I'm interested in Reflection Principles but I can't find any references of works around criteria to classify properties well-behaved relatively to reflection, or at least features that properties must have in order to be reflected.

In a passage by Incurvati L., Conceptions of Set and the Foundations of Mathematics (2020) I read:

[...] equating the hierarchy’s absolute infinity with its not being uniquely characterizable by any property of a certain type K. [...] Little has been said, however, by way of a positive characterization of K. Gödel took the Reflection Principle to hold when K is the class of structural properties, but a full-fledged account of the notion of a structural property remains wanting.

where references to Gödel can be find in Wang H., A logical journey (1996) at pages 283-285, but even there the notion of structural property is left obscure.

The difficult notion is of course that of a structural property. Gödel's association of Ackermann's idea with the inclusive principle suggests that, for him, the properties in Ackermann's axiom (A) are examples of structural properties. He seems to detect certain distinctive features in the properties used in (A), apart from the explicit condition that they do not contain V. He seems to say that any property that shares these features is a structural property. The problem is to give a moderately precise account of these features which lends some credibility to the belief that properties with such features define sets. [...]

Of these observations, 8.7.13 gives the best indication of Gödel's concept of structural properties. [...] I am not sure whether this elaboration agrees with Gödel's intention. In any case, there remains, I think, the problem of applying this vague characterization to arrive at precise characterizations of some rich classes of structural properties of the desired kind.

I ask for any suggestion and reference around this topic (I am currently studying connections between reflection principles and large cardinal axioms on works of Reinhardt, Koellner, Welch among others).

I'm interested in Reflection Principles but I can't find any references of works around criteria to classify properties well-behaved relatively to reflection, or at least features that properties must have in order to be reflected.

In a passage by Incurvati L., Conceptions of Set and the Foundations of Mathematics (2020) I read:

[...] equating the hierarchy’s absolute infinity with its not being uniquely characterizable by any property of a certain type K. [...] Little has been said, however, by way of a positive characterization of K. Gödel took the Reflection Principle to hold when K is the class of structural properties, but a full-fledged account of the notion of a structural property remains wanting.

where references to Gödel can be found in Wang H., A logical journey (1996) at pages 283-285, but even there the notion of structural property is left obscure:

The difficult notion is of course that of a structural property. Gödel's association of Ackermann's idea with the inclusive principle suggests that, for him, the properties in Ackermann's axiom (A) are examples of structural properties. He seems to detect certain distinctive features in the properties used in (A), apart from the explicit condition that they do not contain V. He seems to say that any property that shares these features is a structural property. The problem is to give a moderately precise account of these features which lends some credibility to the belief that properties with such features define sets. [...]

Of these observations, 8.7.13 gives the best indication of Gödel's concept of structural properties. [...] I am not sure whether this elaboration agrees with Gödel's intention. In any case, there remains, I think, the problem of applying this vague characterization to arrive at precise characterizations of some rich classes of structural properties of the desired kind.

I ask for any suggestion and reference around this topic (I am currently studying connections between reflection principles and large cardinal axioms in works of Reinhardt, Koellner, Welch among others).

I'm interested in Reflection Principles but I can't find any references of works around criteria to classify properties well-behaved relatively to reflection, or at least features that properties must have in order to be reflected.

In a passage by Incurvati L., Conceptions of Set and the Foundations of Mathematics (2020) I read:

[...] equating the hierarchy’s absolute infinity with its not being uniquely characterizable by any property of a certain type K. [...] Little has been said, however, by way of a positive characterization of K. Gödel took the Reflection Principle to hold when K is the class of structural properties, but a full-fledged account of the notion of a structural property remains wanting.

where references to Gödel can be find in Wang H., A logical journey (1996) at pages 283-285, but even there the notion of structural property is left obscure.

I ask for any suggestion and reference around this topic (I am currently studying connections between reflection principles and large cardinal axioms on works of Reinhardt, Koellner, Welch among others).

I'm interested in Reflection Principles but I can't find any references of works around criteria to classify properties well-behaved relatively to reflection, or at least features that properties must have in order to be reflected.

In a passage by Incurvati L., Conceptions of Set and the Foundations of Mathematics (2020) I read:

[...] equating the hierarchy’s absolute infinity with its not being uniquely characterizable by any property of a certain type K. [...] Little has been said, however, by way of a positive characterization of K. Gödel took the Reflection Principle to hold when K is the class of structural properties, but a full-fledged account of the notion of a structural property remains wanting.

where references to Gödel can be find in Wang H., A logical journey (1996) at pages 283-285, but even there the notion of structural property is left obscure.

The difficult notion is of course that of a structural property. Gödel's association of Ackermann's idea with the inclusive principle suggests that, for him, the properties in Ackermann's axiom (A) are examples of structural properties. He seems to detect certain distinctive features in the properties used in (A), apart from the explicit condition that they do not contain V. He seems to say that any property that shares these features is a structural property. The problem is to give a moderately precise account of these features which lends some credibility to the belief that properties with such features define sets. [...]

Of these observations, 8.7.13 gives the best indication of Gödel's concept of structural properties. [...] I am not sure whether this elaboration agrees with Gödel's intention. In any case, there remains, I think, the problem of applying this vague characterization to arrive at precise characterizations of some rich classes of structural properties of the desired kind.

I ask for any suggestion and reference around this topic (I am currently studying connections between reflection principles and large cardinal axioms on works of Reinhardt, Koellner, Welch among others).