Re: infinity ...

albstorz@xxxxxx said:
>
> David R Tribble wrote:
> > David R Tribble said:
> > >> So I'm giving you set S, which obviously does not contain any
> > >> infinite numbers. So by your rule, the set is finite, right?
> > >
> >
> > Tony Orlow wrote:
> > > If it doesn't contain any infinite members, it's not infinite. Those terms
> > > differ by more than a constant finite amount, but rather a rapidly growing
> > > amount greater than 1. There is no way you have an infinite number of them
> > > without achieving infinite values within the set.
> >
> > Yes, you and Albrecht keep saying that repeatedly. Please demonstrate
> > why it must be so, because it's not.
>
>
> Your argumentation is not fair, but I don't wonder about that.
> _You_ has to show, that in the case of the whole set there is no
> natural number as big as the whole set.
> You argue: there is no infinite natural number since the peano axioms
> don't allow an infinite natural number.
> That's right. I agree with you.

Alas, Albrecht, it is not true. There is nothing in the Peano axioms that
states any such thing. The inductive proof of the finiteness of the naturals is
flawed in that it applies an increment to each successor, noting that adding 1
does not turn a finite into an infinite, but ignores the intrinsic nature of
inductive proof as a recursively defined infinite concatenation of logical
implications, over which an infinite number of increments does indeed produce
an infinite value as successor. Of course, this cannot be achhieved in any
finite or "countably infinite" (finite but unbounded) number of steps.

> But that's no proof about sets. That's only an aspect of the definition
> which contradicts with the fact, that every set has a number of
> elements.
>
> You misinterpret totally when you say, I think there must be an
> infinite natural number. I don't think so. I only argue that, if there
> are infinite sets, there must be infinite natural numbers (since nat.
> numbers are sets).
> I don't say: there are infinite sets. You say: there are infinite sets
> and there is no infinite number. And I say: If there are infinite sets
> there must be infinite numbers.
So, your position is that there are no infinite sets, since there are no
infinite naturals/ Well, we agree on some things, and yet, in others the
standard nonsense is somewhere between us with a half baked notion of infinity.
Mine is ready for the frosting. :)
>
> My argumentation is very easy:
> Every nat. number represents a set. If you look at the first 100 nat.
> numbers, the 100th nat. number "100" represents the set {1, ... , 100}.
> As this holds for every nat. number, if there are infinite nat. numbers
> there must be a infiniteth nat. number representing this set.
> But the definition of the nat. numbers with complete induction leads to
> the consequence, that there could not be an infinite nat. number.
Um, no, it doesn't. But do go on.
>
> That's the contradiction.
You resolve it by rejecting the infinity of the set. I resolve it by rejecting
the finitude of the elements.
>
> So either the definition of nat. numbers must be changed or there is no
> infinite set of natural numbers.
The Peano axioms can be adjusted to generate the infinites while generating the
finites. Tat's easy.

> Or infinity must be interpreted in a completely other way. Not as a
> size like you do. Infinity is just an unability to count it with
> numbers because it runs out of all what we can know.
>
> All this is shown very expressive in my sketches at the start of this
> thread.
I thought so. Others don't always see what I see, though.
>
> Why do you misinterpret all the time? Maybe my ability to express my
> thoughts in english is too bad.
> But why do you misinterpret Tony also? I think he is native english
> speaker and you should be able to understand him.
LOL! :D Albrecht, while am indeed a native English speaker from New York with a
pretty decent command of the language, I also sometimes feel like I am from
Planet Xorxon, and find it difficult to communicate certain ideas to many of my
apelike family members. I don't think this is a language issue. Your diagram is
pretty language-independent. It's a matter of thinking visually, rather than
axiomatically and linguisitically.
>
> In this state there is no real problem with all this. aleph_0 is just
> onother symbol for infinity.
> The problems occure in that moment if someone declares, that aleph_0 is
> a size, which is greater than any nat. number.
I agree wholeheartedly.

> But there is no "greater" or "less than" or something like this. There
> is just something other, something out of the things we could measure,
> wigh or count.
> The possibility of bijection don't say anything about the size of
> infinity, since infinity is something sizeless, endless, countless.
> That's all.
But, Albrecht, wouldn't you say that the size of the set of naturals is twice
the size of the set of even naturals, since the latter comprise 1/2 of the
former? And wouldn't you consider [0,2) as containing twice as many points as
[0,1)? If you don't believe in infinity at all, how many reals ARE in [0,1)?
>
> Regards
> AS
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

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