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
189 views

problem of a cardinality of a union

Let $\lambda$ a cardinal and $\delta<\lambda^+$. I want to proof there exists a increasing chain $$\{A^i_\delta : i< cf(\lambda)\}\subseteq[\delta\times\delta]^{<\lambda}$$ converging to ...
2
votes
1answer
210 views

Can we get uncountable ordinal numbers through constructive method?

As we know, $2^{\aleph_0}$ is a limit ordinal number, however, it is greater than $\omega$, $\omega+\omega$, $\omega \cdot \omega$, $\omega^\omega$, $\omega\uparrow\uparrow\omega$, and even $\omega ...
2
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2answers
962 views

Do $\omega^\omega=2^{\aleph_0}=\aleph_1$?

As we know, $2^{\aleph_0}$ is a cardinal number, so it is a limit ordinal number. However, it must not be $2^\omega$, since ...
6
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3answers
240 views

Axiom of Choice (for example in the Snake Lemma)

If we have to make a choice, but in the end it doesn't matter what choice we made, did we really make a choice to begin with? More explicitly, somewhere in the standard diagram-chasing proof of the ...
2
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4answers
325 views

Arithmetic on real numbers

So far, I have studied elementary set theory and I have some questions. I know how to add or multiply natural numbers and ordinals, but how do I subtract or divide or root or log? Is there any ...
3
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1answer
464 views

Question about Halmos' Naive Set Theory

Halmos proves shortly before the cited paragraph that finite subsets are not equivalent to themselves. He then says the following: The number of elements in a finite set E is, by definition, the ...
10
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4answers
769 views

Advantage of ZF over other set theories such as New Foundation

What would be the advantage of adopting ZF over other set theories such as New Foundation? I am very curious, since it seems that there is no reason just to stick with ZF. Edit: What about set ...
2
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2answers
187 views

Cofinality of infinite cardinals

I am looking for an answer to the question: "Show there exists an infinite cardinal $\kappa$ with $2^{cf(\kappa)}$ < $\kappa$ " Where $cf(\kappa)$ is defined as the least $\alpha$ such that there ...
0
votes
1answer
86 views

Factor in ultraproduct

The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be ...
3
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2answers
393 views

Importance of cover (topology)

What would be importance of cover in topological space? And how is the concept of cover being used in other areas, more specifically set theory?
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2answers
422 views

Unnecessary property in definition of equivalence relation [duplicate]

Possible Duplicates: Symmetric, Transitive and reflexive Why isn't reflexivity redundant in the definition of equivalence relation? Dependence of Axioms of Equivalence Relation? Let ...
2
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1answer
381 views

Mostowski collapse lemma

Can anyone explain what Mostowski collapse lemma is? The Mostowski collapse lemma states that for any such $R$ there exists a unique transitive class (possibly proper) whose structure under the ...
5
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2answers
234 views

Does the assertion that every two cardinalities are comparable imply the axiom of choice?

If, for any two sets $A$ and $B$, Either $|A|<|B|, |B|<|A|$ or $|A|=|B|$ holds, does the axiom of choice holds? Why?
3
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2answers
288 views

How to show Tukey's Lemma proves Zorn's lemma?

I heard that Zorn's lemma is equivalent to Tukey's lemma. Now I've proved that Zorn's lemma implies that Tukey's lemma but I cannot prove that Tukey's lemma implies Zorn's lemma. How to show this?
2
votes
1answer
244 views

Countability in first-order logic is relative to what exactly?

