Black hole riddle

General relativity tells us that an infinite number of houses can be built at fixed altitudes in the one meter (say) above the horizon, as measured in Euclidian geometry.

What do you mean by "as measured in Euclidean geometry"?

In a continuous classical world it would also be possible to build an infinite number of ever-thinner, ever-more-closely-packed houses in any given meter of flat space--what role does the event horizon play in this problem?

Anyway, the core idea of my last post remains, I think... it is not possible to uniquely identify "the last house" in such a situation.

On the way to the horizon, houses are passed. Upon reaching the horizon, there are no more houses. Then there must have been a last house, uniquely identifiable.

In order for all the houses to maintain a constant distance from the horizon, it will have to expend large amounts of energy--would the energy density of the ever-more-tightly packed houses (in some coordinate system at least, I don't know which system would be the relevant one) cause the houses to form a black hole themselves, or the horizon of the original black hole to expand to include them?

Do you think there is a uniquely identifiable smallest positive fraction of the form 1/n, where n is a positive integer? After all, as you approach zero when moving along the real number line in the negative direction, fractions of this form are passed, but upon reaching zero there are no more such fractions.

Such details can be ignored in a thought experiment like this. These are not ever-more-tightly packed houses, at least not in their own frames. They can all have the same proper sized lots.

True, but it doesn’t answer the riddle. It doesn’t explain, for example, how there is not a uniquely identifiable last house. Have you ever passed a row of houses, and upon coming to the end of the row, found that there was no last house in the row?

On the way to the horizon, houses are passed. Upon reaching the horizon, there are no more houses. Then there must have been a last house, uniquely identifiable.

But in terms of the formation of a new black hole, does GR say...

Well, even if this thought-experiment is possible in principle, won't a freely-falling observer see the houses more and more tightly packed as he approaches the horizon, rather than an equal distance apart? If so, he is experiencing the same thing you would experience if you walked along a number line towards zero with each number of the form 1/n marked off--at some point the markings would become too close to distinguish.

That is indeed the way one of my books explains how an observer free-falling across a horizon can travel an infinite distance (an infinite number of houses) in a finite time. But it still does not answer the riddle. I pass an infinite amount of infinitesimally-sized stuff with every step I take. But that doesn’t explain how, when I come to the end of a row of houses, there is no last house.

That's certainly intuitive, but is it true? Unituitive things happen when you're dealing with infinities, which is why Zeno is remembered so fondly.

Sure it does. If there are infinity houses, there is no uniquely identifiable last house, just like there is no uniquely identifiable last number. What's left to explain?

Such details put the cart before the horse, so they can be ignored.

That is indeed the way one of my books explains how an observer free-falling across a horizon can travel an infinite distance (an infinite number of houses) in a finite time. But it still does not answer the riddle. I pass an infinite amount of infinitesimally-sized stuff with every step I take. But that doesn’t explain how, when I come to the end of a row of houses, there is no last house.