Re: infinity

Tony Orlow says...

>> A sequence of length aleph_0 has a domain equal to the set of all
>> ordinals less than aleph_0.
>So, you are placing aleph_0 directly after the largest finite?

How many times do you have to be told: There *is* no largest
finite! aleph_0 is placed after *all* finite ordinals.

Here's a way to visualize countable ordinals that might
make sense to you.

Take the real number line, and label the point 0 with the
ordinal 0. Label the point 1/2 with the ordinal 1. Label
the point 2/3 with the ordinal 2. In general, label the
point n/(n+1) with the ordinal n. Finally, label the
point 1 with the ordinal aleph_0.

Note: the finite ordinals are the labels for reals of
the form n/(n+1). There is no largest real of that form,
but the real 1 is larger than *any* of them.

>That's what it seems like. Each of those sets you mention
>has a largest element 1 less than its size.

Right. That pattern holds for every finite set. That's
because finite sets of naturals have a largest element.
Infinite sets of naturals *don't* have a largest element.

That's the breakdown: if a set of naturals has a largest element,
then its cardinality is less than or equal to 1 more than its largest
element. If it *doesn't* have a largest element, then it isn't
a finite set.

>> Yes, it is. The size of a set S is the smallest ordinal alpha
>> such that there is a bijection between S and the set of ordinals
>> less than alpha.
>
>Prove it.

That's the *definition* of size.

>Make it make sense. Why does it work in such
>a warped way?

Look, it's a *definition*. As I said, the goal of
mathematics is to rigorously work out the consequences
of definitions. The consequences of the Cantorian
definition of "size" are there whether you want to
call it "size" or not. You can call it something
different, but Cantorian size is a perfectly well-defined
mathematical property.

Go through the definitions. Where is there any
contradiction, or ambiguity, or undefinedness?
There isn't any. You can bury your head in the
sand and say you don't want to consider the
consequences of basic mathematical definitions,
but that doesn't change those consequences. Infinite
sets *do* work that way, whether you like it or not.

At best, you can make up different terminology that
you like better. But that isn't mathematics. It doesn't
matter what you call something.

>> Yes. The position of a limit ordinal is after all the ordinals
>> that are smaller than it.
>And so, you decalre some smallest infinity, aleph_0, as the number
>that is LARGER than all the finites, and say this is the size of
>the set, even though the size and largest element are always
>inductively equal to each other?

A proof by (natural) induction establishes facts
about the *naturals*. It doesn't imply anything
about infinite ordinals.

>What a great system. It stinks.

You have trouble with it because you are basically
an idiot. You don't have any kind of mathematical
discipline, and you think that your *opinion* has
some kind of importance. It doesn't. Mathematics
is not about people's opinions, or about what you
like to believe is true. It is about rigorously
working out the consequences of precisely stated
assumptions and definitions.

If you want to do mathematics, you can make up
your own definitions, your own assumptions, and
you can work out the consequences of those. But
you can't negate what has been worked out for
*other* assumptions and definitions. If you want
to defeat classical mathematics, then you need
to show that its assumptions lead to a contradiction.
That means you have to work with the definitions
and axioms used by classical mathematics---*not*
your definitions.

What you are doing is complete bull***. You are
trying to impose new and different assumptions on
*top* of structures defined in classical mathematics.
You can't do that. You can make up your own structures,
but you can't claim that they are the same objects
(naturals, reals, etc.) of classical mathematics.
They are different objects, and any proof you make
about them has no relevance to the objects of classical
mathematics.

>> It has a successor, but it has no predecessor.
>
>Except for all those "ordinals" that are less than it.

Usually, predecessor(x) means some ordinal y < x such that
y+1 = x.

>This is a bad system.

The concept of "bad" is meaningless in mathematics. If
you think there is something wrong with the definitions,
show that those definitions lead to a contradiction.

>You can keep it, but I see no use for it or justification for it.

That's because you are basically an idiot.

>> The consequences are perfectly consistent, and are not paradoxical.
>
>Uh, yeah, sure. How many paradoxes have grown out of set theory? 20? 30?

Zero. There are no paradoxes in modern set theory.

>> No, you are bullshitting.
>If that's the way you see it, I am sure that makes it hard to pay
>attention.

What makes it hard to "pay attention" is that you are spouting
bull***, with no rigorous definitions, no rigorous axioms, no
rigorous rules of inference. Your posts are complete and utter
crap.

You pick and choose axioms from classical mathematics when
you like them, and reject them when you don't like them. Or
sometimes, you keep the axioms and throw away the *theorems*
if you don't like them. For instance, you pretend to accept
the induction axiom, but then you reject statements that
are provable using induction.

You are full of ***. Your several thousand posts contain
approximately one day's thought worth of ideas, and most
of those ideas are nonsensical if you looked at them with
any rigor.

--
Daryl McCullough
Ithaca, NY

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