Re: An uncountable countable set

*** T. Winter wrote:
In article <4506dbd2@xxxxxxxxxxxxxxxxxxx> Tony Orlow <tony@xxxxxxxxxxxxx> writes:
...
> Yes, it should, and Wolfgang's and my position is that N is unbounded > but finite, given that it only contains finite values, and only one per > unit of value range.

By the definition of "finite", "unbounded" and "finite" are in contradiction
with each other. So a number can not be both.

Given the standard definition of an infinite set, and its "equivalence" with finite quantities, that's true.


> Since the value range is finite, given that no two > naturals are infinitely different, and given that only a finite number > of elements can be fit sparsely in a finite range, the set is finite, > even though it's unbounded and infinite by the Dedekind definition.

Not the Dedekind definition, but the standard definition. There are
models with sets that are infinite, but Dedekind finite. (But in that
case you do not have AC, because AC implies the two kinds are the same.

Do you mean finite but Dedekind infinite? That's what I'm proposing. It sounds like you're talking about some opposite approach.


> At which bit position can the string achieve an infinite value? None > that exists in that string.

Indeed. And that is why you need to do something to give actual meaning
to such strings. In the 2-adics such is done, and in that case,
...111 is sum{n = 0 -> oo} 2^n = lim{n -> oo} (1 - 2^n)/(1 - 2) = -1
because in 2-adic metric lim{n -> oo} 2^n = 0. (The metric is defined
as d(a, b) = 1/2^n if the n-th digits of a and b are the highest order
digits where they differ.)

That would appear to support my position that the number line CAN be viewed as a circle, that there is a generalization from 2's complement to n-base systems where numbers form a circle. Having looked up p-adic arithmetic, I gleaned that using prime numbers solves some problems. But, the T-riffics, solve some additional problems. :)


> > This should be a rather big clue that something's wrong with your
> > concept, and that it does not, in fact, "work as a natural".

It does indeed not "work as a natural", but it works as a 2-adic.

If all p-adics are provably finite, then the distinction becomes transparent.


> > Which presumably requires some extended Peano axioms in order
> > to exist?
> > No, it just requires a loosening of the requirement that we only look at > the minimal set satisfying those axioms.

But it is useful to look at the minimal set satisfying those axioms. There
are too many sets that satisfy those axioms so that it is difficult to
state anything with sense when the sets are not minimal (unless they are
special).

Howmany other sets can you formulate, which satisfy those axioms?

Take the Gaussian integers. They satisfy the axioms when we
state the successor function as S(k + i.l) = k + 1 + i.l Take the normal
integers, they also do satisfy the axioms with the standard successor
function (+1), take the rational numbers, they also do satisfy the axioms
when we define the successor of p/q as p/q + 2.

Yes, it's very useful to classify number systems according to the operations which they allow and the relations between those operations.


> I don't see that it contradicts > them as they stand.

It does not contradict them, but it is the minimality that makes things
provable that otherwise would not be provable.

That depends on the acceptance of the limit ordinals as a model for the naturals, really, when it comes right down to it. To say the least ordinal for all finites is infinite is a mistake, in my opinion.


> Extensions of the naturals such as this also fit > into that model.

Yes, like a host of other extensions.

Well, then, if it doesn't contradict basic notions of the natural number, then it should be allowed as a model, and objections are non-logical, eh?

Tony
.

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