So if k is the number of tapes, is a multi-tape Turing machine allowed to have k = ∞ tapes.
I'd assume not since this would give an infinite transition function?
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You need a finite number of tapes. The proofs that show the equivalence between multi-tape TM and one-band TM rely on the fact that the number of tapes is bounded.
Notice that it especially the number of heads should be bounded. Sure, you can use a 2d TM, however, there is still only one head in this model. If you would allow an infinite number of heads, then the configuration of a TM would be infinite. This will cause a lot of problems and would give a quite different model of computation.
You can have infinitely many tapes. Suppose that your transition function is of the following form:
Each state $s$ is associated with a head number function $h(s,x)$.
When at state $s$, if the symbol under head $h(s,x)$ is $\alpha$ then change the symbol to $\beta(s,\alpha)$, move the tape $n(s,\alpha)$ steps to the right, update the variable $x$ by the amount $c(s,\alpha)$, and switch to state $t(s,\alpha)$.
The variable $x$ is potentially unbounded, so we access an unbounded number of tapes.
This can be implemented in C and so results in a computable model. One-tape Turing machines can be simulated, so this model is Turing-complete.
You can think of many other models of this sort. For example, and finite number of tapes can be consulted at a time, any finite number of tapes can be modified, any number of heads moved, and so on. We can use several variables rather than just $x$, we can allow more indirect reference, and so on. We can even allow $d$-dimensional tapes. Use your imagination.
One could object (following other answerers and commenters) that the correct formulation of a Turing machine with infinitely many tapes allows referencing all of them at the same time; our model can support that to some extent, for example operations of the following sort:
Write $\alpha$ on all tapes.
Make a certain transition only if all but finitely many symbols under all heads are $\alpha$.
And so on. As long as the description of the machine is finite and the operations reasonable, the machine can be simulated by a program in C or by a Turing machine. What is not allowed is a machine with an infinitary description, but as far as I'm concerned "infinitary description" is an oxymoron.