# T Sequences - Explicit Enumeration of $\epsilon_0$

#### [email protected]

You don't get it and you don't want to listen.

I want you to write one single thing. Like Rule 1. Forget all the metaphysics about natural continuations of sequences or whatever it is you're doing. Give me the raw mathematical formalism, starting with the very first thing there is to know.

Then let me digest that and ask questions.

One idea. A few lines of text or a definition of some formula. Like Rule 1, but now that you've retracted your ellipses notation.

I'll make this concrete for you.

I will not read any post you write that is over 150 words in length.

You do recall I hope that a few posts back I asked you to "clarify or retract" the ellipses. I see you've retracted them. But please remember how often and how vehemently you have thought that it's "obvious what they are," and then later "obvious that there's no difference," then finally abandoned altogether.

So get some self-awareness and some humbleness.

Your lengthy posts are not tethered to rationality in my opinion. I don't seem to be able to communicate to you that your exposition is incoherent and meaningless; and your belief that it's practically obvious is a delusion. Not a mathematical confusion of some sort. A genuine break with reality that should generate in you -- I'll be blunt -- concern.

This is how I interpret his post.
You essentially recursively build up a function as follows:
Take $\mathcal{A}_0 = \{1,2,3\}$.
Assume that $\mathcal{A}_n$ is defined, then we define according to rule 1,2,3:
1) $x\in \mathcal{B}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $a, a-1, a-2\in\mathcal{A}_n$ and such that $x=a-3$.
2) $x\in \mathcal{C}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $x = a+1$
3) $x\in \mathcal{D}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $a, a+1, a+2\in \mathcal{A}_n$ and such that $x=a+\omega$.

Then we define
$$\mathcal{A}_{n+1} = \mathcal{A}_n\cup \mathcal{B}_n \cup \mathcal{C}_n\cup \mathcal{D}_n$$

Example:
$$\mathcal{A}_0 = \{1,2,3\}$$
We have
$$\mathcal{B}_0 = \{0\},~\mathcal{C}_0 = \{4\},~\mathcal{D}_0 = \{\omega\}$$
Thus $\mathcal{A}_1 = \{0,1,2,3,4,\omega\}$.

Next, once we take for an ordinal $x$, the level of $x$ to be defined as the least $n$ such that $x\in \mathcal{A}_n$. There are finitely many ordinals of level $n$. We then order the ordinals on their levels.

For example:
Of level 0: 1,2,3
Of level 1: 0, 4, $\omega$

Final sequence: 1,2,3, 0, 4, $\omega$, (here come the ordinals of level 2),...

I think this is what he WANTS to do since what he wrote makes little sense. But he writes things very differently and I think he makes several mistakes in his post.

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#### AplanisTophet

Yes [email protected], that is the basic approach to $T$ sequences and I assume now Mascke understands too.

For any ordinal $\alpha$, there are $\alpha^3$ triplets that can be made from the elements of $\alpha$ (if allowing triplets where two or more elements of the triplet can be the same ordinal). This gives us an exact number of triplets to choose from for each $\alpha$ if comprising a $T$ sequence that inserts one and only one ordinal $\alpha$ into the sequence per rule in a fashion where each rule is of the form: â€œif the triplet (a,b,c) can be formed from some initial segment of a $T$ sequence defined using all rules less than rule $\alpha$, where a,b,c < $\alpha$, then $\alpha$ gets inserted into the sequence.â€

How about that, still with me?

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#### AplanisTophet

Your lengthy posts are not tethered to rationality in my opinion. I don't seem to be able to communicate to you that your exposition is incoherent and meaningless; and your belief that it's practically obvious is a delusion. Not a mathematical confusion of some sort. A genuine break with reality that should generate in you -- I'll be blunt -- concern.
I keep telling you that I don't know much when it comes to mathematical notation and my overall knowledge of mathematics is limited. It's as though you think I'm just an idiot in all aspects though as opposed to just my half-baked mathematical musings in this forum. I naturally assumed you're just trying to irritate me because I happily play along, which is fun and all because continually saying something is incoherent after it's been explained to you numerous times in numerous different ways is funny.

We're on the same page then? I'm saying that I think you're capable. I'm also saying that if you truly couldn't begin to make sense of my enumeration of $\omega^2$, then that's my mistake for thinking you would be a helpful and well meaning mathematician who could assist me by pinpointing breaks in my notation so as to try and define the $T$ sequence model in a nice way that makes sense to everyone. Let me know if I should treat you like a not-so-good (or at least not-so-willing) mathematician that is incapable of helping another not-so-good mathematician like me.

I don't understand your T sequence. I mean, I know what you want to do...
Here [email protected] does what you never do, which is say simply, "yeah, I get what you want to do." Sure, my notation needed a little help or whatever, but if [email protected] wanted to, he/she could have helped me make sense of it because [email protected] is clearly capable. It's not clear to me that you are capable, or if you are capable, that you are not intentionally playing stupid to toy with me. Do you not see that? I don't want to read another novel from you about how incoherent I am because they will never help. You need to say why it's incoherent to you.

Ok, all that said, do you really want to do this 150 words at a time? I have fun either way so I'm happy to.

#### AplanisTophet

This is how I interpret his post.
You essentially recursively build up a function as follows:
Take $\mathcal{A}_0 = \{1,2,3\}$.
Assume that $\mathcal{A}_n$ is defined, then we define according to rule 1,2,3:
1) $x\in \mathcal{B}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $a, a-1, a-2\in\mathcal{A}_n$ and such that $x=a-3$.
2) $x\in \mathcal{C}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $x = a+1$
3) $x\in \mathcal{D}_n$ if and only if there is some $a\in \mathcal{A}_n$ such that $a, a+1, a+2\in \mathcal{A}_n$ and such that $x=a+\omega$.

