Unbounded number of tapes of Turing Machine

Turing Machine with multiple tapes can be encoded such that its computational power is equivalent to Turing Machine with single tape. My question is if we have unbounded number of tapes, just like the length of a tape, would it change the computational power of a Turing Machine.

My intuition: No, since we could always encode the symbols on the tapes "diagnonally" just like the method we use to prove rational numbers are countable. For the sake of convenience we could even assume that each tape is a one-way tape. More precisely if we write this as a matrix with unbounded dimension, $(a_{i,j}), i,j\geq 0$, and the encoded symbols on the new tape are $(a_{k,0},a_{k-1,1},\dots,a_{1,k-1} ,a_{0,k})$ for each $k\in N$.

What do you think of the argument?

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 I think when you are simulating one machine by another, you have to write code, taking advantage of known macros. The idea is good, there is a good deal of detail missing. – André Nicolas Oct 4 '12 at 22:17 @AndréNicolas: The problem here is when simulating the machine with finite tapes,say $k$ for example, we could encode $(\alpha_1, \dots, \alpha_k)$ to a new symbol in the one-tape machine and there are only finitely many new encoded symbols. However in the argument above, there might be infinitely many symbols. – Jing Zhang Oct 4 '12 at 23:07