Afraid the complex numbers are the same order of infinity as the reals. And even if you could successfully argue that the set of restaurants is countably infinite because, like the primes, you cannot exhaust them, there is still no way to get past \aleph_0.
Last time I looked, Manhattan was not all of New York City or state. Restaurants open and close all the time, so one could not eat at them all anyway. However, it is not clear to me that one could not do so over many decades, depending on the specific rules, which might include lunch and dinner, visiting only one of a chain, etc.
As an aside, some say the best way to choose a restaurant, or anything, is to look at the first 1/e (36%) of them, than choose the next best one better than those (or possibly equal). I don’t know if this is mathematically true but think it might be. Perhaps someone smart could enlighten me. Could still take some time.
This thread reminds me of an old SD column from many years ago.
Something about if you had every Chinese person stand in line and walk by you, the line would never end bc babies are being born at too fast a rate for you to ever get to the end of it.
Or something like that. It’s been so long since I read the article.
If that were true, that means people would have to stop marching.
As long as they can hurry up and catch the back of the slowly moving line after they finish … pausing … the system would still work.
I’d first heard a version of that as the Chinese walking four-abreast into the sea. So we can adjust the attrition rate by specifying an abreast factor. So, what is that factor? Of course, if we assume exponential growth as more babies eventually begat more babies, the abreast factor itself will be ever increasing.
I’d think this number here would tell you that your options will never run out. At between 700-1000 new restaurants in three months, that alone is enough for one new restaurant a day for the year, even if half of them close in less than a year. Even if there’s a bad year in which they all close in less than a month, the existing base of restaurants will give enough of a buffer to smooth out such fluctuations.
Add in restaurants opened at other times of the year, and there’s not even a question.
I had a hunch that the answer for “all of NYC” would be a resounding ‘yes’ which is why I restricted it to Manhattan. I was inspired to this question by my visit last week in which despite my best efforts over 4 days I wasn’t even able to sample all of the cuisines I wanted to try, let alone all the restaurants. “Even if you lived here you could eat at a different restaurant every day and never get through all of them” I said, then I wondered if it were literally true.
Just to set my own parameters a bit more tightly, I would definitely not want to eat at each location of a chain, and I would want to eat at food carts, but again only one in each cabal.
It’s surprisingly difficult to find data for Manhattan. Even the sites that claim to have it are incorrectly using the “all of NYC” numbers.
I can’t find a cite. My guess would be that around half of the restaurants in NYC (including food trucks and street vendors) are in Manhattan. If that were true, you’d still have an infinite (aleph null) number, right?
Well, the population of all of NYC is about 8.5 million, and Manhattan is about 1.6 million, so less than 25% of the total. But you’d expect the more expensive neighborhood to attract more new restaurants, so maybe one third to one half? Particularly when you consider how many people commute into Manhattan every day, which would drive up the lunch and early dinner crowds.
So, yeah, probably 250-500 new restaurants in the three month period we have data from (above). Probably at least double that for the rest of the year, so 500-1000 per year seems reasonable. That again puts you pretty comfortably in the “Probably never run out” region.
That’s going to depend in part on how you define “restaurant”. Does a takeout-only place count? What about a food truck? Food court in a mall? Does “restaurant” mean a particular address , so that the address that has been a restaurant of one type or another for 30 years counts as one restaurant? Does it mean a particular type of restaurant at a particular address, so that the location that has been a Chinese restaurant for 30 years counts as a single restaurant no matter how many names and owners it had, or is each change of name/ownership a different restaurant? It’s going to make a difference, because if each change of name/ownership is a separate restaurant, then you will never be able to eat at all of them, not even just the ones in Manhattan. Even if there are only 2000 establishments that count as “restaurants” in Manhattan* , by the time you got to the end of the list after about 2 years of eating 3 meals a day in restaurants, some of the locations will have closed and reopened as a new restaurant so they have to be added to the list.
Now, if the place that has been ten different restaurants in the last 30 years counts as a single restaurant , you might be able to finish the list.
* And there are definitely more - these are the number of outdoor seating permits approved in 2020 to the Manhattan neighborhoods that made the top ten
Batt. Park City/Greenwich Vlg./Soho 1,089
Chelsea/Clinton/Midtown Bus. Dist. 1,047
Chinatown/Lower East Side 887
Murray Hill/Gramercy/Stuy. Town 634
https://www.osc.state.ny.us/files/reports/osdc/pdf/nyc-restaurant-industry-final.pdf
Let’s go to that new Korean place, Schrödinger’s.
Don’t go there at 9pm. That’s when they half close.
I suspect that the continuum hypothesis is somewhere hidden in this formulation as a variant of La Grande Bouffe.
FTR: Many years ago I had the same idea as the OP. Only not for restaurants, but for bars, and not in Manhattan, but in Berlin. So far: yes, they seem infinite.
And, you’ll keep trying to prove this, no matter how many nights you need to go out for a beer.
Not all heroes wear capes. If you can prove the Infinite Berlin Bar hypothesis, maybe you’ll win a Fields medal! You could use the money to buy a new liver.