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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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) \...
7
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1answer
117 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 ...
33
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7answers
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
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1answer
64 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 give ...
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
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2answers
26 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
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1answer
50 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 club,...
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1answer
29 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
32 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
32 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 ...
3
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1answer
96 views
+50

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 ...
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1answer
42 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
32 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
45 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
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0answers
85 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 $2^{<\lambda}=\lambda$ ...
6
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1answer
85 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
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1answer
117 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
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2answers
78 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
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1answer
34 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 ...
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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
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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
37 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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1answer
41 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
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3answers
74 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 ...
11
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1answer
126 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 haven'...
0
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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
30 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 ...
0
votes
1answer
49 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 $2^{\aleph_0} \leq \aleph_{\aleph_\beta}$?
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1answer
54 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 ...
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1answer
33 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
98 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
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2answers
80 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 ...
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1answer
61 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
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1answer
64 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
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1answer
87 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
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1answer
40 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 first-...
3
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1answer
71 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
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0answers
23 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 ...
4
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0answers
55 views

Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, ...
3
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3answers
155 views

In which order should I learn the foundations of mathematics? [closed]

I know from Wikipedia that those are the four pillars of the foundations of mathematics: Proof theory Aximatic Set theory Model Theory Recursion Theory and I want to learn all of them, the problem ...
20
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3answers
805 views

Is there a set theory in which the reals are not a set but the natural numbers are?

Is there any known axiomatization of set theory in which the real numbers are not a set, but the natural numbers and other infinite sets do exist? Such a set theory would have an Axiom of Infinity, ...
1
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1answer
27 views

Range of a P-name

I am working on a set theory problem from Kunen's Set Theory book, and it involves knowing $\text{ran}(\tau)$ where $\tau$ is a $\mathbb{P}$-name. The entire section loves to talk about the domain of ...
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0answers
25 views

Applications of the Axiom of Regularity to non-set-theoretical Mathematics [duplicate]

In the beginning of my mathematics studies at university, we have learnt that nearly all of ordinary mathematics not dealing with proper classes can be formalized within ZFC, which is a famous ...
2
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0answers
41 views

Is this an accurate layman's description of the Anti Foundation Axiom

I'm writing an article that covers as one of its topics hypersets/non-well founded sets. In order to do so I have to describe what the anti-foundation axiom (AFA) is my description is currently as ...
2
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1answer
52 views

Show that $3\cdot |A|<9^{|A|}$ for every $A$

I am trying to prove this question which came up in my university's set theory exam last year. A few similar questions have been asked over the last few years and I cannot figure out the method to ...
1
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1answer
52 views

Non-principal ultrafilters on a set [duplicate]

Let $X$ be a set. If $X$ is finite then all ultrafilters on $X$ are principal, i.e. have the form $\{A \subseteq X : x \in A\}$ for some $x\in X$. But now suppose $X$ is infinite, say $X=\mathbb N$. ...
5
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1answer
85 views

Forcing and violation of the $GCH$ at $\aleph_\omega$

In page 295 of Kunen's Set theory the author asserts that if $M$ is a countable transitive model of the axiom of constructibility $V=L$ then no forcing extension of $M$ can satisfy the theory $$ZFC\,\...
5
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1answer
292 views

Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?

I hope this is not too off-topic / soft for math.stackexchange. My basic question is: why does the Axiom of Choice allow for some of the weirdest constructions in math? I'll make a list of the weird ...
0
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1answer
55 views

Are there any large cardinals that are not ordinals? [closed]

In ZF, are there any useful large cardinal that cannot be well-ordered? I think that some of the partition cardinals are that way, since with AC, we cannot have $\kappa \to (\omega)^{\omega}$. Are ...