Then we define
$$\mathcal{A}_{n+1} = \mathcal{A}_n\cup \mathcal{B}_n \cup \mathcal{C}_n\cup \mathcal{D}_n$$

Example:
$$\mathcal{A}_0 = \{1,2,3\}$$
We have
$$\mathcal{B}_0 = \{0\},~\mathcal{C}_0 = \{4\},~\mathcal{D}_0 = \{\omega\}$$
Thus $\mathcal{A}_1 = \{0,1,2,3,4,\omega\}$.

Next, once we take for an ordinal $x$, the level of $x$ to be defined as the least $n$ such that $x\in \mathcal{A}_n$. There are finitely many ordinals of level $n$. We then order the ordinals on their levels.

For example:
Of level 0: 1,2,3
Of level 1: 0, 4, $\omega$

Final sequence: 1,2,3, 0, 4, $\omega$, (here come the ordinals of level 2),...

I think this is what he WANTS to do since what he wrote makes little sense. But he writes things very differently and I thinks he makes several mistakes in his post.
Since youâ€™ve demonstrated your ability to understand what Iâ€™m saying, perhaps you can try and write rule 4 the same way you did rules 1-3? I wonâ€™t ask you to rewrite any of the rules for me other than these, but the whole reason I use the notation that I do is because your approach wonâ€™t work if you donâ€™t have an $a \in A_n$ to base your notation on like you do for rules 1-3.

PS - Feel free to put the ellipses back in there too, or leave them off, as itâ€™s still clearly the same either way.

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#### Maschke

I naturally assumed you're just trying to irritate me ...
The chip on your shoulder makes it unpleasant to interact with you. You are factually mistaken in your assumption.

#### AplanisTophet

The chip on your shoulder makes it unpleasant to interact with you. You are factually mistaken in your assumption.
Ok, but think of it this way. Pretend I'm a kid skateboarding at a local park and you, a grumpy old man (remember, I said we're just pretending) walk up to me and say, "hey, punk, your skateboarding... it's all wrong. It's just not the way you're supposed to do it." I would rightfully look at you, laugh, and go about my merry way. Let's say you also happen to be a grumpy old pro skateboarder and could give me some pointers if I had correctly interpreted you and not went on my merry way. Well, that would be nice, but there is no way I could have known. You would have had to say something different, like "try picking your back foot up more" (or whatever, I don't actually skateboard). Well then I might have listened to you, or at least tried to do what you were saying. Now let's say you were a really good pro skateboarder that actually went to the park to help kids and made it a point to do so. Well, then we wouldn't be having this conversation.

#### Greens

... walk up to me ...
But the problem is that's not what happened, it was more you coming up to a group of skateboarders and saying "Hey, look at this kickflip I can do", and then doing some strange kickflip-esque thing. You then get annoyed when the group of skateboarders provide criticism of your kickflip.

#### [email protected]

Since youâ€™ve demonstrated your ability to understand what Iâ€™m saying, perhaps you can try and write rule 4 the same way you did rules 1-3? I wonâ€™t ask you to rewrite any of the rules for me other than these, but the whole reason I use the notation that I do is because your approach wonâ€™t work if you donâ€™t have an $a \in A_n$ to base your notation on like you do for rules 1-3.

PS - Feel free to put the ellipses back in there too, or leave them off, as itâ€™s still clearly the same either way.
Sorry no. You got to make a huge effort to learn the proper notations and proper way of communicating in mathematics. I don't care what excuses you have for not learning this, but I have absolutely zero interest in exploring your ideas until you learn the proper way of doing things.

#### AplanisTophet

Sorry no. You got to make a huge effort to learn the proper notations and proper way of communicating in mathematics. I don't care what excuses you have for not learning this, but I have absolutely zero interest in exploring your ideas until you learn the proper way of doing things.
Um, learning is what I'm trying to do here. I just want you to take a crack at rule 4 because I would have defined rules 1-3 the same way as you if the later rules didn't require a different approach. It's not like I haven't made an effort and it's not like I don't appreciate learning how to properly communicate in mathematics (I'm certified when it comes to accounting, the language of business...), I'm just confused. That's why I'm here.

I completely understand if you have better things to do. But no, you will not be telling me I'm full of excuses for not trying to do the very thing I am clearly here trying to do: learn the proper notation for writing rules $\geq$ 4.

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#### [email protected]

Um, learning is what I'm trying to do here. I just want you to take a crack at rule 4 because I would have defined rules 1-3 the same way as you if the later rules didn't require a different approach. It's not like I haven't made an effort and it's not like I don't appreciate learning how to properly communicate in mathematics (I'm certified when it comes to accounting, the language of business...), I'm just confused. That's why I'm here.

I completely understand if you have better things to do. But no, you will not be telling me I'm full of excuses for not trying to do the very thing I am clearly here trying to do: learn the proper notation for writing rules $\geq$ 4.
You can't learn by asking random questions on a forum. I mean, you can, it is useful, but also if you do some outside reading. So I can recommend you some books that you can go through in order to learn proper notation and argumentation. If you go through these books I am definitely willing to help you through it and to check your work. But if you only "learn" through forum posts then I don't think it's going to work out.