Well, for starters, it's physically impossible to have an infinite number of speakers playing an infinite number of waveforms simultaneously, so this silly idea does require mathematical abstraction to be meaningful. That shouldn't be too surprising because there are many places in the physical world where we use infinite series to calculate simple finite physical quantities, e.g. when we integrate to find the area of a region.

My point is that if you really sum every possible waveform, the resulting value may or may not converge depending on the order in which you sum them; in fact, it's a well-known property of such conditionally-convergent series that you can actually get any limiting value you want based on how your order them [0]! (let's ignore the fact that the fourier coefficients can take on a continuous set of values). For example, even if you were only allowed to play a single frequency sound wave sin(x) at volumes that are the inverse of some integer value multiplied by a max volume of 1 (in arbitrary units), you may or may not have them cancel depending on how you group the terms in the sum:

  sum = 1*sin(x) + -1*sin(x) + (1/2)*sin(x) + -(1/2)*sin(x) ...

were the ith term in the sequence (starting at i=1) is

  a_i = (2/n-1)*sin(x) for odd x
  a_i = -(2/n)*sin(x) for even x

This is a conditionally-converging series that will hit all positive and negative harmonic coefficients 1/n and -1/n: the even terms cancel each preceding odd term, and the Nth partial sums therefore alternate between 0 and 2 * sin(x)/(N+1), which itself tends towards zero. But you can group these terms in a different order and get a different limit for the sum; in fact, you can group them to get whatever final value you want!

Now, if you extend this thinking to every frequency of sinusoidal wave, you can start summing every pure tone in arbitrary order to get an arbitrary coefficient for each frequency. By picking your limit for each frequency correctly, you can sum your sine waves in a fourier series [1] to get any song you could ever want! And this is while limiting ourselves to discrete frequencies and alternating harmonic coefficients (since it allows us to take a discrete infinite sum).

So the unexplained punchline to my previous comment is that the problem is ill-defined, or rather, that you can view any song as just a specific ordering of an infinite series of other sounds. (You don't have to use sine waves as your basis, by the way; you can use a bunch of different waveforms that look more like "noise" as long as their combination spans the same infinite-dimensional linear space as pure sine waves; you just end up with different coefficients. For example, in quantum mechanics, you can get a sine wave (momentum eigenstate) by summing energy eigenstates (non-sine waves with a specific form) with the correct coefficients.)

[0] https://en.wikipedia.org/wiki/Riemann_series_theorem#Alterna...

[1] http://mathworld.wolfram.com/FourierSeries.html

Isn't the inverse of a sound the same sound?

You are incorrectly assuming sound is additive, but it isn't.

Sound is non-linear as sound gets louder - sound wave volume is physically limited because the low of the sound wave can't be lower than vacuum.

Another non-linearity is air cannot transmit frequencies higher than some limit.

Another is that sound has a noise floor depending on the temperature of the gas (noise like rain on a roof?).

There are surely other gross non-linearities.

Those non-linearities mean you can't add or subtract some sounds, and you can't assume commutativity.

This would make sense if it were physically possible to play every noise at once. (Where would you place the infinitely many emitters? If they have any displacement at all, then there will be points that would not experience destructive interference at all frequencies.)

There is no empirical result because you can't actually play all sounds at the same time.