pH crash course

Herein, in as few words as possible, I will attempt to describe what pH is and why every aqueous solution has a pH.

pH is mathematically defined as the negative log of the hydronium concentration. Or:

pH = –log[H₃O⁺]

So, what is H₃O⁺? Simply put, it is the active ingredient of acid. A more concentrated acid will produce more H₃O⁺.
Water spontaneously reacts with itself to form hydronium (H₃O⁺) and hydroxide (OH^-):

2H₂O <----> H₃O⁺ + OH^-

At 25°C, the concentration of hydronium times the concentration of hydroxide is always 1x10^-14 moles per liter (M). Mathematically:

[H₃O⁺][OH^-]= 1x10^-14

To make a long story short, [H₃O⁺] cannot be zero and will never be zero. So, any and all aqueous solutions must have a pH value. Besides, the log of 0 is really big. I mean really, really big.

A neutral solution has equal amounts of hydronium and hydroxide:

[H₃O⁺] = [OH^-] = 1x10^-7 M (at 25°C)

The pH of a neutral solution is 7 since -log(1x10^-7) = 7.

So, it is inaccurate to say a solution does not have a pH.

In addition:

A solution can have a negative pH. For example a pH of -1.0 would correlate to a hydronium concentration of 10M.

Every common usage of pH is at 25°C. pH is temperature dependent. For example a neutral solution of pure water at 100°C has a pH of 6.14!!! Even though the pH is <7, this is neutral solution!!!!