# T Sequences - Explicit Enumeration of $\epsilon_0$

#### AplanisTophet

Introduction: Making a Sequence $T$ based on â€œThe Rule of Threeâ€

The goal of this thread is to develop the basic model for what I refer to as a "$T$ sequence" and then expand on the basic model so as to create an enumeration of the ordinal $\epsilon_0$.

Each $T$ sequence is a listing of ordinals that is iteratively generated based on a list of rules. After developing a basic $T$ sequence model with only three rules that happens to be an enumeration of $\omega^2$, I will then add additional rules so as to (try to) create the explicit enumeration of $\epsilon_0$.

The primary means of generating the sequence $T$ is through the use of a function $f$. In general, function $f$ is going to be a function that takes as input a three-member sequence of ordinal numbers (an ordered triplet) and returns the set of all ordinals that are a continuation of any recognizable pattern or limit formed by the triplet. The term â€˜recognizable pattern or limitâ€™ is defined for this purpose as one that appears in a sequence of rules (see â€œRulesâ€ below).

Brief Explanation of Function $f$:

To get started, consider the ordered triplet $1, 2, 3$ as the initial segment of some pattern of ordinals that could be continued or a limit that could be approached: $1, 2, 3, \dots$. What would come next? The number $4$ is a continuation of the ordinalsâ€™ pattern, so we might then say that $4$ is implied by $1, 2, 3, \dots$ or, similarly, $1, 2, 3, \dots \implies 4$. The number $4$ is not the only thing that may be implied by $1, 2, 3, \dots$, however. We can also say that $1, 2, 3, \dots \implies \omega$. If there are any other ordinal numbers that happen to be universally agreeable continuations or limits for the three-member sequence $1, 2, 3, â€¦$, we could consider them too. In this case, I am unaware of any. The triplet $1, 2, 3$ is being used as an example because it implies two ordinals, $4$ and $\omega$ (and so $f((1,2,3)) = \{4, \omega \}$), whereas other triplets may imply one or none. The order of the triplet is important too. If we instead consider $3, 2, 1$ we find that $0$ is a continuation of the ordinalsâ€™ pattern (instead of $4$), so we say $3, 2, 1 \implies 0$. To summarize, we are starting to form some basic rules that collectively are going to define the term â€˜recognizable pattern or limitâ€™.

Rules: Where $a, b, c, \dots$ are ordinals, the following rules are used to define function $f$.

$$\text{Rule 1 : }a, a-1, a-2, \dots \implies \{ a-3 \} \text{, where } a \geq 3$$
$$\text{Rule 2 : }0, 1, a, \dots \implies \{a+1\}$$
$$\text{Rule 3 : }a, a+1, a+2, \dots \implies \{ a+3, a + \omega \}$$

Define function $f$:

$$f((a,b,c)) = \bigcup \{ x : a, b, c, \dots \implies x \}, \text{ where } a, b, \text{ and } c \text{ are ordinals and } x \text{ is a set defined by the above rules}$$

Define function $g$:

For any set of ordinals, $A$, let $g(A)$ be the set of all ordered triplets that can be made from the elements of $A$:
$$g(A) = \{ (a,b,c) : a,b,c \in A \}$$

Define $X_{Ord}$:
$$X_{Ord} = X \setminus \{ x \in X : x \text{ is not an ordinal} \} \text{ for any set } X$$

Define the sequence $T$:

Define a sequence $T = t_1, t_2, t_3, \dots$ via iterations where:

Step 1) $t_1 = 1, t_2 = 2,$ and $t_3 = 3$.

Step 2) Each $t_n$, where $n \geq 4$, is defined by the previous elements of the sequence. Starting with $n = 4$:

a) Let $A = \{ t_i \in T : i < n \}$. E.g., $A = \{1, 2, 3 \}$ on the first iteration.

b) Let $B = \{ f((a,b,c))_{Ord} : (a,b,c) \in g(A) \}$. Using the previous elements of the sequence $A$, this step creates a set $B$ of all the new sets of ordinals implied by letting function $f$ range over $g(A)$. The use of the $X_{Ord}$ function in the definition of $B$ may be unnecessary for this particular $T$ sequence.

c) Let $C = \bigcup B \setminus A$ and let $c_1, c_2, c_3, \dots$ be an enumeration of $C$ that is also well ordered if $|C| \in \mathbb{N}$. This step shaves off any redundant elements of $B$ before potentially well ordering them so that we can add them to $T$.

d) If $|C| \in \mathbb{N}$, then set $t_n = c_1, t_{n+1} = c_2, t_{n+2} = c_3, \dots, t_{n+j-1} = c_j$.

e) If $|C| = |\mathbb{N}|$ (not applicable for this particular $T$ sequence), then let $Tâ€™ = tâ€™_1, tâ€™_2, tâ€™_3, \dots$, be a subsequence of the remaining undefined elements of $T$ and set $tâ€™_n = c_1, tâ€™_{n+2} = c_2, tâ€™_{n+4} = c_3, \dots$.

