Consider these definitions:

I think, there is a mistake in this definition of well-ordering which makes the latter impossible.

Am I right?

1 hour ago, Genady said:

Consider these definitions:

I think, there is a mistake in this definition of well-ordering which makes the latter impossible.

Am I right?

I'm more and more rusty on these things with every passing month, but doesn't the latter require S to be a closed set to be true?

At the very least it requires a topology, if I remember correctly. Open subsets don't necessarily have minimal elements.

I remember I still have a quiz/riddle on divisibility that you posed months ago pending. I'm sorry about that. Maybe you published the solution?

24 minutes ago, joigus said:

I'm more and more rusty on these things with every passing month, but doesn't the latter require S to be a closed set to be true?

At the very least it requires a topology, if I remember correctly. Open subsets don't necessarily have minimal elements.

I remember I still have a quiz/riddle on divisibility that you posed months ago pending. I'm sorry about that. Maybe you published the solution?

No topology here. This is pure set theory, more specifically, ZFC.

There is no requirement that a set has to be well-ordered. It is only a definition, when it is. I claim that the definition as stated is meaningless, i.e., no set so well-ordered exists.

(I don't remember which quiz/riddle on divisibility I posed months ago ☹️ )

P.S. I like your "Location." 🙂

20 hours ago, Genady said:

Am I right?

No. (Check your notes or references about basics of algebra.)

19 hours ago, joigus said:

I'm more and more rusty on these things with every passing month, but doesn't the latter require S to be a closed set to be true?

does it mention or tell about closed or open set?

19 hours ago, joigus said:

Open subsets don't necessarily have minimal elements.

also consider, does a well ordered set have to be open set?

19 hours ago, joigus said:

if I remember correctly.

I opinionate in negative.

Edited Tuesday at 07:11 PM1 day by ahmet

1 hour ago, ahmet said:

  22 hours ago, Genady said:

Am I right?

No. (Check your notes or references about basics of algebra.)

I've checked. Turned out that I am right.

Can you find what is mistaken in the quoted definition? (Read it carefully in the OP.)

@Genady, is the mistake you see that subset S requires at least two elements and not merely be non-empty?

4 minutes ago, KJW said:

@Genady, is the mistake you see that subset S requires at least two elements and not merely be non-empty?

Ha, that's too. Still, there is another one, related.

I think I see it: y has to be not equal to x₀.

3 minutes ago, KJW said:

I think I see it: y has to be not equal to x₀.

That's it!!

7 minutes ago, Genady said:

  4 minutes ago, KJW said:

I think I see it: y has to be not equal to x₀.

That's it!!

Just to clarify, the definition says, "for each y in S", which includes x₀, whereas y must not be equal to x₀.

Edited Tuesday at 10:28 PM1 day by KJW

2 minutes ago, KJW said:

Just to clarify, the definition says, "for each y in S", which includes x₀, whereas y must not be equal to x₀.

Right. As they say in the first part of the definition, xRy and x=y are mutually exclusive.