An example of a Doom would be the number of variations of code that could be used with any computing language in existence that could produce the behavior of the game Doom including the game’s initial creation, but none prior.
Would such an irrational number be of practical use?
In this specific case, "the number of variations of code", "any computing language in existence" and "the behavior of the game Doom including the game’s initial creation" are three separate vague concepts that can only be formalized in complex and arbitrary ways.
If you did, however, you would find yourself on the familiar ground of enumerating "programs" (or something similar, like Turing machines or Gödel numbers) and considering predicates about them. This means dealing exclusively with integers, speaking of irrational numbers doesn't make sense in this context.
Doom is a real number under any circumstance as there is only one value it can be of the set produced by f(d, o) and that value is a real number.
Doom cannot always be represented by a fraction, because it is an indeterminate constant when d is higher than t and o is not high.
I don't think that's the case. `i` is not irrational, it's just perpendicular to 1.
It’s possible that I cannot define an irrational number as an real number member in a sometimes unknown location within a result set of a function, where the size of the set being exactly one results in the possibility of the number being known exactly.
But it has a lot in common with irrational numbers, aside from it looking to meet the qualifications of being irrational.
If you assume total omniscience, then the result of the function would always result in a real, rational number, and that is the number I’m trying to define as irrational, because the answer as defined is always an integer, but depending on the values of the parameters to the function, which number it is would be unknown, and that cannot be defined as a fraction.
At all times, with sufficient knowledge and defined state you could instantaneously determine the Doom value, so it is a real number.
With the example, prior to creation of the game Doom, Doom value would be 0. Around the point of creation, the value would probably be 1, because prior to that point Doom did not exist, and then it would.
After creation, there could be a limit given the available hardware, languages, etc. and code for Doom in those languages. A large number of variations of the code in various languages could still produce Doom. Given sufficient knowledge, there could be a finite value of Doom. But practically if we had to feed an actual set of numbers that could equal Doom into a machine given what we could ascertain, it would be a finite set.
In our example, since there exists a defined value for Doom, it is real.
Since that defined value can only conditionally be known, and otherwise is a set of unknown length, it is conditionally not a constant that is representable by a fraction.
There is a maximum length of all Doom programs in all languages. Therefore, if I understand your definition correctly, Doom is always finite and hence not irrational. That the length of the set is unknown is irrelevant - we don't know how many books there are in the world, but we know that it is a finite number. Not knowing what fraction represents Doom is very different from saying there is no such fraction.
It’s sometimes constant and sometimes not, depending on conditions, but is a set of finite numbers, where that set may be of unknown length or may be a known set of a single number.
It’s a real number as it could be plotted, but is not definable as a fraction, as conditionally it could be of unknown set length.