Step 3) Proceed to the next undefined index $j$ in $T$, set $n = j$, and repeat step 2.

The first few elements of $T$ with the above three rules would be:
$$T = 1,2,3,0,4,\omega,5, \omega + 1, 6, \omega + 2,7,\omega+3,\omega \cdot 2,8,\omega+4,\omega \cdot 2 + 1, \dots$$

Where $T$ is generated one iteration at a time, the first iteration takes each ordered triplet that may be comprised from the ordinals $1, 2,$ and $3$ (there are six possible triplets) and tests each triplet to see if its ordering matches one of the rules. We have $f((3,2,1)) = \{0\}$ by Rule 1 and $f((1,2,3)) = \{4, \omega \}$ by Rule 3. The iteration process then takes the union of $\{0\}$ and $\{4, \omega \}$, orders the union to produce a countable sequence that is also well ordered (because it is a finite sequence), and adds the sequence to the initial undefined elements of $T$. We then start the second iteration by taking each ordered triplet that may be comprised from the ordinals $1,2,3,0,4,\omega$ and seeing which new ordinals the rules imply. Here we get the same old $f((3,2,1)) = \{0\}$ and $f((1,2,3)) = \{4, \omega \}$ in addition to $f((0,1,4)) = \{5\}, f((2,3,4)) = \{5\},$ and $f((0,1,\omega)) = \{\omega + 1\}$. Any duplicate ordinals are removed during the iteration process so as to refrain from adding them to $T$ more than once.

If we keep going with these three rules, the $T$ sequence will become an enumeration of $\omega^2$. Note that no rule exists that is capable of taking three ordinals less than $\omega^2$ and implying a set containing an ordinal greater than or equal to $\omega^2$. Rule #1 could produce $\omega^2$ given the triplet $(\omega^2 + 3, \omega^2 + 2, \omega^2 + 1)$, but since there are no ordinals greater than $\omega^2$, Rule 1 is not capable of inserting $\omega^2$ into the sequence either.

Rule 4 will lead to $T$ becoming an enumeration of $\omega^2 \cdot 2$ (assuming my calculations are correct for all of these):

$$\text{Rule 4 : }a \cdot b, a \cdot (b+1), a \cdot (b+2), \dots \implies \{a \cdot (b + \omega) \}$$

Rule 5 will lead to $T$ becoming an enumeration of $\omega^{\omega}$

$$\text{Rule 5 : }a+b \cdot c, a+b \cdot (c+1), a+b \cdot (c+2), \dots \implies \{a+b \cdot (c+\omega)\}$$

Rule 6 will lead to $T$ becoming an enumeration of $\omega^{\omega} \cdot 2$:

$$\text{Rule 6 : }a^b, a^{b+1}, a^{b+2}, \dots \implies \{a^{b+\omega}\}$$

Rule 7 will lead to $T$ becoming an enumeration of $\omega^{\omega^2}$:

$$\text{Rule 7 : }a+b^c, a+b^{c+1}, a+b^{c+2}, \dots \implies \{a+b^{c+\omega}\}$$

Rule 8 will lead to $T$ becoming an enumeration of $\omega^{\omega^{\omega}} \cdot 2$:

$$\text{Rule 8 : }a + b^{c + d \cdot e}, a + b^{c + d \cdot (e+1)}, a + b^{c + d \cdot(e+2)}, \dots \implies \{a + b^{c + d \cdot(e+\omega)}\}$$

Rule 9 will lead to $T$ becoming an enumeration of $\epsilon_0 = \omega^{\epsilon_0}$:

$$\text{Rule 9 : }a + b^{c+d^e}, a + b^{c+d^{e+1}}, a + b^{c+d^{e+2}}, \dots \implies \{ a + b^{c+d^{e+w}}\}$$

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#### AplanisTophet

The calcs do get a little tricky but hopefully the above is an enumeration of $\epsilon_0$. If anyone sees an error or is unable to make sense of the $T$ sequence model I would love to hear about it.

If the above looks good, the goal will be to consider $T^{\alpha}$ for any (countable) ordinal $\alpha$. The ordinal $\alpha$ will represent the number of rules employed by the $T$ sequence. An uncountable $\alpha$ would obviously require some clarification as $T$ would have to become a transfinite sequence, etc.

