[effie]

But why do you assume two objects of same mass but different composition will add the same mass to the black hole? Different composition means different interactions during the fall and different amount of radiated energy. Infinite number of bits can be encoded in single real number. It is hard to measure more than few digits of it, but so it is hard to measure all the information.

[pdonis]

> why do you assume two objects of same mass but different composition will add the same mass to the black hole?

Because that's what the physics says. See below.

> Different composition means different interactions during the fall and different amount of radiated energy.

All of that can be taken into account before the object falls into the hole; the observer outside can measure it all and deduct it from the mass he expects to be added to the hole.

We are talking about the mass that gets added after all that; and for any given mass added to the hole after all those things are taken into account, there are many different possible combinations of objects falling into the hole that can add that mass.

> Infinite number of bits can be encoded in single real number.

We are not talking about math, we are talking about physics. The number of bits that can be stored in an object of finite size is finite as far as physics is concerned.

[effie]

> "for any given mass added to the hole after all those things are taken into account, there are many different possible combinations of objects falling into the hole that can add that mass."

Approximately, sure, best scales can do around 5 significant digits and null measurements can get us few more digits. But we can't verify equality of mass to arbitrary precision. For elementary particles of same kind, we can assume their masses are the same. But there is infinity of digits available. Perhaps there are no two differently composed bodies that have the same real number as mass (too many options to be different). Then maybe any mass addition to mass of the black hole can encode all the information there is about the body.

[pdonis]

> Approximately, sure

This is irrelevant to the argument; our finite ability to measure masses is not what we are talking about. We are talking about what masses are physically possible, whether or not we can measure all of them with unbounded accuracy.

> there is infinity of digits available

You can't have it both ways. If it is physically true that there are an infinite number of digits available to specify an object's mass, then it is also physically true that there are multiple possible combinations of objects whose masses can sum to that same mass (in fact there will be an infinite number of them).

Conversely, if it is not physically true that there are multiple possible combinations of objects whose masses can sum to a given mass, there cannot be an infinite number of digits available to specify an object's mass: there must be only a finite number of possible masses, and the numbers specifying the possible masses must be such that no two such numbers add up to another such number.

[effie]

It is relevant because the limitation to our measurement capability means we can't know most of the digits. We can't confirm experimentally that a given mass can be composed of "multiple possible combinations of objects whose masses can sum to that same mass". The mass number for an object can exist and yet it may be impossible to duplicate it with other objects. Imagine every object having unique ID with infinity of digits.

[pdonis]

> Imagine every object having unique ID with infinity of digits.

And then, as I said, there will be an infinite number of possible combinations of other masses that will add to that mass. The fact that we can't verify that experimentally is irrelevant; your model allows it and that means that, in your model, the unique ID of a given black hole's mass would not uniquely identify the original pieces of matter that formed it, and therefore would not provide the information that you originally claimed it could provide, in the post of yours that started this subthread.