ω is N in its usual order. — fishfry

w = N. No matter what order.

That does not contradict that also w is the order-type of <w standard-ordering-on_w> = the order-type of <N standard-ordering-on_N>

ω+1 is N in the funny order: — fishfry

That is plainly incorrect.

w+1 is not N, no matter what order.

w+1 = w u {w} = N u {N}

That does not contradict that w+1 is the order-type of <N "funny-order"-on_N>.

ω+1 as an alternate ordering of the natural numbers — fishfry

Again, w+1 is not an ordering. Rather w+1 is an ordinal, and it is the order-type of <w "funny-order"-on_N>.

is the change from ω-street to ε-street a "can't get theah from heah" transition? I see the language that says you just add a successor, but what successor would that be? — tim wood

I think fishfry addressed that. epsilon_0 is a limit ordinal, not a successor ordinal. epsilon_0 is the union of the set of ordinals of the form w^x, where x is a finite sequence of ascending 'w' exponents. That is, epsilon_0 = U{w, w^w, (w^w)^w ...}.

By "successor" I understand some number, as 3 is the successor to 2, 4 to 3, and so forth. — tim wood

'successor' for ordinals is simply this, by definition:

successor of x = x u {x}.

Defintions!

we can both agree that mere familiarity with terms doesn't get a person very far. — tim wood

My point was not that understanding definitions is sufficient, rather that understanding definitions is necessary.

all the successors have already been used — tim wood

Within any limit ordinal, there is no last successor. For example, w is a limit ordinal, and there is no member of w that is the last successor.

With zero and 1, I take it a person can get to any number in {0, 1, 2,.., n}, though perhaps not efficiently. The limit of that being ω. — tim wood

Just to be clear, in set theory, the existence of a set that has all the natural numbers as members is not proven by taking a limit or a union.

Rather, from the axiom of infinity and axiom of separation it follows that there is a unique inductive set (a set that has 0 and is closed under the successor operation) that is a subset of any inductive set. That unique set is the set of natural numbers = w.

It is important to mention this so that it does not appear as if w is conjured in some non-rigorous way as "just gather together all the numbers you get starting from 0 and adding 1".

Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem. — TonesInDeepFreeze

I responded to all of your points but then thought better of it.

My challenge was to write something that would give a high-level overview of a difficult subject to casual readers without much math background. Necessarily not everything was perfectly pedantic. So you missed that point entirely.

A number of your statements were flat out wrong, such as claiming that a bijection of a well-ordered set to itself is necessarily another well-order. I had already given the counterexample of the naturals and the integers.

It's just not worth getting into it with you. Your several posts to me seemed not just pedantic, but petty, petulant, and often materially wrong. You either misunderstood the pedagogy or the math itself, often both at the same time. It's just like what you wrote a few days ago, giving a long list of topics to be studied before one can read my article, including a year's worth of abstract algebra. The challenge is to write something that can be read by casual readers WITHOUT any mathematical prerequisites. You don't seem to understand that. You might try it yourself, it's harder than it looks.

I have to tell you your post left me with a bad taste. You seem to just want to throw rocks, but you couldn't even find pebbles, so you threw grains of sand. I hope you got something out of it.

But to give a specific example just so you understand what I'm talking about, let me take your opening remark here:

"Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem." followed by some picky complaint.

I led with "x is an ordinal iff x is the order-type of a well ordered set" because that's something that I can explain to a casual audience in a couple of paragraphs.

Whereas if I go with the standard textbook definition, "An ordinal is a transitive set well-ordered by $\in$, that would be immune from your petty criticism, but it would lose the entire audience immediately and never get it back. That's why I deliberately, and with thought and consideration, chose the formulation I did. You might give this point some thought. That the idea isn't to be technically pristine; but rather to choose formulations that have a hope of getting through to a casual audience. This entire concept of being understandable to a casual audience went completely over your head.

Likewise your persistent complaint that I omitted the fact that I am talking about total orders (which you called "connected" for reasons I didn't understand). Of course I did. That was a deliberate choice. One of my earlier drafts had a long, boring exposition of order theory, including partial orders, and then I just deleted it and decided to implicitly assume total orders to make the exposition more readable.This never occurred to you, you just kept harping on the point. If you would take a moment to ask yourself, "How would I explain ordinal numbers in a fair amount of depth to a casual audience," you might come to understand some of the tradeoffs involved.

Necessarily not everything was perfectly pedantic. So you missed that point entirely. — fishfry

No, I got your point that your posts are meant only as an overview. But that doesn't entail that I can't mention clarifications and some more exact formulations myself.

