views

Is ZFC ω-consistent over ZF?

Gödel proved that if ZF is consistent, then ZFC is also consistent. He did this by showing that his constructible universe is a model of ZFC. Gödel also introduced the notion of an $\omega$-consistent ...
2
votes
1answer
71 views

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
4
votes
1answer
63 views

Show forcing condition compatibility by induction

Say that we have a countable support forcing iteration $ \mathbb{ P}_{ \alpha }$ ( using Jech's definition ) where $ \alpha $ is a limit ordinal, and consider two conditions $ f , g \in \mathbb{ P}_{ ...
0
votes
0answers
27 views

Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. [duplicate]

Let $A \subseteq \mathbb{R}$ and $|A|=\omega$, prove $|\mathbb{R}\setminus A| = 2^\omega$. Hint: $\mathbb{R}\sim \{0,1\}^\mathbb{N}$ I'm somewhat stuck trying to prove this, could someone ...
0
votes
1answer
37 views

If $ \bigcup N_\alpha $ is stationary, then $ \{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $ \kappa $ and a disjoint family ...
2
votes
2answers
24 views

Unboundedness of the set of subgroups of a cardinal

I'm trying to figure out why the following is true: Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ ...
2
votes
1answer
43 views

Checking whether some sets are clubs in $ \aleph_2 $ and $ \aleph_1 $

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a ...
1
vote
2answers
105 views

Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?

I am studying axiomatic (ZF) set theory and in particular the construction of the natural numbers from axiomatic set theory. Let me start by giving some introduction to my question. The axiom of ...
-3
votes
1answer
28 views

Relation between limit ordinals and alephs. [duplicate]

I was wondering what the relation is between a limit ordinal and the alephs. Are all limit ordinals alephs and if so can it be proven.
-2
votes
1answer
25 views

Is the set of all limit ordinals a set? [closed]

It could be a limit ordinal too, then would be a class, but I'm not sure about is that an ordinal number
-1
votes
1answer
31 views

Proof involving well ordering

Let $A,B$ be well ordered sets with corresponding well orderings $\leqslant $ and $\leqslant '$. If $A$ is order isomorphic with $B$ initial segment $B '$ and $B$ is order isomorphic with $A$ initial ...
-1
votes
1answer
35 views

Comparing infinite cardinals [closed]

I have a question concerning infinite cardinals which I found on an old exam paper: Let $c=2^{\aleph_0}$, $x=2^c, y=2^{2^c}, z=2^{2^{2^c}}$. Put $x^{y^z}, x^{z^y}, y^{z^x}$ in ascending order. I'm ...
1
vote
1answer
31 views

Cardinality of the set of all Hamel bases.

If we look at $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a Hamel basis $B$ has cardinality $\mathcal{c}$ (assuming the continuum hypothesis). What about the cardinality of the set of ...
1
vote
1answer
43 views

Diagonal union of non-stationary sets

I have a family of non-stationary sets $A_{\alpha}$ for $\alpha < \kappa,\ A_{\alpha}\subset\kappa$. The exercise is to show that $\triangledown A_{\alpha}$ is also non-stationary. I've been ...
6
votes
3answers
239 views

Why is establishing absolute consistency of ZFC impossible?

Why is establishing the absolute consistency of ZFC impossible? What are the fundamental limitations that prohibit us with coming up with a proof? EDIT: This post seems to make the most sense. In ...
2
votes
1answer
111 views

Prove $\bigcup \omega_1 = \omega_1$

This is a question regarding ordinals and probably requires some background knowledge of ordinal arithmetic. I will list what I hope is the relevant information for this question: $\omega_1$ is ...
3
votes
0answers
77 views

Existence of $\lambda^+$ Aronszajn trees when $\lambda$ is regular and $2^{<\lambda}=\lambda$

While I was dealing with Aronszajn trees I found the following exercise from Kunen's old book. If $\kappa=\lambda^+$ and $\lambda$ is a regular uncountable cardinal and $2^{<\lambda}=\lambda$ ...
3
votes
2answers
87 views

A class that contains itself as an element

Can a class be defined which contains itself as an element? I know its forbidden for a set to contain itself, and I have seen arguments that suggest it's possible for a class to do so but none of ...
5
votes
0answers
58 views

Examples of provably${}^n$ unprovable statements

Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement ...
2
votes
2answers
73 views

Can number 2 be defined as a formula in set theory?

Natural numbers can be represented as sets, however there are more than one representation of natural numbers in set theory (for example von Neumann's and Zermelo's). But all the representations of ...
2
votes
1answer
54 views

Large Cardinal Extension Property

I have been reading Kanamori's Higher Infinite and I am trying to understand that a cardinal $\kappa$ is $\Pi^1_1$-indescribable iff it has the extension property. We say that $\kappa$ has the ...
2
votes
1answer
33 views

Help with a problem about club sets

Let $\kappa$ be a regular, uncountable Cardinal and let $f:\kappa\rightarrow\kappa$. I'm trying to show that $\{\alpha<\kappa\mid f''\alpha\subseteq\alpha\}$ is club in $\kappa$. I can see why it's ...
8
votes
2answers
688 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 ...
1
vote
0answers
26 views

Is the Axiom of Choice required to prove $|\alpha|=|L_\alpha|$, for infinite ordinals? [duplicate]

The proofs of this fact I've seen all rely on appealing to AC at some point or another. But is this required? Is there a choiceless proof?
1
vote
1answer
34 views

A question to $\Diamond$ implies the existence of a Suslin tree

I'm reading the Proof that $\Diamond$ implies the existence of a Suslin tree in Jech, Set Theory (2003), p.241. The nodes in the constructed tree are countable ordinals, so $T=\omega_1$, and every ...
0
votes
1answer
78 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
5
votes
2answers
431 views

Notation for surreal numbers

On the sound of sounding ridiculous, but in the line of "There are no stupid quetsions": Is there a way to express $\omega_1$ (and in general $\omega_k$ with $k >= 1$ as a Conway game (that is ...
0
votes
0answers
20 views

Ordinals recursed

Consider any limit ordinal $\alpha$. It seems intuitively obvious to me that for any such ordinal there is $f:ORD \to ORD$ (with $ORD$ being the class of ordinals) with $f(1)=\alpha$ and $f$ ...
9
votes
1answer
109 views

Independent Transcendental Numbers

I've been thinking about numbers which have not yet been proven nor disproven to be transcendental, such as $e + \pi,\, \pi - e,\, \frac{\pi}{e},\, \gamma,\,\zeta(3),$ etc. Some of these numbers ...
5
votes
3answers
64 views

How can an $\omega_1$-tree possibly be normal and yet not have any $\omega_1$-branch?

An $\omega_1$-tree is a tree of height $\omega_1$. An $\omega_1$-tree $T$ is normal if: $T$ has a unique least point (the root); every level of $T$ is at most countable; if $x \in T$ then there ...
6
votes
2answers
73 views

Original proof of Zorn's Lemma

On the wiki page for Zorn's Lemma it says that this lemma was Proved by Kuratowski in 1922 and independently by Zorn in 1935 but then it says: Zorn's lemma is equivalent to the well-ordering ...
0
votes
1answer
42 views

Which rules of inference does Suppes use?

I'm reading Axiomatic Set Theory by Suppes, and I'm having a bit of trouble understanding which rules of inference (logical system) he is using, here's an example (capital letters are used for sets): ...