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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1answer
16 views

Relation between wave equation and wave functions

I would like to know answers to these questions: Can we say that the wave equation is a predicate that wave functions (solutions to the wave equation) satisfy? Can we say that the wave equation is a ...
1
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1answer
46 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
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1answer
59 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, ...
2
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1answer
26 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 $b\to a$. ...
0
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1answer
36 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 ...
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0answers
47 views

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

Working on an excersise asking me to (in part) find such a sentence. Having a hard time. Any hints?
1
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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 $M$?
3
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1answer
55 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 ...
1
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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 $[P\to P]$ ...
1
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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
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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 ...
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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
101 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 ...
29
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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 ...
4
votes
1answer
62 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}_{ ...
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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
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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
42 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 ...
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1answer
24 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
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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.
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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 ...
2
votes
1answer
70 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 ...
-1
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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
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1answer
30 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
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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 ...
3
votes
0answers
75 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 $2^{<\lambda}=\lambda$ ...
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 ...
3
votes
1answer
105 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 ...
2
votes
2answers
72 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
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 ...
1
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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?
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 ...
1
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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 ...
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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 $f$ ...
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 ...
9
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1answer
108 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 ...
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): ...
1
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1answer
28 views

Simple question about stationary sets in transitive models of ZFC

Let $\kappa$ be a regular cardinal, and suppose $X\subseteq\kappa$ is stationary in $\kappa$. Furthermore, let $\mathcal{M}$ be a transitive class model of ZFC with $X\in\mathcal{M}$. I'm trying to ...
-1
votes
1answer
46 views

Least ordinal $\beta$ such that it is provable that $2^{\aleph_0} \leq \aleph_{\aleph_\beta}$

What is the east ordinal $\beta$ such that it is provable in $\mathsf{ZFC}$ that $2^{\aleph_0} \leq \aleph_{\aleph_\beta}$?
1
vote
1answer
52 views

Need this transfinite construction reach a closed set in the absence of the Axiom of Choice? Or Hartogs' theorem?

In this question I'm primarily concerned with the workings of a set theory that lacks both Foundation and Choice. Given a set $X$ and a function $\sigma:\mathcal{P}(X)\to\mathcal{P}(X)$, we have a ...
1
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1answer
30 views

Filters and antichains in preorders

In a preorder $\mathbb{P}$ (reflexive and transitive), what means to say that a subset $C\subset\mathbb{P}$ is not contained in any filter of $\mathbb{P}$? For example, it is obvious that if $C$ ...
3
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2answers
93 views

How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
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 ...
1
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1answer
57 views

Ernst Zermelo's counterexample

According to the book Real Analysis by Royden page 6(=1+2+3): Given an equivalence relation on a set X, it is often necessary to choose a subset C of X which consists of exactly one member from ...
5
votes
1answer
58 views

If two posets have same dense open sets, are they equivalent as notions of forcing?

Suppose that $\mathbb{P}_0=(P,\leq_0)$ and $\mathbb{P}_1=(P,\leq_1)$ are partial orderings (in the weak sense, i.e., reflexive and transitive relations) on the same underlying set $P$, and such that ...
5
votes
1answer
85 views

Statements independent of ZF that quantify over the real numbers

(This question is a bit vague, because I probably haven't aquired all the logical tools needed to express it in a more concise way) I've seen a few examples of statements in set theory that can ...
4
votes
1answer
38 views

What is “non-simple applied first-order functional calculus” (60's set theory)

Azriel Lévy says in his 1960 paper Axiom Schemata of Strong Infinity in Axiomatic Set Theory, that the $\sf{ZF}$ set theory is formalized with a finite number of axioms in "non-simple applied ...
3
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1answer
66 views

Why does $\epsilon_0 = \omega^{\epsilon_0}$

I saw this on wikipedia and wasn't sure why. It seems obvious given the definition but how can we be sure? Also, what would be $\epsilon_0 \bullet \epsilon_0$? Would it be $\epsilon_0$? Thanks
0
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0answers
22 views

Can epsilon induction be derived from the transitive closure induction?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations. The induction of the transitive closure ...