Notably, this is why we focus on the arbitrarily large memory and time case, where we can assume that the machine has arbitrarily large memory and time to work with.
The key question here is whether a finite physical computer can always be extended with more memory and time without requiring us to recode the machine into a different program/computer, and most modern computers can do this (modulo physical issues of how you integrate more memory and time).
In essence, the key property of modern computers is that the code/systems descriptor doesn’t change if we add more memory and time, and this is the thing that leads to Turing-completeness if we allow unbounded memory and time.
Notably, this is why we focus on the arbitrarily large memory and time case, where we can assume that the machine has arbitrarily large memory and time to work with.
The key question here is whether a finite physical computer can always be extended with more memory and time without requiring us to recode the machine into a different program/computer, and most modern computers can do this (modulo physical issues of how you integrate more memory and time).
In essence, the key property of modern computers is that the code/systems descriptor doesn’t change if we add more memory and time, and this is the thing that leads to Turing-completeness if we allow unbounded memory and time.