How exactly is "true randomness" defined in the realms of cryptography? Especially in relation to stream ciphers, I frequently read about (sometimes theoretical, sometimes practical) attacks that are able to "distinguish a ciphertext from a truly random stream".

What's logical to me is that - just because a ciphertext looks random, it isn't necessarily random. Looking around, the general consensus is that "ciphertext needs to be indistinguishable from a stream of truly random bits". 

This got me thinking: what exactly is "true randomness"? According to (cryptographically related) definitions I found, "true randomness is unpredictable". So far, so good... but that's also marks the exact point where I've lost it. 

"Unpredictable" would practically mean that we have nothing to compare the ciphertext with, because we can not predict what output "true randomness" might produce. Also, there is a (minimal) chance that "true randomness" might output the exact same series of bits as a ciphertext. Meaning: a ciphertext might read 63F1t49X43 and there's a (minimal) chance "true randomness" might produce exactly the same output 63F1t49X43. No one can tell, because "true randomness is unpredictable".

Not being able to predict any true randomness, how can we compare, distinguish, or even claim that a ciphertext is not truly random? Obviously not by comparing it with "true randomness" as that would be impossible due to "true randomness being unpredictable". 

Now, I'm pretty sure cryptography is not philosophy and — as a result — I'm absolutely sure I'm missing something obvious in relation to the cryptographic meaning of "true randomness". I'm guessing the details are to be found in the cryptographic definition of "true randomness", which leads to my question: 

How exactly is "true randomness" defined in the realms of cryptography?

Practically, I guess you could say that I'm not really sure I correctly understand how someone can provide (cryptographically sound) proof to the claim that a series of bits is truly random when "true randomness" is considered to be unpredictable. So, if you think it's not the definition that might be confusing me here, please feel invited to set my head straight by pointing me to whatever I might be interpreting incorrectly.

EDIT

To avoid misunderstandings: when talking about "true randomness", I'm thinking along the lines of "True Random Number Generators" and not "Pseudo Random Number Generators". That's why I'm asking about "true randomness" and not pseudo-randomness.
Randomness is not a property of strings of bits (or characters of any sort).  Rather it is a property of the process that generates those strings.  However, it is convenient to conflate the string with the thing that produced the string, and thus to speak about strings being “random” or “not random”.

The string 00000, for example, is random if it was the outcome of “random process”, such as a coin being tossed five times and landing on tails five times in a row.  Similarly, the string 1,2,3,4,5,6 is random if it was the outcome from rolling a die six times.  Note, the random process does not need to be “fair” to be random, though processes that deviate substantially from the uniform distribution are not as useful for cryptographic purposes.

What is a “random process”?  As I think about it, a random process is either an indeterministic process (if any actually exist), or a deterministic process for which the entropy of what we don't know (that is germane to the outcome) is greater than the entropy of the generated string.  There is a lot we don't know (and can never know, given the Uncertainty Principle) about the state of the flipped coin – e.g. the exact position and momentum of every particle in the coin, the air around it, and the hand that flips it (all of which are germane, in that we presumably would need to know them to accurately predict the outcome of the flip). 

If the string 00000 was the outcome of a number-generating algorithm, where the entropy of what was unknown and germane about the algorithm was less than 5 bits (e.g. we knew the algorithm and all but four bits of information about the seed), then the string would be “not-random”. At best it could be “pseudorandom”, meaning computationally difficult to distinguish from random.