On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) is weaker in consistency strength than $\mathsf{PA}$. (Incidentally, while I believe I can see an argument that $\mathsf{PA}$ interprets $\mathsf{NFU}$, I don't really see how to show that it is strictly weaker.) In this Math SE answer, Randall Holmes states that $\mathsf{NFU}$ interprets Robinson arithmetic and furthermore says that it probably interprets bounded arithmetic with exponentiation (which I think means $\mathsf{I}\Delta_0 + \mathrm{exp}$, although I'm not completely familiar with the semi-formal terminology people use for these things).

This would put $\mathsf{NFU}$ right in the range of theories for which ordinal analysis should be well-behaved (i.e., those which are at least as strong as $\mathsf{I}\Delta_0$) but also should be feasible to actually do (e.g., those which are as strong as or not too much stronger than $\mathsf{PA}$). So this raises a question that I have been unable to find any discussion of.

Question 1. What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?

More generally, a lot is known about consistency strength and interpretation of theories of first- and second-order arithmetic between $\mathsf{I}\Delta_0$ and $\mathsf{PA}$. Proof-theoretic ordinals are supposed to be a rough measure of consistency strength, but in principle this is a more fine grained issue.

Question 2. Where does bare $\mathsf{NFU}$ sit relative to well-known theories of arithmetic in terms of interpretation and consistency strength?

On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) is weaker in consistency strength than $\mathsf{PA}$. In this Math SE answer, Randall Holmes states that $\mathsf{NFU}$ interprets Robinson arithmetic and furthermore says that it probably interprets bounded arithmetic with exponentiation (which I think means $\mathsf{I}\Delta_0 + \mathrm{exp}$, although I'm not completely familiar with the semi-formal terminology people use for these things).

This would put $\mathsf{NFU}$ right in the range of theories for which ordinal analysis should be well-behaved (i.e., those which are at least as strong as $\mathsf{I}\Delta_0$) but also should be feasible to actually do (e.g., those which are as strong as or not too much stronger than $\mathsf{PA}$). So this raises a question that I have been unable to find any discussion of.

Question 1. What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?

More generally, a lot is known about consistency strength and interpretation of theories of first- and second-order arithmetic between $\mathsf{I}\Delta_0$ and $\mathsf{PA}$. Proof-theoretic ordinals are supposed to be a rough measure of consistency strength, but in principle this is a more fine grained issue.

Question 2. Where does bare $\mathsf{NFU}$ sit relative to well-known theories of arithmetic in terms of interpretation and consistency strength?

On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) is weaker in consistency strength than $\mathsf{PA}$. (Incidentally, while I believe I can see an argument that $\mathsf{PA}$ interprets $\mathsf{NFU}$, I don't really see how to show that it is strictly weaker.) In this Math SE answer, Randall Holmes states that $\mathsf{NFU}$ interprets Robinson arithmetic and furthermore says that it probably interprets bounded arithmetic with exponentiation (which I think means $\mathsf{I}\Delta_0 + \mathrm{exp}$, although I'm not completely familiar with the semi-formal terminology people use for these things).

This would put $\mathsf{NFU}$ right in the range of theories for which ordinal analysis should be well-behaved (i.e., those which are at least as strong as $\mathsf{I}\Delta_0$) but also should be feasible to actually do (i.e., those which are as strong as or not too much stronger than $\mathsf{PA}$). So this raises a question that I have been unable to find any discussion of.

Question 1. What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?

More generally, a lot is known about consistency strength and interpretation of theories of first- and second-order arithmetic between $\mathsf{I}\Delta_0$ and $\mathsf{PA}$. Proof-theoretic ordinals are supposed to be a rough measure of consistency strength, but in principle this is a more fine grained issue.

Question 2. Where does bare $\mathsf{NFU}$ sit relative to well-known theories of arithmetic in terms of interpretation and consistency strength?

On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) is weaker in consistency strength than $\mathsf{PA}$. (Incidentally, while I believe I can see an argument that $\mathsf{PA}$ interprets $\mathsf{NFU}$, I don't really see how to show that it is strictly weaker.) In this Math SE answer, Randall Holmes states that $\mathsf{NFU}$ interprets Robinson arithmetic and furthermore says that it probably interprets bounded arithmetic with exponentiation (which I think means $\mathsf{I}\Delta_0 + \mathrm{exp}$, although I'm not completely familiar with the semi-formal terminology people use for these things).

This would put $\mathsf{NFU}$ right in the range of theories for which ordinal analysis should be well-behaved (i.e., those which are at least as strong as $\mathsf{I}\Delta_0$) but also should be feasible to actually do (e.g., those which are as strong as or not too much stronger than $\mathsf{PA}$). So this raises a question that I have been unable to find any discussion of.

Question 1. What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?

More generally, a lot is known about consistency strength and interpretation of theories of first- and second-order arithmetic between $\mathsf{I}\Delta_0$ and $\mathsf{PA}$. Proof-theoretic ordinals are supposed to be a rough measure of consistency strength, but in principle this is a more fine grained issue.

Question 2. Where does bare $\mathsf{NFU}$ sit relative to well-known theories of arithmetic in terms of interpretation and consistency strength?