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
116 views

Width and height of partial ordered sets

The width $w$ of a partial ordered set(poset) is defined as the cardinality of the maximum antichain. By Dilworth Theorem, we know it is equivalent to the minimum number of chains in any partition. ...
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0answers
46 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that I found that $\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $. However I found ...
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vote
1answer
63 views

If $X$ is a subset of $\omega_{\alpha}$ such that $|X| < \aleph_{\alpha}$, then $|\omega_{\alpha} - X| = \aleph_{\alpha}$

If $\alpha=0$, then $\omega_0=\mathbb{N}$ and $\aleph_0=$ countable. So $|\omega_{\alpha} - X| = \aleph_{\alpha}$ becomes $|\mathbb{N}-X|=|\mathbb{N}|$ which is true (the function $f:\mathbb{N} ...
1
vote
1answer
66 views

Product of a Suslin tree with itself

From Kunen "Set Theory", Chapter II, Exercise 36: If $T, T'$ are $\kappa$-trees, the product, $ T \times T' $ is the $\kappa$-tree whose $\alpha$-th level is $\mbox{Lev}_\alpha(T) \times ...
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votes
3answers
114 views

Showing a set of indexes where restriction is bijection is a club

I'm trying to show some general statement: I have a regular cardinal $ \kappa $ and an increasing continuous family of sets $ \langle X_\alpha \mid \alpha < \kappa \rangle $ with $ ...
0
votes
1answer
74 views

Godel Universes [closed]

Can somebody give me a nice and clear definition of what these are at different levels and different ordinals. I have read the wikipedia page and talked with peers, but am still confused about the ...
3
votes
1answer
92 views

ZF Set Theory and Law of the Excluded Middle

I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the ...
1
vote
1answer
35 views

A filter $G$ on $P$ is generic iff whenever $W$ is a partition of $P$, $G \cap W \not = \emptyset$

I'm finally struggling a little bit with the notion of forcing, so I decided to do some of the exercises in Jech's book (2nd Edition) in order to sharpen a little bit my intuitions about the whole ...
0
votes
2answers
60 views

Show that $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ and $n \cdot \aleph_{\alpha} = \aleph_{\alpha}$

a) Give a direct proof of $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ by expressing $\omega_{\alpha}$ as a disjoint union of two sets of cardinality $\aleph_{\alpha}$. b) Give a direct proof ...
2
votes
2answers
83 views

Embedding of linear order into $\mathcal{P}(\omega)/\mathrm{fin}$

I try to prove following problem (in Kunen): Assume $\mathrm{MA}(\kappa)$ and $(X,<)$ be a total order with $|X|\le\kappa$, then there are $a_x\subset \omega$ such that if $x<y$ then ...
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0answers
65 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
2
votes
1answer
50 views

Introducing a new element to make a new model of set theory

Say we have a model of set theory $V$ and a partially ordered structure $\mathbb{P}$, and I want to talk about a $V$-generic filter $G$. A $V$-generic filter is a filter such that for every $D\in V$ ...
10
votes
3answers
1k views

Best book on axiomatic set theory.

Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.
5
votes
1answer
39 views

What's a good introductory text to ZF set theory? [duplicate]

I've done the usual undergraduate coursework and am interested in learning about ZF set theory. What are some texts that would be accessible to me, and what are the most popular texts in this ...
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0answers
37 views

Proving Replacement in $ZF^-$ without Replacement but with Collection

I was told in lecture, that in $ZF^-$ the replacement axiom scheme follows from adding the collection axiom scheme (without proof). So I tried proving it, but since I'm new to set theory, I need ...
0
votes
0answers
18 views

Logarithms of Cardinals [duplicate]

Given any infinite cardinal $\lambda\neq\omega$, is it the case that there's a cardinal $\kappa$ such that $2^{\kappa}=\lambda$? Does this depend on whether the Continuum Hypothesis is true? Clearly, ...
6
votes
2answers
110 views

Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?

I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot ...
3
votes
2answers
62 views

Proving that a set that's not finite is infinite.

Call a set finite if there is a bijection of the set with some natural number, and call a set infinite if there is an injection of the set of natural numbers into that set. How do you prove that sets ...
1
vote
0answers
55 views

How to make a large cardinal unique?

A general form of questions regarding large cardinals is the following: Let $A(x)$ be the formula asserting "$x$ is a large cardinal of type $A$" then is the following true? ...
1
vote
1answer
53 views

Confusion in set-theory: Definition of formulas needs set

I am confused about some definitions in logic/ axiomatic set theory: We stated in our logic lecture the ZFC axioms and called the members of a ZFC-model "sets". But to define formulas and structures, ...
4
votes
0answers
75 views

Do We Need Non-Constructible Sets?

I was reading about Godel's Constructible Universe in which the Continuum Hypothesis and Axiom of Choice are true. It made me wonder, what kind of math would we be unable to do if every set were ...
0
votes
1answer
56 views

Prove that $|A| < |A| + h(A)$ for all $A$

Prove that $|A| < |A| + h(A)$ for all $A$, where $h(A)$ is the Hartogs number of $A$. Attempt: By definition, $h(A) > 0$ because it is the least ordinal number which is not equipotent to any ...
0
votes
1answer
35 views

For an infinite set $S$ , is $|S| < |$Sym $(S) |$?

