Re: infinity

Randy Poe said:
>
> Tony Orlow wrote:
> > Virgil said:
> > > Not if there is any finite range, but if there either is no range or the
> > > range is not finite then, and only then, is there room for all the Peano
> > > naturals.
> > What a sharp piece of logic that is!
>
> It follows from definitions you insist on.
Yeah, right. You wanna explain that spewage?
>
> > There are values but there is no range of values.
>
> If one insists on defining range in terms of a maximal element
> of a set, then there is no range UNDER THAT DEFINITION for
> sets with no maximal element.
If the set is boundless, then the range is strictly less than LUB-GLB. This has
been stated. Wake up.
>
> This is a simple logical consequence of the definition you
> insist on. If you don't like what follows from YOUR DEFINITION,
> if you find the conclusions defective, then YOUR DEFINITION
> is defective.
>
> It's easy enough to extend the concept of "range" (or, equivalently,
> diameter) to cover sets with no maximum.
Yeah, it's been done, and repeated several times. Pay attention. You don't
rmember discussions of the range of [0,1) as <1, or of the finites as <oo?
Please!
>
> > Or, the range is not finite,
>
> Yes, extending the definition in a natural way will lead to
> that conclusion.
The diameter is not based on differences WITHIN the set. It has the rule,
basically, that if the set is boundless, then diameter is EQUAL TO LUB-GLB,
which is strictly larger than any difference within the set. That is not the
same definition, is it?
>
> > even though every difference between any
> > two values is always finite.
>
> Well, that carries no particular evidentiary nature about
> the range. What does carry information is the ">" relationship.
> Since any finite value is not big enough to serve as range
> for the set, we conclude the range is not a finite value.
The range is DEFINED based on differences WITHIN the set. The salient part of
the range is the "<", where the range is <oo, or less than any infinite number,
which means some finite number. The range comes from the set of finite numbers,
and just because you cannot identify any as the largest, doesn't make it
infinite.
>
> > In one of these cases, suddenly we can fit an
> > infinite number of unit intervals in a finite space.
>
> Nope. Nobody has ever said that but you.
That's exactly what's being said. The range doesn't exist and the point is
moot, or the range is infinite, because we define it based on something other
than one of a set of differences within the set. No element has either an
infinite number of predecessors or of successors, and all elements are a
successor or predecessor of every other element. You have a finite but
unboundedly large sequence of finite naturals.
>
> This is a defective conclusion you have drawn from true statements
> other people have said. I don't remember what false TO-axiom
> you used for it. Probably that old one about "a sequence of
> finites must be finite in length".
No, I am looking at your arguments. You either want to claim a set of values
has NO range, or that the range is infinite, even though there is no infinite
difference in the set. These are your last two flailing attempts to cling to an
infinite set of finite whole numbers, and the cause of my reference to
Berkeley's stupid tree. If a tree falls in the woods and you're there, but
you're deaf, but your dog can hear, but the tree falls on a mountain of
feathers, and you're on the moon so there's no air, does it make a sound? It's
alla bunch of hogwash. Sorry. The range is finite, the sequence is finite, the
set is finite.
>
> But whatever it was, this false conclusion, like all of yours,
> was created by injecting a false TO-axiom into a previously
> valid argument.
No, it was my interpretation of your defenses. If that interpretation is wrong,
explain again how you defend ytour claim that the range is infinite, when range
is defined as the greatest possible difference between values in the set.
>
> - Randy
>
>

--
Smiles,

Tony
.

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