Re: Cantor and the binary tree

David Kastrup said:
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
>
> > Virgil said:
> >
> >> Where do those extra unreal, unrational infinite naturals come
> >> from?
> >
> > From the number line and the definition of natural numbers, which
> > does not specify that they be finite.
>
> Last time I looked, the induction axiom quite definitely stated that
> the operation of succession, seeded at 0, and implying for each number
> only its successor (and a finite number trivially has a finite
> successor), exhausts the naturals. Since no induction step can leave
> the finite realm starting from a finite number, this means that the
> _definition_ of natural numbers _clearly_ causes every natural number
> to be finite.
That is incorrect. There is an inductive proof to that effect, but it violates
the mathematics of infinite series, when it tries to maintain a property of
finiteness over an infinite range of numbers that are constantly increasing.
One has to be careful that they are not using a method of proof which violates
the idea they are trying to prove. While the successor to a finite number is
always finite, that's because the successor represents the addition of a finite
value to that finite value. The sum of two finite values is always finite, but
the sum of a finite value and an infinite one is always infinite.
>
> > Virgil sure is dumb, that he needs this repeated over and over.
>
> Not being able to word things in a way that even the most stubborn
> idiot can't at times avoid admitting them is not a sign of dumbness.
> It is an inadequacy, but so is every attempt of the near impossible.
Well, then, there is an immense amount of inadequacy here.
>
>

--
Smiles,

Tony
.

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