2
votes
1answer
80 views

Definition in Kunen

In Kunen's second edition of set theory he gives the following definition Let $(\mathbb{Q},\leq_\mathbb{Q},\mathbb{1}_\mathbb{Q})$, and $(\mathbb{P},\leq_\mathbb{P},\mathbb{1}_\mathbb{P})$ be forcing ...
4
votes
1answer
82 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
2
votes
2answers
101 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
6
votes
1answer
83 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
0
votes
0answers
27 views

Addition on well ordered sets not-commutative by showing $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$

My goal is to show that addition on well ordered sets are non-commutative by showing that, $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$ Some definitions (let A and B be ...
1
vote
1answer
61 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
2
votes
1answer
68 views

A well order on $\mathbb Z$ that respects addition?

Does there exists any well-ordering on $\mathbb Z$ that respects addidtion that is if $a < b$ then $a +c < b+c$ for all $c$ in $\mathbb Z$?
3
votes
2answers
53 views

Embedding $\omega_1$ into the direct sum/product of $\Bbb R$'s and $\Bbb N$'s

I'm trying to find a way to visualize the first uncountable ordinal $\omega_1$. This is rather difficult, as the visualization tactic that I often use for countable ordinals - namely, the "matchstick" ...
3
votes
1answer
73 views

Zorn's Lemma related statement

Consider the following statement: If $X$ is partially ordered set such that every chain in $X$ has un upper bound, then for every $x \in X$ there is a maximal element $m$ in $X$ such that $x \le m$. ...
2
votes
1answer
66 views

A well ordering on $\mathbb{R}$ and bigger sets

Consider the set of sequences $S = \{f:\mathbb{N}\to\mathbb{N}\}$, define an order on $S$ by the following: Based on the well-ordering of $\mathbb{N}$ and induction, either $f_1 = f_2$ or there is a ...
7
votes
1answer
78 views

Order-preserving injections of ordinals into $[0,1]$

The usual "matchstick" representation of an ordinal number can be thought of as an order-preserving injection of that ordinal into the interval [0,1]. For example, here's a representation of ...
5
votes
2answers
95 views

Old exam question about well ordering on $\mathbb{R}$

I am not sure how to start tackling this question and would love a hint: Let $<$ the regular order relation on $\mathbb{R}$, and $<_w$ well ordering on $\mathbb{R}$. We define a coloring ...
1
vote
2answers
86 views

Either two sets have the same cardinality, or one has cardinality greater than the other

This is a problem (10.11) from Munkres, Topology, 2 ed. Problem: Let $A$ and $B$ be two sets. Using the well-ordering theorem, prove that either they have the same cardinality, or one has cardinality ...
2
votes
2answers
104 views

structure of the hypernaturals

I want to understand the structure of the hypernaturals a little better. Let me recall the ultraproduct construction of the hypernaturals. On the set of all sequences of $\mathbb{N}$, we define an ...
2
votes
1answer
63 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
3
votes
1answer
103 views

A set that is transitive but not well-ordered by $\in$?

I am trying to find a transitive set which is not well-ordered by $\in$. This question raises when I read Jech's Set Theory, in which an ordinal number is defined as a transitive and ...
7
votes
0answers
256 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
1
vote
1answer
122 views

Hausdorff theorem for any linear orders

I read this question: Any good decomposition theorems for total orders? and the answers. I like very much the Hausdorff theorem for scattered linear order. I repeat it here : Theorem (Hausdorff). A ...
1
vote
1answer
99 views

Show that $X$ is separable.

I'm working through Kunen's Set Theory and I'm not sure how to proceed on part of one exercise. Let $X$ be compact Hausdorff and $\mathbb{O}_X$ be the poset of nonempty open sets of $X$ ordered by ...
2
votes
2answers
119 views

detecting cofinality in ordinals

Suppose K is an unbounded well ordered set, and suppose K is minimal in the sense that each proper initial segment of K has strictly smaller cardinality. Suppose J is an unbounded well ordered set of ...
4
votes
1answer
80 views

Is the class of countable posets well-quasi-ordered by embeddability?

The question is in the title. Here "$P$ embeds into $Q$" means there is a function $f : P\to Q$ such that for all $p,p'\in P$, $p \le_P p'$ if and only if $f(p) \le_Q f(p')$. A well quasi order $W$ ...
14
votes
1answer
155 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
4
votes
2answers
162 views

Dense sets and Suslin Lines

I'm studying Infinitary Combinatorics and need a hint for proofing that every Suslin Line has a $\omega_1$ dense set. I've tried to do something with the set of all subsets of the line of cardinality ...
0
votes
1answer
116 views

The limit elements of well-ordered proper classes

For any well-ordered class $W$ with no maximum element, we can define a successor function $f_W : W \rightarrow W$ by asserting that $f_W(w) = \mathrm{min}\{x > w \mid x \in W\}.$ This allows us to ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
8
votes
3answers
356 views

Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$?

Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable. Question: can we show the stronger result that if $X$ is ...
3
votes
2answers
138 views

Good resource to learn transfinite induction and/or recursion?

I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow. What's a good resource for learning ...
4
votes
2answers
147 views

An uncountable famliy in $2^\omega$ linearly ordered by inclusion without passing to $2^{\Bbb Q}$.

I was solving the problem of finding an uncountable, linearly ordered subset of $2^\omega$ ordered by inclusion. I was having a lot of trouble before I realized that I can substitute anything ...
3
votes
3answers
120 views

Which ordinals embed in $2^\omega$ ordered by inclusion?

Which ordinals can be embedded in the power set of $\omega$ ordered by inclusion? I see that $\omega\cdot\omega$ can (and therefore anything less than that): we can partition $\omega$ into $\omega$ ...
5
votes
1answer
140 views

What is cofinality of $(\omega^\omega,\le)$?

Let us consider the set $\omega^\omega$ of all maps $\omega\to\omega$ with the pointwise ordering. By cofinality of $(\omega^\omega,\le)$ I mean the smallest cardinality of the subfamily $\mathcal B$ ...
4
votes
1answer
87 views

Presheaves over sieves and posets

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found ...
5
votes
3answers
143 views

How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?

We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
3
votes
2answers
249 views

Why does every countable limit ordinal have cofinality $\omega$?

According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
7
votes
1answer
112 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
2
votes
1answer
112 views

Two questions of set theory: necessary and sufficient conditions for a subset of a poset to be centered, posets with noncentered linked subsets.

$(1)\hspace{4pt}$ Let $\left\langle X,\leq\right\rangle$ be a poset, and let $Y\subseteq X$ with $Y\ne\emptyset$. I’m trying to prove that $Y$ is centered—i.e., $Y$ is a filterbase on ...
3
votes
1answer
92 views

Existence of a minimal set of linear order that is equivalent to the given partial order

Let $\{(P, \leq_{\alpha})\}_{\alpha \in A}$ be a set of linear order on set $P$, $(P,R)$ be a partial order on the same set $P$. We say that the former is equivalent to the later, iff: $$\forall a, b ...
7
votes
3answers
241 views

Ordinal exponentiation - $2^{\omega}=\omega$

This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if ...
3
votes
1answer
118 views

Is every discrete topological space orderable?

I apologize for asking a question in topological terms when it's not really about topology, but here goes: If $S$ is a set, is it always possible to totally order $S$ so that every non-minimal ...
6
votes
3answers
144 views

Incomparable subsets of real numbers with usual order relation

Consider linear order $(R,<)$, would there be two sub-orders $(X,<), (Y,<)$ of it such that $X,Y$ are uncountable, which are incomparable in the following sense: There does not exist order ...
0
votes
1answer
348 views

Every finite partially ordered set has a maximum length chain.

Every finite partially ordered set, $(A, \leq)$, has a maximum length chain. A chain is a sequence of distinct elements $a_1 \leq a_2 \leq .......\leq a_n$ with relation $"\leq"$ where $a_i \in A$ ...
5
votes
2answers
252 views

If every nonempty subset of a set $S$ has a least and greatest element, is $S$ finite?

Some sets are well-ordered; all of their nonempty subsets have least elements. You can also have sets where all of their nonempty subsets have greatest elements. Some sets have both of these ...
1
vote
3answers
106 views

Question regarding well-ordering theorem

The well-ordering theorem states that every set can be well-ordered. That is, there is a total order on the set in which every non-empty subset has a least element. I was wondering whether it's also ...
4
votes
2answers
156 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
2
votes
1answer
245 views

Question about Hausdorff Maximal principle and antichain

I have this prob and still haven't figured about Let (A, $\leq$) be a partially ordered set. A subset B of A is called an antichain if and only if for any two distinct elements x and y in B, ...
2
votes
2answers
70 views

Reaching elements with finite applications of the successor relation in well-ordered sets

Given an well-ordered class $(A,\leq)$ we want to proof that every element $a \in A$ can be reached by applying the successor relation on a the smallest element of $A$ or on an limes element of $A$ ...
5
votes
1answer
165 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}) $ ...
2
votes
1answer
183 views

The relation between order isomorphism and homeomorphism

Let $X$ be a set and let $<_1,<_2$ be order relations on $X$. Let $T_1,T_2$ be the topologies induced on $X$ respectively. If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that ...
2
votes
1answer
151 views

Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
4
votes
1answer
96 views

Generalization of ordinals to well-founded sets?

In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, ...
9
votes
1answer
179 views

Atoms necessary for the existence of a generic filter?

I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me. Definitions We fix some partially ordered set $(P,\preceq)$. A ...