We're not saying we would have wanted to live anywhere in 1905, since we're pretty attached to our smartphones—but if we did have to time-travel back a century or so, we'd definitely take New York. You have to admit it sounds like a magical fairyland: carriages, snow, twinkling lights, lampposts. All it's missing is a bunch of talking animals.
So you can understand why New York City becomes Paul's beacon of hope. The setting in "Paul's Case" is split between a (mostly) dreary Pittsburgh and a (mostly) dreary New York. Let's check out exactly how Cather plays these two places against each other.
Half of Paul's Pittsburgh is his school; his church; his street, Cordelia Street; and, most of all, his house. These are dreary, ugly, soul-killing places, and they all smell a little like cooking—probably the worst kind, too, all old soup and stale frying oil. Gross, gross, gross.
But then the other half is a magical wonderland of music and song. This is Carnegie Hall and the theater, and to Paul they're full of "unimaginable splendor":
It was at the theatre and at Carnegie Hall that Paul really lived; the rest was but a sleep and a forgetting. This was Paul's fairy tale, and it had for him all the allurement of a secret love. (1.29)
Look, we already know there's a problem here, because the thing about fairy tales? They're not real. All those queenly actresses? "They were hard-working women, most of them supporting indigent husbands or brothers, and they laughed rather bitterly at having stirred the boy to such fervid and florid inventions" (1).
In other words, they're just a bunch of actors in greasy makeup performing in front of cheap sets. This isn't even Broadway. This is so far off-Broadway that, well, it's in Pittsburgh. But to Paul, it's magical. Notice how abstract the descriptions of the theater are—all blazing light and music, but no concrete details—in comparison with his house:
[H]is ugly sleeping chamber, the cold bath-room, with the grimy zinc tub, the cracked mirror, the dripping spigots. (1.20)
No wonder he wants to escape. And when he does, he goes somewhere even more magical than the theater.
Compared to Pittsburgh, New York is beautiful. Check it out:
The snow had somewhat abated, carriages and tradesmen's wagons were hurrying to and fro in the winter twilight, boys in woollen mufflers were shovelling off the doorsteps, the avenue stages made fine spots of color against the white street. Here and there on the corners were stands, with whole flower gardens blooming under glass cases, against the sides of which the snow-flakes stuck and melted; violets, roses, carnations, lilies of the valley, somehow vastly more lovely and alluring that they blossomed thus unnaturally in the snow. The Park itself was a wonderful stage winter-piece. (2.11)
Notice how Paul just thinks of New York as a bigger theater. It's a "wonderful stage winter-piece." All he has for comparisons is plays, and so he can't even experience New York directly. He has to funnel it through his own expectations of what it would be like. And it's awesome. It gets even better at night:
When he returned, the pause of the twilight had ceased, and the tune of the streets had changed. The snow was falling faster, lights streamed from the hotels that reared their dozen stories fearlessly up into the storm, defying the raging Atlantic winds. A long, black stream of carriages poured down the avenue, intersected here and there by other streams, tending horizontally. There were a score of cabs about the entrance of his hotel, and his driver had to wait. Boys in livery were running in and out of the awning that was stretched across the sidewalk, up and down the red velvet carpet laid from the door to the street. Above, about, within it all was the rumble and roar, the hurry and toss of thousands of human beings as hot for pleasure as himself, and on every side of him towered the glaring affirmation of the omnipotence of wealth. (2.12)
Here is Paul in the very thick of things. If Pittsburgh to him seemed like a backwater horror of a city, New York is the center of the world. All the activity and energy of the nation is concentrated here, and it all has to do with wealth.
This is a good place to point out that, while people may have been spending their money in New York, they were making a lot of it in places like Pittsburgh. Pittsburgh was a city built around steel. Andrew Carnegie founded a steel factory there in 1875, and the industry's success turned the city into a bustling metropolis by the 1910s. This association with steel is probably part of why Paul hates it so much. It was a city of workers, a city about making money rather than spending it.
It's also key to note that the U.S. in 1905 was just coming off of the high of the Gilded Age, a period that saw a few people make vast fortunes—and a lot more people suffer in dreadful conditions in order to produce all those fortunes. Paul and his father aren't factory workers, but they're still part of a system that rewarded a very few at the expense of many.