# Questions tagged [well-orders]

For questions about well-orderings and well-ordered sets. Depending on the question, consider adding also some of the tags (elementary-set-theory), (set-theory), (order-theory), (ordinals).

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### Why does proof of Zorn's lemma need to use the fact about ordinals being too large to be a set?

I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma. The Hypothesis in Zorn's lemma is Every chain in the set Z has an upper bound in Z Then ...
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### What does the cardinality alone of a totally ordered set say about the ordinals that can be mapped strictly monotonically to it?

For any cardinal $\kappa$ and any totally ordered set $(S,\le)$ such that $|S| > 2^\kappa$, does $S$ necessarily have at least one subset $T$ such that either $\le$ or its opposite order $\ge$ well-...
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### Using the Principle of Well Order, show that for all $n \in N$ it holds that $4^n -1$ is divisible by 3.

Using the Principle of Well Order, show that for all $n \in N$ it holds that $4^n -1$ is divisible by 3. I have already defined the set of counterexamples $C$, then I proved that for $n=1$ the ...
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### Well-ordering theorem and cardinality of real numbers

If we assume that the axiom of choice is right, the well-ordering theorem can be verified. So, the set of real numbers also can be constructed by well-ordering property. By the well-ordering property, ...
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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 ...
1 vote
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### Intuitionistic well-orderings of uncountable sets

The well-ordering principle has always been considered to be highly unconstructive, as far as I know. However, I think intuitionistic mathematics can be compatible with the existence of a well-...
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### If i have an a well ordered set in which every chain admits an upper bound then the maximal element is unique

It is clear that the Zorn lemma guarantees the existence. I prove that the minimal element is unique, and obviously the set is totally ordered. So because the uniqueness of the successor of all ...
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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 ...
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### Are there any guessing principles in set theory which guess well-orders of sets?

$\Diamond(\omega_1)$ was introduced by Jensen and is known to imply $\mathsf{CH}$ and the existence of a Souslin tree. It is the statement that there is a sequence (in particular a $\Diamond$-sequence)...
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### Does a total or well ordering has a countable cofinal

I read a post sometime earlier asking about the cardinality of total ordering on a set, which I forgot about the detail but led me to this question. For a total/well-ordered set $A$ does there exist ...
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### Why can't well ordered sets have infinite decreasing subsequences?

Chapter 2 of Mathematics for Computer Science presents the following: Define the set $\mathbb F$ of fractions that can be expressed in the form $\frac{n}{n+1}$. Define $\mathbb N$ as the set of ...
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### Countable versus uncountable dense linear orders

Let $(\Omega,\leq)$ be a dense linear order without endpoints. If $\Omega$ is countable, we know by Cantor that $(\Omega,\leq)$ is order-isomorphic to $(\mathbb{Q},\leq)$. Suppose that $\Omega$ is not ...
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### Does this beginningless-past thought experiment result in several possible non-well-ordered sets?

Assume there is an infinitely large universe with a beginningless past in which two eternal particles move at a constant speed of 1 km per “day” with respect to each other. Intuitively (to me), it ...
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### Why do we need "canonical" well-orders?

(I asked this question on MO, https://mathoverflow.net/questions/443117/why-do-we-need-canonical-well-orders) Von-Neumann ordinals can be thought of "canonical" well-orders, Indeed every ...
1 vote
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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 ... 125 views

### What is the meaning of "induction up to a given ordinal"?

Given an ordinal $\alpha$, what does it mean: "induction up to $\alpha$"? When $\alpha=\omega$, is this is ordinary mathematical induction? Also, Goodstein's Theorem is equivalent to "...
1 vote
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### Question about a proof that any well-ordered set is isomorphic to a unique ordinal

I am studying a proof that every well-ordered set is isomorphic to a unique ordinal. However, I don't understand why $A = pred(\omega)$ (see yellow). One direction is clear: Let $x \in pred(\omega)$, ...
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### non-wellordered linear order that doesn't contain a copy of $\omega^*$ in ZF?

Obviously a wellorder cannot contain a copy of $\omega^*$ (the dual order of $\omega$), and this can be proven in ZF. In ZFC, it is easy to prove any linear order which is not a wellorder does contain ...
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### Intersection of a chain of closed sets

I was recently attempting proving a conjecture of mine about the existence of certain minimal nonempty closed sets in a topological space. I opted to use Zorn's Lemma, and the proof would go through ...
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### Product of Countable Well-Ordered Set with $[0,1)$ is Homeomorphic to $[0,1)$

As part of a proof that the long line is locally Euclidean, I'd like to prove the following: Proposition. If $A$ is a countable well-ordered set, then $A \times [0,1)$ with the dictionary order is ...
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### Order type of this order over naturals

For no specific reason other than curiosity, I find myself wondering of a way to formalize a specific order of the natural numbers based on a sort of "recursive prime number factorization", ...
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### Partial order, well order and initial segment

I saw the following definition for initial segment in https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs If $\leq$ is a partial order in a set $X$, then a chain ...
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### Riddle: finite set that contains one of the three numbers

A colleague told me this two-part riddle: PART $\mathbb{N}$: Three players sit in a circle, each having a positive integer number $x_i$ written on their hat. Every player can see only the numbers of ...
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I am trying to define a Matrix $M_n$ of dimension $2^n \times n$ where each row corresponds to a element of the set $\{0,1\}^n$. To avoid ambiguity, I using an order relation over the set $\{0,1\}^n$ ...