A number of your statements were flat out wrong, — fishfry

If they are, then I'm happy to correct them.

such as claiming that a bijection of a well-ordered set to itself is necessarily another well-order. — fishfry

I specifically said the permutation is not a well-ordering. What is a well-ordering is the ordering induced. Again:

If R is a well ordering on S, and f is permutation of S, then R* is a well ordering on S as R* is defined by:

R*f(x)f(y) <-> Rxy

Clearly, by basic set theory, R* is a well ordering on S. (Indeed, <S R> is isomorphic with <S R*>.).

I have no idea why you would deny that.

I had already given the counterexample of the naturals and the integers. — fishfry

Whatever you said about the naturals and integers couldn't refute the theorem I just mentioned above.

Your several posts to me seemed not just pedantic, but petty, petulant, and often materially wrong. — fishfry

There is good reason for the various points I mentioned - they keep things clear, not merely pedantic. 'petulant' is psychologizing that happens to be incorrect. And I am happy to correct any errors I wrote.

You either misunderstood the pedagogy or the math itself — fishfry

I don't claim to be pedagogically expert. I posted to clarify certain points and to keep my mind focused a little bit on math occasionally.

a long list of topics to be studied before one can read my article. — fishfry

I said no such thing that your post couldn't or shouldn't be read without first studying anything. I said that a clear understanding requires understanding certain definitions. That does not preclude that one can first read your post then go on to learn the definitions. You are reading into what I wrote things that I did not write.

The challenge is to write something that can be read by casual readers WITHOUT any mathematical prerequisites. — fishfry

That's fine; I didn't write anything that begrudges you from doing that.

"Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem." followed by some picky complaint. — fishfry

It wasn't a complaint. I merely wished to add a clarification, as I said "just to be clear" that the theorem you mentioned doesn't happen to be the definition. That is relevant as someone might misconstrue that it was intended as a definition.

I led with "x is an ordinal iff x is the order-type of a well ordered set" because that's something that I can explain to a casual audience in a couple of paragraphs. — fishfry

That's fine, and I didn't fault you for it. I merely added a point of clarification.

Likewise your persistent complaint that I omitted the fact that I am talking about total orders (which you called "connected" for reasons I didn't understand). — fishfry

I didn't complain. I merely added the information.

And I used 'connective' in line with the notion of a connected relation.

I [...] decided to implicitly assume total orders to make the exposition more readable. — fishfry

I don't begrudge you striving for readability. I just wanted to add the point of clarification, which I I did by saying if x not equal y then either Rxy or Ryx. (The reason I used 'connective' rather than connected is that I wasn't clear in the moment whether 'connected' is an adjective applied to relations or to the set that the relation is on. No harm though, as I mentioned that 'connective' might be ersatz and so I defined it explitiy). If you wish to leave certain things as implicit in your own posting, then that should not preclude me from making them explicit.

If you would take a moment to ask yourself, "How would I explain ordinal numbers in a fair amount of depth to a casual audience," you might come to understand some of the tradeoffs involved. — fishfry

I think of tradeoffs virtually every time I post, since in such a cursory context of posting, I too have to take shortcuts. The fact that I added clarifications and information doesn't entail that I don't understand that an overview can't cover every technicality.

That's fine, and I didn't fault you for it. I merely added a point of clarification. — TonesInDeepFreeze

I appreciate your corrections, some of which were on point and some, in my opinion, perhaps not. If I was too sensitive when I read your criticisms, I'll try to grow a thicker skin. Honestly I don't know how people write books, then sit back while critics throw rocks. I could never take it. Actors too.

I prefer not to engage on each point you raised, because the back-and-forth would get in the way of the overall intent of the thread, which is to introduce the ordinals to a casual audience.

I posted to clarify certain points and to keep my mind focused a little bit on math occasionally. — TonesInDeepFreeze

That's certainly fair, and it was uncharitable of me to ascribe ignoble motives to you.

Thanks for your comments.

I think of tradeoffs virtually every time I post, since in such a cursory context of posting, I too have take some shortcuts. The fact that I added clarifications and information doesn't entail that I don't understand that an overview can't cover every technicality. — TonesInDeepFreeze

You're right, I was defensive today. I was going to write "uncharacteristically" defensive but of course that would be a lie :-)

I looked this up, evidently connected relationship is another word for a total relationship. "Today I learned."

Thanks very much. I appreciate it.

I like this idea a great deal, even though I don't follow the equations. It's a sort of relative relativity. I've been fascinated by this since infancy, but there was no-one around to help me with it.