So with several examples I’ve managed to prove what nobody really questioned, that it’s possible to imagine a complicated curve like the route of the New York Thruway and assign to all the points on it, or at least to the center line of the road, a unique number that no other point on the road has. And, more, it’s possible to assign these unique numbers in many different ways, from any lower bound we like to any upper bound we like. It’s a nice system, particularly if we’re short on numbers to tell us when we approach Loudonville.

But I’m feeling ambitious right now and want to see how ridiculously huge, positive or negative, a number I can assign to some point on the road. Since we’d measured distances from a reference point by miles before and got a range of about 500, or by millimeters and got a range of about 800,000,000, obviously we could get to any number, however big or small, just by measuring distance using the appropriate unit: lay megaparsecs or angstroms down on the Thruway, or even use some awkward or contrived units. I want to shoot for infinitely big numbers. I’ll start by dividing the road in two.

After all, there are two halves to the Thruway, a northern and a southern end, both arranged like upside-down u’s across the state. Instead of thinking of the center line of the whole Thruway, then, think of the center lines of the northern road and of the southern. They’re both about the same 496-mile length, but, it’d be remarkable if they were *exactly* the same length. Let’s suppose the northern belt is 497 miles, and the southern 495. Pretty naturally the northern belt we can give numbers from 0 to 497, based on how far they are from the south-eastern end of the road; similarly, the southern belt gets numbers from 0 to 495, from the same reference point.

Continue reading “Searching For Infinity On The New York Thruway”

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