It's not true that all possible espresso-to-milk ratios have Italian-sounding names. The proof relies on 3 assumptions:
1) The possible espresso-to-milk ratios are uncountable (or, to put it more simply, any positive real number could be an espresso-to-milk ratio).
2) Italian-sounding names may be arbitrarily large, but any particular Italian-sounding name must be finite in length.
3) Italian-sounding names are composed from a countable set of characters/phonemes/whatever symbols.
The proof is this: we show that the set of Italian-sounding names is countable, and then assumption 1 necessarily implies that there must exist some
ratios without unique, Italian-sounding names. To complete the first half of the proof, consider the following:
Assumption 3 implies that we may write down a list of our "alphabet" of symbols, with each symbol corresponding to a unique, positive integer. It doesn't matter if this list is finite, as long as it is countable. We denote elements of this alphabet by s
with a subscript.
Assumption 2 implies that any Italian-sounding name may be written as a unique string S
for some integer n
depends on the particular string S
, of course).
denote the k
-th prime number, ie p1
= 2, p2
= 3, p3 = 5
The unique factorization theorem implies that f(S
) = p1s1p2s2p3s3
is an injection from Italian sounding names into the natural numbers, and hence that Italian-sounding names are countable.
There can be no injection from an uncountable set to a countable one, so application of assumption 1 completes the proof. QED.
Now, let's analyze the assumptions:
1) is interesting from a physics perspective. If we simply count the whole, fundamental particles
composing the espresso and milk as being relevant, then this assumption actually fails. But then again, pi and i are not valid ratios in this case. If we look at volume
ratios, then we have to know if space itself is quantized to decide if the assumption is valid. Lacking a quantum theory of gravity, we can only speculate. If we look at mass
ratios, then, by E=mγc², we must include the kinetic and potential energy contributions of the espresso and milk's subatomic constituents. In an open system (ie, one coupled to the rest of the universe), the energy spectrum ought to be continuous, and thus any real mass ratio is possible. On the other hand, a perfectly isolated cup of coffee in a Schrödinger's box is composed of bound, disentangled particles, and will therefore have a discrete energy spectrum, rendering the possible mass ratios countable. If I had even less going on in my life, I could probably get a full academic paper out of this question alone.
2): While Italian (or any foreign language) may seem like an endless stream to the untrained ear, if one listens very closely, one will find that every human, no matter how verbose, eventually stops speaking
. This cessation may only occur at the point of death, of course, but it inevitably comes. Therefore, an infinite name, while badass, probably doesn't resemble any human language, let alone Italian. Of course, these names can be very long, for example: https://en.wikipedia.org/wiki/Titin#Lin ... gnificance
3) The concept of a language composed of an uncountable alphabet is interesting. If spoken, the tone of the signal, for example, would affect meaning out to arbitrary precision. The only way that I can conceive of such a language being understandable is if, past a certain point of accuracy, additional information only conveys nuance
rather than dramatically altering the meaning -- that is, the mapping from phonemes to meaning is a continuous function. For example, a particular sound would convey anger, and then the level of anger would be specified precisely by the exact tone of the sound. The problem is that any finite
signal, when Fourier transformed, will have a spread in frequency. The only way to emit a pure tone is to have an infinitely long sound, and so we again run into the finite lifespan problem. I am not a linguist, so I will leave further discussion on this matter to them.
This is my first post to the forum, and it took a little while to get registered. Therefore, I apologize for the tardiness of this post. If anyone attempted to open a coffee shop with an uncountably infinite menu over the weekend, I am deeply sorry for your lost investment of time and capital.