# Tagged Questions

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### Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
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### Set Theory and Zorn Lemma

Prove that there is a set $B\subseteq P(\mathbb N)$ such that for all $n\in \mathbb N:\mathbb N- n\in B$, every finite intersection of elements in B is not empty and for all $C\subseteq\mathbb N$ such ...
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### Noninital ordinal of a set

If $\mathbb R$ is well ordered by an order $\ll$ such that the ordinal of R and that order $\alpha$ is not initial. How can I show that there is a closed interval $[a,b]$ ($a<b$) such that the ...
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### Uniqness of sets with certain property

let $\kappa$ be a cardinal and let $A \subseteq \{X \subseteq \kappa: |X| = 2 \}$ be a set with the propety: "for each disjoint pair of sets $B,C \subseteq \kappa$ with cardinality < $\kappa$ ...
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### A proof of Zorn's Lemma [closed]

Prove Zorn's Lemma is equivalent to the statement: For all $(A, \leq )$ the set of all chains of $(A, \leq)$ has an $\subseteq$ - maximal element. Zorn's Lemma: If every chain in a partially ...
In her text Introduction to Modern Set Theory , Judith Roitman defined a filter of a set $X$ as a family $F$ of subsets of $X$ so that: (a) If $A \in F$ and $X \supseteq B \supseteq A$ then $B \in ... 1answer 97 views ### Prove that the Lexicographical Order is Partially Ordered Let$X$be a partially ordered set under$\le$. Let$FIN(X)$be the set of finite sequences (including the empty sequence) whose members are elements of$X$. If$\sigma = (x_1 x_2 ... x_n)$, ... 1answer 72 views ###$H(\kappa)$-absoluteness of a formula Let$\varphi(x,y)$be an$\in$-formula which is absolute between transitive models of ZF minus powerset axiom. Then$\exists x\, \varphi(x,y)$is$H(\kappa)$-absolute, where$H(\kappa)$is the set ... 2answers 93 views ### Stationary sets Currently learning about stationary sets, came across this problem: Let$\{S_\alpha : \alpha<\omega_1\}$be disjoint pairwise sets with$S_\alpha \subseteq \omega_1$non-stationary for each ... 1answer 203 views ### Infinite combinatorial games Hercules vs. Hydra: Recall the story where every time Hercules cuts of a head, two more heads grow instead. Now suppose the following: The hydra starts off with one head, but every time Hercules cuts ... 1answer 85 views ### Similar to Fodor lemma Let$\lambda>\aleph_0$be a regular cardinal such that$S \subseteq \lambda$is not a stationary subset. Prove that there exists a regressive function$f:S \to \lambda$such that ... 1answer 171 views ### Exercise 24.13 of T. Jech's *Set Theory* Having struggled my way through most of chapter 24 of Jech's Set Theory, I'm stuck on the very last part of the very last question, 24.13: Let$I=I_{NS}$be the nonstationary ideal on$\omega_1$, ... 2answers 78 views ### The existence of the set of all functions on A into B I need to prove that the set of all functions on A into B, denoted by$B^{A}$, exists. I think if$|A| = m$and$|B| = n$then$|B^{A}|=n^{m}$because for each element of$A$there are$n$... 1answer 38 views ### Isomorphisms between Orderings If$h$is an isomorphism between$(P,<)$and$(Q,\prec)$then show$h^{-1}$is an isomorphism between$(Q,\prec)$and$(P,<)$DEFINITION:$h$is an isomorphism between$(P,<)$and ... 1answer 54 views ### Example of a$\kappa$-long sequence of disjoint club subsets of regular cardinal$\kappa$I'm self-studying set theory and got stuck on this exercise: Let$\kappa$be a regular cardinal. Give an example of a sequence$\langle C_\alpha\mid\alpha<\kappa\rangle$such that$C_\alpha$is ... 2answers 85 views ### Problem on filters$\mathcal{F}$is filter on$\mathcal{I}$, but not ultrafilter. Prove that$\exists$$\mathcal{X, Y}\notin\mathcal{F} | \forall\mathcal{Z}\in\mathcal{F} \mathcal{X}\cap\mathcal{Z}\neq ... 1answer 125 views ### Prove the Recursion Theorem Let g be a function on a subset A\times\Bbb N into A, and a \in A Prove there is a unique sequence, f of elements of A such that a) f_{0} = a b) f_{n+1} = g(f_{n}, n) for all n ... 