Skolem's Paradox tells us that countability in first-order logic is relative. Relative to what? Below is what I've gathered. Countability it relative to: 1. what a model takes to be $\mathbb N$ 2. ...
9
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2answers
615 views

Using Replacement to prove transitive closure is a set without recursion

In the course on set theory I'm doing, I'm told that one of the main motivations behind the axiom of replacement is that the Axiom of Infinity asserts the existence of an infinite set, namely $\omega ...
3
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3answers
372 views

Importance of free ultrafilter

What would be the usage of free ultrafilter? And how is it important? BTW, I know the concept of free ultrafilter, so I only need explanation on usage and importance.
2
votes
1answer
214 views

Regressive injective function in a set of infinite cardinals

This is also from Kunen, Set Theory, ch. II: Let $A$ be a set of infinite cardinals such that for each $\lambda$ regular $A\cap\lambda$ is not stationary in $\lambda$. Show that there is an ...
6
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2answers
306 views

Is this a good way to explicate Skolem's Paradox?

Skolems Paradox shows an ostensible conflict between Cantor's Thoerem (CT) and the downward Löwenheim–Skolem Theorem (ST). CT: for any set $A$, the powerset of $A$, $P(A)$, has a strictly greater ...
7
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2answers
196 views

Topological restrictions on cardinality

From what I know, a Polish (completely metrizable separable) space has a cardinality at most of $\mathbb R$. Completeness assumption can be omitted here, because a completion of a metrizable separable ...
2
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2answers
435 views

If ZFC has a model, must it be at least a countable model?

(1) must ZFC have an infinite model? (2) if so, why? (3) is it because of the replacement schema? (4) if so, is it because we have a finite language and so we can only satisfy or describe ...
0
votes
2answers
126 views

Real line, field of real numbers and $\omega_1$ topological space difference

Real line is separable. Then, why is $\omega_1$ topological space not separable? IF this is true, doesn't this settle continuum hypothesis? Also, does the field of real numbers have anything to do ...
2
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0answers
171 views

Confusion over Trees in Set Theory

Let $T$ be a normal Suslin tree, so among other things for each $x \in T$, there is some $y > x$ at each higher level less than $\omega_1$, and every branch in $T$ is at most countable. My question ...
5
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2answers
111 views

about a club set

I hope someone could help me with my question. I have to prove that $$C=\{ \alpha < \omega_1 : \alpha \in LIM\text{ and }j\upharpoonright \alpha \times \alpha : \alpha \times \alpha ...
5
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2answers
286 views

Constructing the reals from fractions of ordinals

We can construct the positive rationals from ratios of positive integers (and thus from pairs of finite ordinals). Can we analogously construct the reals from pairs of countable ordinals?
5
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0answers
186 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
175
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1answer
6k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
2
votes
1answer
128 views

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in ...
4
votes
2answers
216 views

A decreasing sequence of stationary sets.

I'm having trouble with exercise 45 page 91 of Kunen's book: Let $\kappa > \omega$ be regular. Show that there are stationary sets $S_\alpha \subset \kappa$ for $\alpha < \kappa$ such that ...
10
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3answers
353 views

An inconsistent version of Jensen's principle $\diamondsuit$

In page 93 of Kunen's book, exercise 55 says: Show that the following version of $\diamondsuit$ is inconsistent: There are $A_\alpha \subset \alpha$ for $\alpha < \omega_1$, such that for all ...
2
votes
1answer
404 views

Distribution Functions of Measures and Countable Sets

Let $\mu$ be a continuous probability measure on $[0,1]$. Then, the function $g:[0,1] \to [0,1]$ defined by $g(x) = \mu([0,x])$ is called the distribution function of $\mu$. I have proved that $g$ is ...
1
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1answer
181 views

The collection of continuous functions between any 2 compact Hausdorff spaces forms a set

I would like to show precisely what I have stated in the title (assuming that it is correct; I have reason to suspect it is, thanks to a tricky past exam paper I'm trying to surmount); namely, that ...
27
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4answers
1k views

Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem. As I have ...
0
votes
1answer
361 views

the sup of a set of ordinals

Let $A$ a set of ordinals. We know that $\sup A:=\bigcup A$ is an ordinal. Frequently, in proofs, one use that it is a limit ordinal. I would want to know when it is. To show that it is limit : let ...
2
votes
1answer
190 views