Let $S$ be an infinite set ; does there exist any surjection of $S$ onto $A(S)$ ( the set of all bijections on $S$ ) ? I have atmost been able to prove that if $C( S)$ is the set of all countable ...
1
vote
1answer
102 views

Consistency of restricted forms of Martin's Axiom with the negation of the Continuum Hypothesis

Consider $\mathsf{MA}(S)$, the forcing axiom for all ccc posets which preserve a Souslin tree $S$. is $\lnot \mathsf{CH}$ consistent with $\mathsf{ZFC}+\mathsf{MA}(S)$? Does there exist a model for ...
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1answer
77 views

Continuous functions on a Suslin line

This question is motivated by Brian Scott's answer in this thread. It looks to me that continuous functions on Suslin lines may have remarkable properties (from my perspective). Convention. I am ...
3
votes
1answer
67 views

If $2^{\aleph_0}$ is weakly inaccessible, can every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfy $2^\kappa = 2^{\aleph_0}$?

Question. Is the following consistent with ZFC? $2^{\aleph_0}$ is weakly inaccessible Every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfies $2^\kappa = ...
4
votes
1answer
92 views

When are extensional equivalence classes still sets?

Let $\sim$ denote extensional equivalence. That is, $y\sim x \Leftrightarrow \forall z(z\in y \leftrightarrow z\in x)$. Given a set $x$, let $[[x]] := \lbrace y:y\sim x\rbrace$. Clearly, ...
3
votes
1answer
99 views

Undefinable Real Numbers

Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question Part ...
44
votes
4answers
9k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
2
votes
0answers
49 views

Large Cardinal Consequences of $\kappa$-Suslin Hypothesis

$\kappa$-Suslin Hypothesis ($\kappa$-SH) for the infinite regular cardinal $\kappa$ says that every tree of height $\kappa$ either has a branch of length $\kappa$ or an antichain of cardinality ...
0
votes
1answer
148 views

Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
3
votes
2answers
66 views

$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
6
votes
6answers
1k views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
2
votes
1answer
68 views

How big can the continuum be without choice?

I've heard an argument before (although I can't remember where) that the continuum hypothesis is false, since the powerset operation is a something much more 'powerful' than the mere cardinal ...
4
votes
1answer
74 views

Are there any large cardinal properties of the critical point of a $j: L \longrightarrow L$?

I've recently been thinking a bit about $L$ and $0 \sharp$. As is well known, the existence of $0 \sharp$ is equivalent to the existence of a non-trivial elementary embedding $j: L \longrightarrow ...
0
votes
1answer
63 views

set theory and real numbers

I'm not a set theorist just done some casual reading so please keep the answer simple... ZFC has a countable model M (provided it's consistent). In this model the real numbers R are countable (from ...
0
votes
1answer
26 views

Is elementhood between transitive sets monotonic under successor?

A set $z$ is transitive if $x\in y\in z$ implies $x\in z$. Given a set $x$, we define the successor of $x$, denoted $x^+$, to be $x\cup\{x\}$. Now, let $x$ and $y$ be transitive sets. If $x\in y$, ...
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vote
3answers
353 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
2
votes
1answer
76 views

How to force p<b?

Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are: $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination ...
0
votes
1answer
24 views

algebraic poset

I learn domain theory and stack in definition of algebraic poset. Recall $P$ is algebraic if for every $x\in P$,the set of compact element $y$ below $x$ is directed and has $x$ as least upper bound. ...
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votes
2answers
93 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
0
votes
0answers
44 views

Set theory with urelements

I'm looking for a first-order formalization of set theory (not necessarily one-sorted) which makes a distinction between sets and urelements (objects, that are not sets, but can be elements of a set). ...
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vote
0answers
41 views

Ordinal arithmetic and functions

I have two function $G$ and $F$ defined on ordinals and I know that $$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
2
votes
3answers
87 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 ...
4
votes
3answers
190 views

Reference textbook developing NBG set theory

I'm starting Borceux "Handbook of Categorical Algebra". It starts with a brief discussion of the logical foundations of category theory. He describes two approaches: 1.defining universes and 2. With ...
0
votes
2answers
105 views

Axiomatic Set Theory (ZFC): Intersection

I'm currently reading "Axiomatic Set Theory" by Suppes, and the book gives a proof of the existence of the intersection (which relies on the Axiom of Separation). While I understand the idea of the ...
1
vote
2answers
56 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
13
votes
1answer
493 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
1
vote
1answer
63 views

Axiomatic Set Theory: Why do we need the “Axiom of Union”?

I've been reviewing the axioms of ZFC, and I'm trying to make sense of why the "Axiom of Union" was put in place. While the existence of the intersection (of two) sets seems to be a "Theorem" we can ...
0
votes
1answer
45 views

Definition of continuity of ordinal function

In the book Introduction to Set Theory' by Hrbacek and Jech, chapter $6$ Ordinal Numbers, section $6$ Normal Form, I don't understand the definition of continuity of ordinal numbers. Ordinal ...