0answers 78 views ### rk(x\times y) and rk(x^{y})? My working ground is ZF^{-}, i.e ZF without the Axiom of Foundation. I define V_{0}:=0, V_{\alpha +1}:=\mathcal P(V_{\alpha}) and, if \lambda is limit, V_{\lambda}:=\bigcup_{\beta ... 1answer 92 views ### Sets and Functions prove B^{A} exists [closed] B^{A} is the set of all functions from A into B. Prove B^{A} exists. The hint that is given in textbook is$$B^{A} \subseteq P(A \text{ x } B)$$The question comes from Hrbacek and Jech's book: ... 2answers 132 views ### Closed, unbounded subset of a cardinal. I missed two lectures in my set theory course, and now I don't understand the homework problems. One is this: let \kappa be a regular uncountable cardinal. Show that the following sets are closed ... 1answer 133 views ### Diagonal intersection of club sets Let C_\alpha\subset\omega_1 be a club set for every \alpha<\omega_1. Show that the diagonal intersection of all the C_\alpha's, that is \{\alpha<\omega_1:\forall\beta<\alpha,\alpha \in ... 1answer 48 views ### Club set functions coincide Show that if f,g:\omega_1 \to \omega_1 preserve order (monotone increasing) and are continuous (f(\beta)=\sup\{f(\alpha) : \alpha<\beta\} for all limit ordinals \beta<\omega_1), then they ... 2answers 77 views ### Ordinals closed under functions Let  \{ f_n : n \in \mathbb N \}  be a set of functions f : (\omega_1)^k\to \omega_1  where the k is different between functions. Prove that the set of ordinals \alpha < \omega _1  that ... 2answers 64 views ### Tree, no uncountable antichains Show that if a \omega_1 tree (that is, each vertex has height less than \omega_1 and each level \alpha < \omega_1 is countable and non-empty) has no uncountable anti chains, and in addition ... 2answers 114 views ### Disjoint Refinement Prove that for any countable family of infinite sets from \mathbb{N}, A = \{A_n \colon n \in \mathbb{N}\}, there is a disjoint refinement B = \{B_n \colon n \in \mathbb{N}\} of infinite sets, ... 1answer 58 views ### Why are there 2^{\aleph_0} injections from \omega to \omega_1? [duplicate] I have to prove that there are 2^{\aleph_0} injections from \omega to \omega_1. I can see that there is a bijection between this set and the set of pairs: (permutation of \omega, infinitely ... 2answers 219 views ### Proving the \Delta lemma using Fodor's Theorem Presently trying to solve Exercise 9.10 from Jech's Set Theory (3rd Millenium edition): Prove [the \Delta lemma] using Fodor's Theorem. which contains the following mysterious hint: Let ... 1answer 131 views ### Question about exercise 8.5 in Jech's Set theory The exercise I am asking about is the next one: For every stationary S\subset \omega_1 and every \alpha < \omega_1 there is a closed set of ordinals A of length \alpha such that A\subset ... 1answer 28 views ### Well-founded part of a graph Let (A,R) be a graph. Define by transfinite recursion:  W_{0}=\emptyset \\ W_{\alpha+1}=\{a \in A : ext_{R}(a) \subseteq W_{\alpha +1}\} \\ W_{\alpha}=\cup_{\beta < \alpha} W_{\beta} \text{if ... 1answer 121 views ### Instance of Continuum Hypothesis implying cardinal inequality I'm currently trying to solve Exercise 5.27 of Jech's Set Theory (3rd Millennium ed.), viz: If 2^{\aleph_1}=\aleph_2, then \aleph_{\omega}^{\aleph_0} \ne \aleph_{\omega_1}. The presumption ... 1answer 63 views ### Axiom of pair follows from the weak axiom For any x and y , if there exists z such that x \in z \quad and \quad y \in z then, there exists a unique set {x,y} Can you guys give me a hint? I don't know how to prove this.. 1answer 90 views ### Is this space countably compact Let X be a Tychonoff countably compact space and A is a subapce of X such that for any countable B \subset A we have \overline{B} \subset A. My question is this: Is this subspace A ... 1answer 97 views ### Prove the existence of a set in the Euclidean plane I got stuck on the following problem. Prove that there exists a subset A of \mathbb{R}^2 such that every line in \mathbb{R}^2 goes exactly through two points in A. I know that I should apply ... 1answer 93 views ### The set of limit members of an infinite uncountable well-ordered set isn't countable X is an uncountable infinite well ordered set. I need to show that the set of the limit members in X is uncountable. A limit member is a member of X that is not a successor of any other member in X ... 