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Gerhard Paseman
Gerhard Paseman
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It depends. Do you care about naming or describing such sets?

In the system you propose, you may be able prove the existence of more than finitely many of them, but in a countable (non-infinitary, since you mention a small extension of ZFC) language, you will only be able to describe countable many of these sets or equivalence classes.

It may also be possible to show that there is no surjection from any given set with a cardinal (that you can describe or posit in this system) onto this class, but all that says is that the system you consider is too weak to analyze the class as fully as you want. Back at you: if you were given an answer, what would you hope to do with it?

I am not expert enough to answer the question about consistency strength. Based on the comments of Asaf and Monroe to the question, I suspect an assumption that the class you have is enumerated by ana cardinal of type X is equivalent in strength to the assumption that a cardinal of type X exists.

Gerhard "Philosophers Don't Know What's Wanted" Paseman, 2019.04.27.

It depends. Do you care about naming or describing such sets?

In the system you propose, you may be able prove the existence of more than finitely many of them, but in a countable (non-infinitary, since you mention a small extension of ZFC) language, you will only be able to describe countable many of these sets or equivalence classes.

It may also be possible to show that there is no surjection from any given set with a cardinal (that you can describe or posit in this system) onto this class, but all that says is that the system you consider is too weak to analyze the class as fully as you want. Back at you: if you were given an answer, what would you hope to do with it?

I am not expert enough to answer the question about consistency strength. Based on the comments of Asaf and Monroe to the question, I suspect an assumption that the class you have is enumerated by an cardinal of type X is equivalent in strength to the assumption that a cardinal of type X exists.

Gerhard "Philosophers Don't Know What's Wanted" Paseman, 2019.04.27.

It depends. Do you care about naming or describing such sets?

In the system you propose, you may be able prove the existence of more than finitely many of them, but in a countable (non-infinitary, since you mention a small extension of ZFC) language, you will only be able to describe countable many of these sets or equivalence classes.

It may also be possible to show that there is no surjection from any given set with a cardinal (that you can describe or posit in this system) onto this class, but all that says is that the system you consider is too weak to analyze the class as fully as you want. Back at you: if you were given an answer, what would you hope to do with it?

I am not expert enough to answer the question about consistency strength. Based on the comments of Asaf and Monroe to the question, I suspect an assumption that the class you have is enumerated by a cardinal of type X is equivalent in strength to the assumption that a cardinal of type X exists.

Gerhard "Philosophers Don't Know What's Wanted" Paseman, 2019.04.27.

Source Link
Gerhard Paseman
Gerhard Paseman
  • 13k
  • 3
  • 32
  • 63

It depends. Do you care about naming or describing such sets?

In the system you propose, you may be able prove the existence of more than finitely many of them, but in a countable (non-infinitary, since you mention a small extension of ZFC) language, you will only be able to describe countable many of these sets or equivalence classes.

It may also be possible to show that there is no surjection from any given set with a cardinal (that you can describe or posit in this system) onto this class, but all that says is that the system you consider is too weak to analyze the class as fully as you want. Back at you: if you were given an answer, what would you hope to do with it?

I am not expert enough to answer the question about consistency strength. Based on the comments of Asaf and Monroe to the question, I suspect an assumption that the class you have is enumerated by an cardinal of type X is equivalent in strength to the assumption that a cardinal of type X exists.

Gerhard "Philosophers Don't Know What's Wanted" Paseman, 2019.04.27.