Return to Answer

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $L_1,\ldots, L_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $L_1, L_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts as "there's life beyond $V_{\omega+\omega}$".

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

Some of the comments on the first (Sets vs Classes) allude to various constructions in homotopy theory involving long-running transfinite recursions - and even large cardinals - so presumably there is some Replacement involved.

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $L_1,\ldots, L_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $L_1, L_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts as "there's life beyond $V_{\omega+\omega}$".

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

Some of the comments on the first (Sets vs Classes) allude to various constructions in homotopy theory involving long-running transfinite recursions - and even large cardinals - so presumably there is some Replacement involved.

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $L_1,\ldots, L_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $L_1, L_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts as "there's life beyond $V_{\omega+\omega}$".

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $L_1,\ldots, L_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $L_1, L_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts as "there's life beyond $V_{\omega+\omega}$".

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

Some of the comments on the first (Sets vs Classes) allude to various constructions in homotopy theory involving long-running transfinite recursions - and even large cardinals - so presumably there is some Replacement involved.

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-vector spaces defined though duality: $V_1=\mathbb{R}[t]$ and $V_{k+1}=V_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $V_1,\ldots, V_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $V_1, V_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy which seems vastly more intricate.

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?

This very matter is discussed in depth by Mathias, in Chapter 9 of his The Stength of Mac Lane Set Theory https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows that to prove

$\;\;$ "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets"

requires some use of unbounded Separation, and to prove

$\;\;$ "there exists an infinite set of pairwise nonequinumerous infinite sets"

requires some use of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement".

Algebraists might prefer these assertions concerning sequences of $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting,

$\;\;$ "for all $n$ the sequence $L_1,\ldots, L_n$ exists"

requires some use of unbounded Separation, and

$\;\;$ "the sequence $L_1, L_2, \ldots $ exists"

requires some use of Replacement.

Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts as "there's life beyond $V_{\omega+\omega}$".

Meanwhile, some of the motivation of this question resonates with mine in posing these questions:

When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?