Just a piece of the story:
The seminal 19651963 paper of Hartmanis and Stearns, "On the Computational Complexity of Algorithms" introduced the definitions of quantified time and space complexity on the multitape Turing machine model and showed that given more time/space a TM can compute more things.
... The computational complexity of a sequence is to be measured by how fast a multitape Turing machine can print out the terms of the sequence. ...
Where "sequence" is a generic sequence. Then, when defining a T-computable sequence, they restrict the attention to binary sequences:
... For the sake of simplicity, we shall talk about binary sequences, the generalization being obvious. We use the notation $\alpha = a_1 a_2 ...$
...
The class of T-computable binary sequences shall be denoted bu $S_T$, and we shall refer to $T(n)$ as a time-function. $S_T$ will be called a complexity class.
And then in Corollary 2.8:
... Thus, when considering time-functions greater or equal to $n$, the slightest increase in operation speed wipes out the distinction between binary and nonbinary output machines.
With a backward link to Hilbert and Church / Turing's work on the halting problem:
Theorem 5. Given a time-function $T$, there is no decision procedure to decide whether a sequence $\alpha$ is in $S_T$