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I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

, where $``\#"$ is "Hessenberg natural sum"

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^{-1}$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$

Another very closely related pair [due to Randall Holmes] along similar lines is the following:

Define $inc$ as the function sending $(x,n)$ to $(x,n+1)$ for any $x$ and any natural number $n$, and fixing anything not in $V \times N$

Now define $A_x$ as the $\{inc``B \cup \{(x,0)\}: B \in A \}$ This "increments" all elements of elements of $A$ which are in $V \times N$ and adds $(x,0)$ as a further element of each element.

For any function $f$ with domain the set of all double singletons, define $tuple(f)$ as the union of all $(f(\{\{x\}\})_x)$

This gives a $T^2(|V|)$ tuple with projections at the same type as the tuple.

I think that the limit on how many projections there are is intrinsic.

I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

, where $``\#"$ is "Hessenberg natural sum"

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^{-1}$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$

Another very closely related pair [due to Randall Holmes] along similar lines is the following:

Define $inc$ as the function sending $(x,n)$ to $(x,n+1)$ for any $x$ and any natural number $n$, and fixing anything not in $V \times N$

Now define $A_x$ as the $\{inc``B \cup \{(x,0)\}: B \in A \}$ This "increments" all elements of elements of $A$ which are in $V \times N$ and adds $(x,0)$ as a further element of each element.

For any function $f$ with domain the set of all double singletons, define $tuple(f)$ as the union of all $(f(\{\{x\}\})_x)$

This gives a $T^2(|V|)$ tuple with projections at the same type as the tuple.

I think that the limit on how many projections there are is intrinsic.

A closely related observation of mine is that we can use similar method to have type level functions of maximally $T|V|$ many entries! For every function $f$ whose domain is a set of singletons, there is a type-level function $t(f)$ corresponding to it defined as:

$t(f)=\{(x,z)| z \in f(\{x\})\}$

Where $(,)$ is the Quine-Rosser type-level pair.

I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

, where $``\#"$ is "Hessenberg natural sum"

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^{-1}$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$

I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

, where $``\#"$ is "Hessenberg natural sum"

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^{-1}$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$

Another very closely related pair [due to Randall Holmes] along similar lines is the following:

Define $inc$ as the function sending $(x,n)$ to $(x,n+1)$ for any $x$ and any natural number $n$, and fixing anything not in $V \times N$

Now define $A_x$ as the $\{inc``B \cup \{(x,0)\}: B \in A \}$ This "increments" all elements of elements of $A$ which are in $V \times N$ and adds $(x,0)$ as a further element of each element.

For any function $f$ with domain the set of all double singletons, define $tuple(f)$ as the union of all $(f(\{\{x\}\})_x)$

This gives a $T^2(|V|)$ tuple with projections at the same type as the tuple.

I think that the limit on how many projections there are is intrinsic.

I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

and were $``\#"$ is "Hessenberg natural sum"

where: $max(\alpha,\beta)=\gamma \equiv_{df} \\(\alpha > \beta \land \gamma=\alpha) \lor (\beta > \alpha \land \gamma=\beta) \lor (\alpha = \beta \land \gamma = \alpha)$

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^-1$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$

I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.

I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would be what could be thought of as being the simplest genuine extension of the Quine-Rosser pairs.

Statement: for any ordinal $\alpha$, there is a tuple of length $T^2 \alpha$, and there will be no limitation (unlike the tuple in the question) on which set can be a projection of it.

Let $D$ be the set of all Ordinals.

Define functions $F_i$ for $i \in D\setminus \{0\}$ as:

$F_i(X)= [X \setminus D] \cup \{d \#(i \# 1)| d \in D \cap X \} \cup \{i\} $

, where $``\#"$ is "Hessenberg natural sum"

Now the tuple $t^{\alpha}(S)$ whose projections are entries $s_{\{\{i\}\}}$ of the sequence $S$ from $\iota^2"\alpha \to V$ is defined as:

$ t^{\alpha}(S) = \{F_i(x)| x \in s_{\{\{i\}\}} \in rng(S) \land \{\{i\}\} \in \iota^2" \alpha \}$

Now to retrieve back the $i^{th}$ projection from the pair, the smallest ordinal member of the elements of the pair would serve as the indicator of the projection, so take the set of all elements of the tuple having ordinal $i$ as the smallest ordinal in them, and taking the converse function $F^{-1}$ over those would yield the $i^{th}$ projection of the tuple. So simply:

$i^{th} (p) = \{ x | F_i(x) \in p \}$

This tuple works in the most general manner that a tuple can work in NF\NFU, as well as in NBG, MK set theories. However, in ZF and its extensions, it is equivalent to the tuple mentioned in the question.

Notation: $\iota^2" X = \{\{\{x\}\} : x \in X \}$ ; $T^2|X| = |\iota^2" X |$