"Oh really?" says Bengt, in a somewhat disdainful tone.
"Really."
"So, if I give you a countable set, you can tell me a real number which is not in my set?"
"Yes... I suppose."
"Bet you a beer you can't."
"You're on."
"Okay, listen to this: There is a set S, which contains all the symbols valid for expressing a number. Digits, decimal points, fraction signs, and so on. You can throw in some arithmetic symbols too if you like. The point is, it's a finite number of symbols."
"Uh-huh."
"With those, you can put together expressions of finite length. There is clearly a countable set of such expressions. Now, we define the set U, which is the set of all numbers that can be defined using such expressions. We can call them the expressible numbers."
"Or we can call them the espresso numbers. Because I like espresso."
"No. First, that doesn't sound mathematical. Second, I don't like espresso."
"Oh."
"So, there you go. Find me a real number that is not expressible."
Alban thinks for a while.
"Umm... pi?"
"Well... I could say that the pi symbol is in S, but that would be no fun. Or I could bring up Lebniz' and Gregory's formula. But that's not necessary. Because if pi is not in S, you have to use some other way to express the number you mean. Can you explain what pi is?"
"Hmm. I guess that should be possible somehow... but we know it's irrational, so it has an infinite non-repeating sequence of digits."
"But so does the square root of two. And that's obviously expressible."
"Damn you and your strange problems."
"Yeah yeah. Sore loser."
- Is there any such number? Since it's well known that the set of real numbers is not countable, there must be some flaw in Bengt's argument - what is it?
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