All Questions
Tagged with well-orders elementary-set-theory
159
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Partial order on sets and application of Zorn's Lemma to construct well-ordered subset
I would appreciate help with the following question:
Let $(A,<)$ a linear ordered set.
a. Let $F\subseteq P(A)$. Prove that the following relation is a partial order in $F$: $X\lhd Y$ for $X,Y\in F$...
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Is this set uncountable? $A = \{A_n \colon n \in \mathbb{N}\}$ where $A_n$ is the set $\mathbb{N}$ with the number $n$ removed from it
The set in the title is presented in this answer as an example of a similar set to the $P(\mathbb{N})$ (in the context of explaining the necessity of the axiom of choice in the existence of a well ...
- 39
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Law of Trichotomy for Well-Orderings
Often in beginning set-theory courses, and in particular in Jech's book Set Theory, it is proved from scratch that given any two well-orderings, they are isomorphic or one is isomorphic to an initial ...
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Hessenberg sum/natural sum of ordinals definition
I was given the following definition of Hessenberg sum:
Definition. Given $\alpha,\beta \in \text{Ord}$ their Hessenberg sum $\alpha \oplus \beta$ is defined as the least ordinal greater than all ...
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146
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Prove that if $A$ is any well-ordered set of real numbers and $B$ is a nonempty subset of $A$, then $B$ is well-ordered.
To prove: if $A$ is any well-ordered set of real numbers and $B$ is a nonempty subset of $A$, then $B$ is well-ordered.
My attempt: Suppose towards a contradiction that given any well-ordered set of ...
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46
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Limit Countable Ordinal - is it a limit of a intuitive sequence of ordinals?
I am studying set theory, ordinal part.
Set theory is new to me.
I know that commutativity of addition and multiplication
can be false in infinite ordinal world.
$ \omega $ = limit of sequence $\, 1,2,...
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467
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Difference between partially ordered, totally ordered, and well ordered sets.
I just started studying set theory and I'm a bit confused with some of these relation properties.
Given a set A = {8,4,2}, and a relation of order R such that aRb means "a is a multiple of b"...
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Using WOP to prove certain naturals can be written a certain way.
$\newcommand{\naturals}{\mathbb{N}}$
Use the Well-Ordering Principle to prove that every natural number greater than or equal to 11 can be written in the form $2s+5t$, for some natural numbers $s$ and ...
- 11
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Order isomorphism on a subset and a segment
I am trying to understand Theorem 1.7.4 of Devlin's Joy of Sets. The Theorem states:
There is no isomorphism of $X$ onto a segment of $X$ (supposing $X$ is a woset).
The difficulty I am having is that,...
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Why is well-ordering needed to define the statement "$\forall i, \, P(i) \implies P(i + 1)$"?
I have learned a proof that the well ordering principle is equivalent to the inductive property for $\mathbb{N}$, and have understood it. However, I am confused as to the following statement my notes ...
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2
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172
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$\mathbb{N}\times\{0,1\}$ and $\{0,1\}\times \mathbb{N}$ not isomorphic
Show that the sets $W=\mathbb{N}\times\{0,1\}$ and $W'=\{0,1\}\times\mathbb{N}$, ordered lexicographically are non-isomorphic well-ordered sets.
Any hints on how to prove this?
- 357
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189
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Is the set of all linear orders on $\mathbb{N}$ linearly orderable?
In studying the issue of linear orders and well ordering in the context of ZF Set Theory (without the Axiom of Choice), I have recently been thinking about the following question:
Is the set of all ...
- 4,331
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Well Ordering Principle Proof without mathematical induction with different approach
Denote $\Bbb Z_0$ be the set of all non-negative integers.
Well Ordering Principle for $\Bbb Z_0$. Every non-empty subset $S$ of $\Bbb Z_0$ has a least element; that is, there exists $m \in S$ such ...
- 447
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Every Non empty subset has a least element implies linear order
Suppose $(A,R)$ be structure where R is a binary relation on $A$.
Suppose $A$ has the property that every Non empty subset of $A$ has a least element w.r.t. the relation $R$. Then $R$ is a linear ...
user1072974
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How do you prove that there exists a highest element of any finite, nonempty subset of Natural Numbers? Is the following algorithmic proof valid?
Since the given set, $C \subset \mathbb{N}$ is non empty, hence by well ordering principle there exists $\alpha \in C$ which is the lowest element in C. Also, since the set $C$ is finite, $\quad \...
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