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>>jeremy+(OP)
I have been thinking about this problem for a year or so now. I was introduced to the concept watching a Numberphille video and several hours later read this article.

I've been taking a somewhat metaphysical approach to thinking about it. Instead of thinking about a Turing Machine as just a head on an endless tape, can we not consider the Universe to represent the upper limits for our calculations? Just as Turing suggested that we can take the two dimensional representation of a mathematical equation and represent it in one dimension, can we represent the entire state of the Universe in a similar way on a hypothetical Turing Machine? The halting problem is then a question of whether or not the Universe has a halting state.

To extend from that, any machine that can be conceived must fit within the bounds of the Universe. Of course BB(Universe) is incalculable, but I think that it defines an upper limit. It is pointless to consider a Turing Machine that features an endless tape if an endless tape is an impossibility.

In not sure if this adds to the discussion, but I haven't had anyone else I could discuss this with until now.

>>chinpo+QI
By postulating an endless tape you're just stalling petty objections based on how long a tape would be required for any given task. The only point that's relevent, when considering Turing machines in the abstract, is that the amount of tape required for any given task be finite.