#### Maschke

If anyone sees an error or is unable to make sense of the $T$ sequence model I would love to hear about it.
Would you entertain elementary questions designed to help readers understand your exposition? In other words, hypothetically speaking, if someone considered your exposition so incoherent they could barely get started trying to understand it; posed simple questions; and asked for clear, coherent mathematical answers; would you try to be helpful, and explain yourself clearly line by line and point by point? Or would the mere fact that they consider your exposition incoherent frustrate you to the extent that you have a hard time doing that? Can you get past your erroneous belief that unpacking your exposition would take "five minutes" as you once opined?

And if you find yourself getting frustrated, can you please criticize the speech and not the speaker? As in: I find your exposition incoherent. As opposed to personal remarks.

If so:

$$\text{Rule 1 : }a, a-1, a-2, \dots \implies \{ a-3 \} \text{, where } a \geq 3$$
$$\text{Rule 2 : }0, 1, a, \dots \implies \{a+1\}$$
$$\text{Rule 3 : }a, a+1, a+2, \dots \implies \{ a+3, a + \omega \}$$
(1) What is the meaning of the ellipses (three dots) in each case?

(2) What if $a$ is a limit ordinal in Rule 1?

(3) In Rule 2, suppose $a = 2$, giving $0, 1, 2, \dots \implies \{3\}$; but in Rule 3, if $a = 0$ then $0, 1, 2, \dots \implies \{3, \omega\}$. How do you account for this inconsistency?

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#### AplanisTophet

Can you get past your erroneous belief that unpacking your exposition would take "five minutes" as you once opined?
Busting my recent "opus" simply took pointing out that I was incorrectly applying the countable union of countable sets theorem under statement #2 because I thought that all the indexes being countable meant they could be sequenced. Of course, this thread has nothing to do with that and it's certainly not (yet another!!! :giggle: ) opus where there is obviously something wrong and I'm just trying to figure out what. Rather, this thread is an honest attempt at defining the $T$ sequence model and enumerating $\epsilon_0$.

(1) What is the meaning of the ellipses (three dots) in each case?

(2) What if $a$ is a limit ordinal in Rule 1?

(3) In Rule 2, suppose $a = 2$, giving $0, 1, 2, \dots \implies \{3\}$; but in Rule 3, if $a = 0$ then $0, 1, 2, \dots \implies \{3, \omega\}$. How do you account for this inconsistency?
1) The ellipses represent the suggestion that the ordered triplet presents a pattern or sequence that could be continued so as to derive a next element or limit. Their inclusion is a semantic preference based on the "Brief Explanation of Function $f$". If you find them confusing others may as well, though I suspect not including them could also be as equally confusing to others.

2) If $a$ is a limit ordinal, then $a-1, a-2,$ and $a-3$ all equal $a$.

3) On the second iteration $g(A)$ will contain the triplet $(0,1,2)$. We will then get $f((0,1,2)) = \bigcup \{ \{3\}, \{3, \omega\} \} = \{ 3, \omega \}$ where $\{3,\omega\} \in B$ and $3, \omega \in \bigcup B$.

#### Maschke

1) The ellipses represent the suggestion that the ordered triplet presents a pattern or sequence that could be continued so as to derive a next element or limit. Their inclusion is a semantic preference based on the "Brief Explanation of Function $f$". If you find them confusing others may as well, though I suspect not including them could also be as equally confusing to others.
That's perfectly incoherent and does not have any meaning at all. Your entire exposition fails until you define your notation. What on earth is a semantic preference?

2) If $a$ is a limit ordinal, then $a-1, a-2,$ and $a-3$ all equal $a$.
By what definition of ordinal subtraction? More incoherence. You can't subtract from a limit ordinal.

When I said earlier (and keep saying) that your exposition is incoherent, I don't mean that it becomes so in the details after it's all done. I mean right from the getgo. Your first three rules have no meaning. I pointed out that two of them are inconsistent. You can't subtract from a limit ordinal. And you haven't defined your ellipses notation except to say that it's a "semantic preference." Whose, exactly?

Sorry, this is all a nonstarter. It's not that your entire exposition falls apart. It's that it never gets started at all. Rule 1 is nonsense as stated since you don't define your notation.

How about drilling down and telling me what the ellipses mean in Rule 1? The standard meaning of the notation 1, 2, 3, ... is the entire sequence of positive integers. If you mean something else, please define it with precision.

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#### AplanisTophet

That's perfectly incoherent and does not have any meaning at all. Your entire exposition fails until you define your notation. What on earth is a semantic preference?
Omitting the ellipses from each of the rules and the definition of $f$ wouldn't change a thing mathematically. Do you not see that?

