The concept of binary description by Ray Solomonoff In this article, http://world.std.com/~rjs/rayfeb60.pdf in page 7 of the PDF, Solomonoff gives an a-priori probability for a string which is $pr(S)=2^{-|S|}$.

My question - why are shorter strings considered more probable? Is there some kind of hidden assumption that justifies this?

e.g. that the machine which produces these strings have a constant probability to stop at a any given point (an inherent preference to shorter strings).
Later in page 10 of the pdf he explains that in fact the probabilities are almost 1/2, as there is a very small probability to stop the at each step.

This stems from the following assumption: There is some mechanism that produces these strings, and it does so sequentially, therefore going through shorter strings on its way to longer ones and having a certain probability to stop at each step.

Another explanation that was more satisfying:

These binary strings are in fact prefixes of infinite random sequences. The probability of a prefix to be 1 or 0 is of course 1/2 etc'. The machine that reads these infinite binary strings is a prefix turing machine, which means that if it produces a certain sequence, it does so by reading a prefix of the random string.