October 9th, 2019, 01:23 AM  #41  
Senior Member Joined: Oct 2009 Posts: 884 Thanks: 340  Quote:
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 $a, a1, a2\in\mathcal{A}_n$ and such that $x=a3$. 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. Last edited by skipjack; October 10th, 2019 at 01:31 PM.  
October 9th, 2019, 05:57 AM  #42 
Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52 
Yes Micrm@ss, 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: “if 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.” How about that, still with me? Last edited by AplanisTophet; October 9th, 2019 at 06:12 AM. 
October 9th, 2019, 11:09 AM  #43  
Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52  Quote:
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 notsogood (or at least notsowilling) mathematician that is incapable of helping another notsogood mathematician like me. Quote:
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.  
October 9th, 2019, 01:02 PM  #44  
Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52  Quote:
PS  Feel free to put the ellipses back in there too, or leave them off, as it’s still clearly the same either way. Last edited by AplanisTophet; October 9th, 2019 at 01:21 PM.  
October 9th, 2019, 06:28 PM  #45 
Senior Member Joined: Aug 2012 Posts: 2,412 Thanks: 755  
October 9th, 2019, 07:10 PM  #46 
Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52  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. 
October 10th, 2019, 12:46 PM  #47 
Member Joined: Oct 2018 From: USA Posts: 99 Thanks: 72 Math Focus: Algebraic Geometry  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 kickflipesque thing. You then get annoyed when the group of skateboarders provide criticism of your kickflip.

October 10th, 2019, 01:11 PM  #48  
Senior Member Joined: Oct 2009 Posts: 884 Thanks: 340  Quote:
 
October 10th, 2019, 04:27 PM  #49  
Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52  Quote:
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. Last edited by AplanisTophet; October 10th, 2019 at 04:30 PM.  
October 10th, 2019, 10:17 PM  #50  
Senior Member Joined: Oct 2009 Posts: 884 Thanks: 340  Quote:
 

Tags 
$epsilon0$, enumeration, explicit, sequences 
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