> > > Any bounded set of naturals is finite, but that does not prove that
> > > every set of naturals is bounded, so it does not prove that every set
> > > of naturals is finite.
> >
> > Any natural defines a bounded set. There are no naturals which have
> > infinite values. Hence, all sets defined by naturals are finite.
>
> WM is saying that any set of naturals that is bounded above by some
> natural is bounded above by some natural. OK!
>
> But whah WM does not say, and cannot prove since it is false, is that
> every set of naturals is bounded above by a natural.
Of course every set of naturals is bounded by a natural. What else
should be in there? A television set? We cannot determine the magnitude
of it, but we know that it is finite and is maximum of its initial
sequence.
Every countable set like N is potentially infinite but not actually
infinite because all of its elements are finite and so is any of its
initial segments.
Re: Cantor and the binary tree ... > Virgil wrote: Since the rest of WM's daydream is based on the ... > It is based on the existence of infinitely many rationals and on the ... > As it has not an infinite member,... > naturals, then there are only elements, which count their initial sets. ... (sci.math)
Re: Cantor and the binary tree ... >>> and still be in any path that starts at the root node....Virgil offers declarations that he ... >>> finite set and that any set that contains all the naturals must ... >> an infinite range of values. ... (sci.math)
Re: Galileos Paradox and the Project of the Reals ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ... (sci.math)
Re: Calculus XOR Probability ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... (sci.math)
Re: Calculus XOR Probability ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering"... (sci.math)