By what definition of ordinal subtraction? More incoherence. You can't subtract from a limit ordinal.
Left subtraction works with ordinals $\beta \leq \alpha$ because there is a unique $\gamma$ such that $\alpha = \beta + \gamma$.

https://en.wikipedia.org/wiki/Ordinal_arithmetic

When I said earlier (and keep saying) that your exposition is incoherent, I don't mean that it becomes so in the details after it's all done. I mean right from the getgo. Your first three rules have no meaning.
The first three rules have no meaning on their own, no. You have to feed a triplet into function $f$ and then see if the triplet matches the pattern given by the rules. How could I possibly give the rules meaning on their own?

I pointed out that two of them are inconsistent.
No two rules are inconsistent with each other as, unlike axioms, that would be impossible for a $T$ sequence. I told you what the result of $f((0,1,2))$ would be. Did you not understand? Each rule has it's own localized variables where setting $a$ equal to $2$ for purposes of applying rule 2 has no bearing on what $a$ is for rule 3. The ordered triplet 0,1,2,... obviously doesn't fit Rule 1, but it does fit Rule 2 and Rule 3. You might then ask, how many rules can one triplet fit? Uncountably many if you like would be the answer, but we can also have uncountably many rules where any single triplet can only fit a finite number though too. I digress though, as that would be a discussion for when we consider $T^{\alpha}$ for any ordinal $\alpha$.

You can't subtract from a limit ordinal.
Rule 1 never does assert that we subtract a finite ordinal from a limit ordinal (though we can as explained above and it's perfectly fine) unless I want to consider triplets like $(\omega, \omega, \omega)$, in which case Rule 1 would then have $f((\omega, \omega, \omega)) = f((\omega, \omega-1, \omega-2)) = \{\omega-3\} = \{\omega\}$ because $\omega = \omega-\beta$ for any ordinal $\beta < \omega$.

And you haven't defined your ellipses notation except to say that it's a "semantic preference." Whose, exactly?
Mine. Function $f$ is defined as $f((a,b,c)) = \bigcup \{ x : a, b, c, \dots \implies x\}$, but if you want it defined as $f((a,b,c)) = \bigcup \{ x : a, b, c \implies x\}$ instead, just take the ellipses out of each of the rules. It makes no difference (just semantics). Clearly, my work doesn't hinge on your ability to make sense of the ellipses.

Sorry, this is all a nonstarter. It's not that your entire exposition falls apart. It's that it never gets started at all. Rule 1 is nonsense as stated since you don't define your notation.
Is there something you need to know other than that the local variables $a, b, c, ...$ for each rule are ordinals? Consider Rule 8, which has not only a local variable $a$ in it, but also $b,c,d,$ and $e$ too. Can you understand how when the input for function $f$ is a mere triplet?

How about drilling down and telling me what the ellipses mean in Rule 1? The standard meaning of the notation 1, 2, 3, ... is the entire sequence of positive integers. If you mean something else, please define it with precision.
If you still don't understand what the ellipses mean after my explanations in the OP, above post, and in this post, just take them all out of the proof and try again. It really won't make a difference, and, since it has no bearing on the underlying mathematics, your assertion that my work is a non-starter because of the ellipses makes no sense to me.

#### Maschke

Omitting the ellipses from each of the rules and the definition of $f$ wouldn't change a thing mathematically. Do you not see that?
No, I don't see that at all. There's a huge difference between "1, 2, 3" and "1, 2, 3, ..."

Left subtraction works with ordinals $\beta \leq \alpha$ because there is a unique $\gamma$ such that $\alpha = \beta + \gamma$.
Nonsense. You can't subtract from a limit ordinal.

If you still don't understand what the ellipses mean after my explanations in the OP, above post, and in this post, just take them all out of the proof and try again. It really won't make a difference, and, since it has no bearing on the underlying mathematics, your assertion that my work is a non-starter because of the ellipses makes no sense to me.

I honestly don't think this is going to be productive for me. I'm going to let it go. Sorry I couldn't be of more help.

#### AplanisTophet

I honestly don't think this is going to be productive for me. I'm going to let it go. Sorry I couldn't be of more help.
Yeah, you tried your best I guess...

#### Maschke

Yeah, you tried your best I guess...
Another way to go here would be for you to challenge yourself to clearly articulate the meaning of your ellipses notation. But you can't get out of your own way.

#### [email protected]

Nonsense. You can't subtract from a limit ordinal.
Aplanis is right here though. The theorem he states is correct. For $\beta\leq \alpha$, there is indeed a unique $\gamma$ such that $\alpha = \beta + \gamma$. This is usually not denoted as subtraction though, but he could define it likes this if he wishes.