> On 20/06/2013 5:04 AM, Alan Smaill wrote: >> Nam Nguyen <firstname.lastname@example.org> writes: ... >>> - In this of (c) you can _verify_ that 0, s(0), s(s(0)) are >>> finite individuals, in your constructed set named "U". >>> >>> - In this of (c) you can _NOT verify_ x is a finite individual >>> given x is in your constructed set named "U". >>> >>> Agree? If not, please refute my above by clearly _constructing a set_ >>> named "U", per the possibility (c), _without_ your '...' symbol. >> >> The question is irrelevant to my argument. > > It is relevant: you just don't realize it.
You are the one claiming *impossibility*. I don't have to prove anything.
> As long as you don't > _cast away_ the informal symbol '...' in your constructed U as > I've previously done on stipulations (1) and (2) [see the below quote] > then your argument would go nowhere, and virtually every question > would be relevant. > > <quote> > > (1) (0 e U) and (s(0) e U) and (s(s(0)) e U) > (2) (x e U) => (s(x) e U). > > </quote> > >> We just *suppose* we are in situation (c). > > But "suppose" does _not_ necessarily grant you the right to > prove a particular statement as true or false.
Of course not -- I'm not claiming that.
>> Is it *possible* that the only elements of U are those that can be >> proved to be in U, using the inductive definition? > > You have changed the subject, the question: your question now no > longer references about "finite elements", i.e. finitely encoded > elements. So let's go back to where we were.
Yes, this is an additional question. But why do you refuse to answer?
> Is it possible that Alan's constructed U (constructed with his '...') > would contain only finitely encoded individuals, where '...' > would refer to the Generalized Inductive Definition? > > The answer is Yes, it's possible. > > Can we prove that Alan's constructed U (constructed with his '...') > would contain only finitely encoded individuals, where '...' would > refer to Generalized Inductive Definition? > > The answer is No, we can not prove that.
Fine, let's go with that.
Now what about the question:
Is it *possible* that the only elements of U are those that can be proved to be in U, using the inductive definition?"