# 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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### Well-founded Relation on infinite DAGs

A well-founded relation on set $X$ is a binary relation $R$ such that for all non-empty $S \subseteq X$ $$\exists m \in S\colon \forall s \in S\colon \neg(s\;R\;m).$$ A relation is well-founded when ...
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### Detemine whether the interval [4,8] is well-ordered. Explain.

I don't think this interval is well-ordered because the subset (4,8) would not have a smallest value. I'm stuck on how to show (4,8) has no smallest value.
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### Order type of N and Q

Studying linear orderings, I learned two theorems. Suppose two linearly ordered sets A and B satisfy the following: (1) countably infinite, (2) dense, i.e. if x<z then there exists y such that x&...
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### The arithmetic of first uncountable ordinal number

I think, I know the proof of 1+ω0 = ω0. (ω0 is countable ordinal s.t ω0=[N]). To prove this, I can define a function f: {-1,0,1,2,...}->{0,1,2,...} by f(x)=x+1. But if ω1 is first uncountable ...
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### Shouldn't the Well Ordering Principle apply only to sets with at least two elements?

From what I've been taught in school, the well-ordering principle states that every non-empty set must have a least element. To me, the least element of some set $X$ is an element $a$ such that, for ...
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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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### Can every statement that can be proved using the well-ordering principle be proved using weak mathematical induction?

The following is problem 30 of chapter 4.4 of Discrete Mathematics with Applications, 3rd ed. by Susanna Epp: Prove that if a statement can be proved by the well-ordering principle, then it can be ...
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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 ...
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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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