On 3/22/2013 6:13 PM, fom wrote: > On 3/22/2013 5:11 PM, William Hughes wrote: >> On Mar 22, 10:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote: >> >> <snip> >> >>> All sets are finite, but not fixed. >> >> I did not claim that all sets in >> Potential infinity are fixed. >> They are finite and thus not >> different from finite sets. >> >> WH: Infinite sets are different from finite sets >> WH: but they do not contain anything >> WH: "beyond any finite set". >> >> WM: Of course. >> >>> There is no upper threshold, >>> contrary to every finite set. >> >> More Wolkenmuekenheim logic >> >> Every finite set has an upper threshold. >> >> A potentially infinite set >> does not have an upper threshold >> >> A potentially infinite set is finite. >> > > And, when properly developed in a constructive > framework, the objects to which that statement > might apply are clearly understood as such. >
Whoops. Misread WH's intent.
Nevertheless, although true for Markov's system, the statements
1) When constructive objects are defined in terms of potential realizability, then the potentially realizable objects are finitely realized.
2) Potential realizability is understood ideally, so that the generation of arbitrary constructive objects is not limited.
3) And, the generation of any potentially realizable constructive object involves finitely many discrete operations.