2answers 192 views ### infinite sequence of sets \{X_i\} that for each i, X_i\in X_{i-1} I need to show that the following infinite sequence \{X_i\} doesn't exist: for all i, X_i \in X_{i-1} I really don't know where to start. The only thing in my mind is the axiom : for every non ... 1answer 277 views ### Dedekind Cuts - Additive and Multiplicative Identities I wanted to receive some help regarding some homework questions; I appreciate any sort of feedback possible. I am certain that any Dedekind Cut A has the following properties: 1) A is not the ... 2answers 178 views ### Well-Ordered Sets and Functions I need some assistance with the following problems. I've come up with some ideas but may need some clarification. 1) Let S be a well-ordered set. a) Need to show S has a first element. ... 2answers 88 views ### Infinite union of internal sets not internal This is homework problem. I need to give an example of internal sets A_n \subset \mathbb{R}^* for which the union \bigcup _{n=i}^\infty A_n  is not internal. Also, this whole internal set ... 1answer 86 views ### A countale partially ordered set that has an uncountable number of maximal chains I'm looking for a countable set S with a partial order < that has an uncoubtable number of maximal chains. I had many ideas but non of then is correct (for example- S= natural numbers, "<" is ... 1answer 143 views ### Exercise on partially ordered sets from Kunen's *Set Theory* This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let  \mathbb{P}  be a partial order. Define  \text{c.c.}(\mathbb{P})  ... 1answer 78 views ### A product of 2 Bourbakian posets, ordered by pointwise ordering. A poset P is called Bourbakian if every order-preserving map P\rightarrow P has a LEAST fixed point. Let P, Q be Bourbakian. Let P\times Q be ordered pointwise, that is (a,b)\le (c,d) if and ... 2answers 202 views ### How do I show there isn't an order isomorphism b/w the two sets \{1, 2, 3,…\} and \{1, 2, 3, …, \omega \} That is, how can I prove there isn't a bijection f from one set to the other such that f(x) < f(y) iff x < y? 1answer 96 views ### Show that for n, m \in \omega, the ordinal and cardinal exponentiations n^m This is an exercise from Kunen's book. Show that for n, m \in \omega, the ordinal and cardinal exponentiations n^m are equal. What I've tried: I want to prove by using induction on m. ... 2answers 151 views ### Show that if \kappa is an uncountable cardinal, then \kappa is an epsilon number Firstly, I give the definition of the epsilon number: \alpha is called an epsilon number iff \omega^\alpha=\alpha. Show that if \kappa is an uncountable cardinal, then \kappa is an ... 2answers 270 views ### Equivalent characterizations of ordinals of the form \omega^\delta Let \alpha be a limit ordinal. Show the following are equivalent: \forall \beta, \gamma<\alpha (\beta+\gamma<\alpha) \forall \beta<\alpha(\beta+\alpha=\alpha) \forall X\subset ... 1answer 100 views ### How to show the two inequalities? This is an exercise from Kunen's book. Show that \alpha < \beta implies that \gamma+\alpha<\gamma+\beta and \alpha+\gamma\le\beta+\gamma. Given an example to show that the "\le" cannot ... 1answer 120 views ### On T_2, first countable, countably compact space As we know, For every T_2, first countable, compact space, its cardinality is not more than 2^\omega. (See chapter 3 of Engelking's book.) However, I want to know whether the result is ... 2answers 95 views ### On Countable chain condition If the topological space X has CCC (= countable chain condition ) with given a countable closed discreted subspace Y of X, could we seperate the points in Y by countable disjoint open sets in ... 2answers 64 views ### Elementary Question Let be f:X\to X a bijection, an A\subset X a invariant subset of X, i.e f(A)\subset A. How can see that$$f(A)=A$$I'm trying to show that$$f(A^{c})\subset A^c but I can not.
Why is the subspace of the ordinal $2^\omega+1$ consisting of all ordinal of countable confinality countably compact, first countable, and has cardinality $2^\omega+1$ ?