Re: abundance of irrationals!)

In article <MPG.1cf2ae05a97c583e989c63@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Randy Poe said:
> >
> > aeo6 Tony Orlow wrote:
> > > I already said you can have your set of finite counting numbers. Just
> > don't
> > > claim the set has an infinite number of members unless you can come
> > up with
> > > some infinite set of symbols to represent them. S^L is finite for
> > finite S and
> > > L, period.
> >
> > What do you think the value of L is for the base-10
> > representation of the naturals? Do you think there
> > is a value L such that there are no naturals with
> > L+1 digits?
> >
> > - Randy
> >
> >
> I think that if there are an infinite number of naturals, then there must be
> some with infinitely many significant digits.

A finite sequence is one with a last term,
An infinite (non-finite) sequence is one that has no last term.
All finite sequences are bounded.
Some infinite sequences are bounded, but even the unbounded ones are
sequences of all finite elements.

According to TO's argument, at some point one adds 1 to a natural with
finitely many digits and gets something with infinitely many digits, but
TO does not explain either how or why this happens.

Is there a largest finite natural to which adding 1 still gives a finite
natural? I don't think so, but apparently TO does.
.

Relevant Pages

  • Re: Cantor and the binary tree
    ... and diagonal traversal does not cover all strings. ... Do the math, and stop playing bad logic games, and declaring nonexistent differences between the finite and infinite. ... Any such list is exponentially longer than it is wide in digits. ... If they are a larger set than the naturals, then that is a valid conclusion, perhaps, but to say they can't be enumerated like the naturals, is just wrong. ...
    (sci.math)
  • "Algorithmic Randomness, Quantum Physics, and Incompleteness"
    ... all finite sequences are to be found infinitely often and ... different members of the infinite set of random numbers. ... Just using random numbers with initial digits 1 thru 9, ... it generated by a shorter input algorithm than the length ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... As we enumerate the naturals starting at one, ... infinite number of times to achieve an infinite set, ... Your "procedure" for incrementing a set is an unending one: ... We define the naturals by strings of digits. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >>> representation of the naturals? ... >> some with infinitely many significant digits. ... > An infinite sequence is one that has no last term. ... > All finite sequences are bounded. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... and diagonal traversal does not cover all strings. ... Any such list is exponentially longer than it is wide in digits. ... of the naturals". ... systems, and showing you that those laws apply for infinite sets as well as finite sets, since the laws governing finiteness apply to the formulas that govern strings. ...
    (sci.math)
Quantcast