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A striking result referenced in the Shulman paper is due to Colin McLarty, establishing that the NF axiomatization of what a set is yields a ${\bf Set}$ that isn't Cartesian closed.

All the extra baggage of universes or inaccessibles is still somewhat of a sledgehammer for the problem at hand, though; all we want is for large categories to 'be like small categories' in enough ways that we can carry out all the constructions we care about with large categories, but inaccessibles or Grothendieck universes also have a plethora of other consequences (like the need to juggle universes*). A solution to these problems comes in the form of reflection principles, which are essentially axioms asserting that proper classes look enough like sets that we don't have to soil ourselves when they appear, but don't endow them with enough independence to give rise to a whole hierarchy of universes we need to ask questions about. All of this is discussed at length in the Shulman paper referenced above, with additional references therein.

Second paper: McLarty, Colin. Failure of Cartesian Closedness in NF. J. Symbolic Logic 57 (1992), no. 2, 555--556. https://projecteuclid.org/euclid.jsl/1183743976

*As Tim points out in the comments, how many universes we have to juggle when taking this route is up to us. Skilled jugglers may use an infinite number, while those new to the approach may use only two.

A striking result referenced in the Shulman paper is due to Colin McLarty, establishing that the NF axiomatization of what a set is yields a ${\bf Set}$ that isn't Cartesian closed.

All the extra baggage of universes or inaccessibles is still somewhat of a sledgehammer for the problem at hand, though; all we want is for large categories to 'be like small categories' in enough ways that we can carry out all the constructions we care about with large categories, but inaccessibles or Grothendieck universes also have a plethora of other consequences (like the need to juggle universes¹). A solution to these problems comes in the form of reflection principles, which are essentially axioms asserting that proper classes look enough like sets that we don't have to soil ourselves when they appear, but don't endow them with enough independence to give rise to a whole hierarchy of universes we need to ask questions about. All of this is discussed at length in the Shulman paper referenced above, with additional references therein.

Second paper: McLarty, Colin. Failure of Cartesian Closedness in NF. J. Symbolic Logic 57 (1992), no. 2, 555--556. https://doi.org/10.2307/2275291

¹As Tim points out in the comments, how many universes we have to juggle when taking this route is up to us. Skilled jugglers may use an infinite number, while those new to the approach may use only two.

A striking result referenced in the Shulman paper is due to by Colin McLarty, establishing that the NF axiomatization of what a set is yields a ${\bf Set}$ that isn't Cartesian closed.

A striking result referenced in the Shulman paper is due to Colin McLarty, establishing that the NF axiomatization of what a set is yields a ${\bf Set}$ that isn't Cartesian closed.

All the extra baggage of universes or inaccessibles is still somewhat of a sledgehammer for the problem at hand, though; all we want is for large categories to 'be like small categories' in enough ways that we can carry out all the constructions we care about with large categories, but inaccessibles or Grothendieck universes also have a plethora of other consequences (like the need to juggle universes). A solution to these problems comes in the form of reflection principles, which are essentially axioms asserting that proper classes look enough like sets that we don't have to soil ourselves when they appear, but don't endow them with enough independence to give rise to a whole hierarchy of universes we need to ask questions about. All of this is discussed at length in the Shulman paper referenced above, with additional references therein.

Second paper: McLarty, Colin. Failure of Cartesian Closedness in NF. J. Symbolic Logic 57 (1992), no. 2, 555--556. https://projecteuclid.org/euclid.jsl/1183743976

All the extra baggage of universes or inaccessibles is still somewhat of a sledgehammer for the problem at hand, though; all we want is for large categories to 'be like small categories' in enough ways that we can carry out all the constructions we care about with large categories, but inaccessibles or Grothendieck universes also have a plethora of other consequences (like the need to juggle universes*). A solution to these problems comes in the form of reflection principles, which are essentially axioms asserting that proper classes look enough like sets that we don't have to soil ourselves when they appear, but don't endow them with enough independence to give rise to a whole hierarchy of universes we need to ask questions about. All of this is discussed at length in the Shulman paper referenced above, with additional references therein.

Second paper: McLarty, Colin. Failure of Cartesian Closedness in NF. J. Symbolic Logic 57 (1992), no. 2, 555--556. https://projecteuclid.org/euclid.jsl/1183743976

*As Tim points out in the comments, how many universes we have to juggle when taking this route is up to us. Skilled jugglers may use an infinite number, while those new to the approach may use only two.