Equivalences to “D-finite = finite”

By a D-finite set, we mean a set admitting no injection from the natural numbers (or equivalently, a set not in bijection with any proper subset). I have encountered a proof that the following are ...
5
votes
1answer
501 views

Do sets whose power sets have the same cardinality, have the same cardinality? [duplicate]

Possible Duplicate: power set cardinal equality Let $X$ and $Y$ be sets, and suppose that $|\mathscr{P}(X)| = |\mathscr{P}(Y)|$ (where $\mathscr{P}$ denotes the power set). Does it follow ...
2
votes
3answers
481 views

Axiom of subsets and Russell's paradox

Russell, with his paradox, proves that the set $\{x:x\notin x\}$ of all sets that are not members of themselves doesn't exist. So, he demonstrates that the set $\{x:p(x)\}$ doesn't exist necessary (it ...
8
votes
1answer
273 views

Definable order types without infinity axiom.

Denote by $ZF^\times$ the theory of $ZF$ without the axiom of infinity. We know that $V_\omega$, the set of all hereditarily finite sets in a model of $ZF$, is a model of $ZF^\times$. We further know ...
14
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1answer
255 views

Are sets constructed using only ZF measurable using ZFC?

Suppose $S$ is a subset of $\mathbb{R}$ which can be defined without using the axiom of choice, i.e. which can be proved to exist using only the axioms of ZF. Does it follow that $S$ is measurable? ...
2
votes
1answer
213 views

proof of diamond

I have many questions about page 2 of this paper http://www.cs.elte.hu/~kope/ss3.pdf. First, on the top, I try to prove that, if $cf(\delta)\neq\kappa$, then we can choose the $\alpha, \beta$ in an ...
2
votes
1answer
220 views

consistency of large cardinal axiom

It is known that if ZFC is consistent, ZFC+"no such large cardinal exists" is consistent. Then, is ZFC+"such large cardinal exists" known to be consistent? This would imply that proving large cardinal ...
11
votes
1answer
1k views

Axiom of choice and calculus

I thought many results in calculus need axiom choice. For example, I thought one needs AC to prove that a bounded sequence in the real line has a convergent subsequence. Recently I was taught that one ...
4
votes
1answer
839 views

Proving the pairing axiom from the rest of ZF

In ZF, the pairing axiom states that for every $x,y$ there exists the set $\{x, y\}$. Wikipedia also tells us we can dispense this axiom: This axiom is part of Z, but is redundant in ZF because it ...
2
votes
3answers
1k views

Using setminus notation with set elements

The "correct" way to write a set without a specific element is as follows: $S \setminus \{s\}$ But in some contexts this is cumbersome to write/type or read, and it detracts from the flow of the ...
21
votes
2answers
2k views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
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vote
1answer
232 views

Formal Definition/counter part in mathematics for “Objects” of Object Oriented Models [closed]

I'm a newbie in both formal mathematics and theoretical computer science, so please bear with me if you find my question is not properly framed. Object Oriented Modeling seems very useful in defining ...
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vote
1answer
177 views

Can PA+Con(ZFC), prove everything ZFC can?

Can PA+Con(ZFC) prove every theorem of ZFC?
30
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9answers
3k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
4
votes
1answer
263 views

What is the smallest possible value of $\omega_1$ in $\mathrm{ZF}$?

It is consistent with $\mathrm{ZF}$ that a countable union of countable sets may be uncountable. As far as I understand it, this is because in absence of $\mathrm{AC}$ we cannot necessarily choose a ...
1
vote
2answers
74 views

A decomposition of an ordinal

Again http://www.cs.elte.hu/~kope/ss3.pdf . After the Remark 2, I have some problem to prove that there exists a increasing decomposition of $\delta<\lambda^+$ ($\delta$ ordinal ? or cardinal ?) ...