My computer is very much a Turing machine since each time it runs out of disk space I can buy more disks. The fact you think a Turing machine has infinite memory rather than finite but arbitrarily large memory tells me all I need to know about the sorry state of your theoretical education.
A Turing machine is a mathematical concept. It has, by definition, infinite memory. A computer is just a real machine that can do similar things as a Linear Bounded Automaton, a Turing machine's little brother who'll never do quite the same things.
There is a physical limit to the number of disks you can add to your CPU. There is a physical limit to the memory you can address in your CPU (48-bits). It is not arbitrarily large.
Also wrong for Turing Machines, it really is infinite. That's a big difference to arbitrarily large. The halting problem is undecidable for TM's but not for arbitrarily large (you'll need precise definitions though).
The machine with 7918-states, Z, stops (well, Z cannot be proven to run infinitly long) iff. ZFC is consistent. For this it needs a finite amount of space but we cannot calculate how much. If we could calculate an upper bound we've proven ZFC is consistent.
