On 2/2/2013 2:19 PM, Virgil wrote: >> Can a potentially infinite list >> of potentially infinite 0/1 >> sequences have the property that > >> if s is a potentially infinite 0/1 >> sequence, then s is a line of L > > One cannot ever actually have a potentially infinite list, any more than > one can have an actually infinite list, one can actually only have a > finite list with some indication of how that list might be indefinitely > extended. > > Which is as much an abbreviation of a truly infinite list as it is an > abbreviation of a potentially infinite list. >
I would guess this is why the Russian school of constructive mathematics focused heavily on algorithms as exemplifying the meaning of potentially infinite. Numbers are algorithms having secondary algorithms guaranteeing convergence of the first.
No abbreviations. Wish I knew more about their work.
On the list for this year's reading.
Tried Sanin's book on constructive reals...
Needed Sanin's paper on constructive logic...
Needed Markov's book on theory of algorithms...
Got them all now. It looks like I will have the necessary symbol definitions. I was afraid I might need something by Kolmogorov.
