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No, it is just saying that the process is unbounded, not infinite. This subtle difference can be very confusing. Unbounded means: for every n, the computation
by ot 12y ago
No, it is just saying that the process is unbounded, not infinite. This subtle difference can be very confusing.
Unbounded means: for every n, the computation can generate more than n values.
Infinite means: the computation generates infinite values.
Talking about infinity is usually very tricky because it is easy to run into paradoxes without a good knowledge of set theory. So many mathematicians are more comfortable using finitary arguments like the unboundedness argument above.
One prominent example is calculus: despite logicians have laid down solid foundations for non-standard calculus using actual infinitesimals/infinites, most analysts prefer to use the Weierstrass epsilon-delta-like arguments, for example:
lim x->0 f(x) = 0 is defined as
for every epsilon there exist a delta such that |x| < delta implies |f(x)| < epsilon
while with infinitesimals you can just write
f(csi) is infinitesimal for every infinitesimal csi
Notice how the epsilon-delta argument is similar to the unboundedness argument for the integer generator.
- ArkyBeagle 12y agoBut the infinitesimals invite treating them as "quantities" in an algebraic "game". That's a drill-lesson in abstraction when they're teaching you ( usually pre)Calculus, before you learn the dF(x)/dt transforms in derivative and integral tables. They also turn up in engineering contexts, which is pretty cool when it happens - it's not all just slide rule and look-up - you get to play with first principles some times when you're lucky... Wasn't me, but I got to watch. I still suspect the generators are infinite sets in the Cantor sense, just stated more politely. We're more of a habit to think in terms of ... series in that way. I recall being exhilarated by the sheer ... hubris of Cantor's approach. So I am probably biased. And as strange as it may sound, if I use infinite sets to derive the Diagonalization, then use that to assist in Godel's Theorem, it feels less like there's a tangled hierarchy when reasoning about Turing machines. That could easily be an elegance that I'm am stubborn about missing.