Given dedicated application of some rather
specious reasoning, one could come to this conclusion.
Now, there is no such thing as a
perpetual motion machine, and the Universe is no exception. Eventually,
the entropy in the Universe will expand to the point
that no meaningful information can be transmitted. That
is, no signal, only
noise. Of course, that's a long way
away.
But why stop there? This reasoning leads us to the conclusion that science is not only
countable, it is
finite: Countable infinities cannot be
represented any more than
uncountable ones can!
We don't have enough time to recite all of the positive
integers, or represent even one number that requires an
infinite decimal (or pick the base of your choice), much less represent all of them!
Ok, so
the number of scientific papers that will be published is finite. Better submit now while you can!
But wait: let's imagine an ideal (but practical) calculating machine. The main purpose of the machine is to continually print out positive integers, each one greater than the last. Of course, all sorts of subsystems are attatched to the machine, dedicated to keeping the machine running: To manufacture spare parts for itself after humanity dies out, and to find energy when the sun goes out. Eventually after all of the stars are extinguished, the machine will find ways of exploiting
black holes until they, too, all evaporate or are impossibly far away.
The machine will doggedly recite number after number, continuing until it, too, must expire because it can't
repair itself fast enough to replace decayed protons.
Eventually the machine will spit out its highest value.
Call it
N.
Does this mean that N+1 does not exist?
In the end, we have to realize that our reasoning is flawed:
- Of course, some people may confuse the terms 'countable' and 'infinite'. It's easy to do,
but so is learning basic mathematical terminology.
We can choose to accept the axiom of infinity, or not. Once we accept the existence of even one infinity, however, all the rest inevitably follow.
- More importantly, we are confusing science, i. e. statements about a field derived from a particular meticulous method, with the field itself. (In other words, we are confusing the doctor with the patient.)
In regard to mathematics, the number of mathematical statements we can practically make is unfortunately finite. If you concatenate all of the mathematical literature likely to be produced from now until the end of time, and then represent it as a number, you will get a very, very large, but nonetheless finite, number. Call it L. Does L+1 exist?. I can't think of a mathematician willing to give up the notion of the natural numbers being closed for the operation of addition.