Re: infinity

Daryl McCullough said:
> 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.
And has no predecessor. Good way to kludge away the largest finite.
>
> 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.
The point 1 is not included unless you allow infinite n, right?
>
> 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.
When n=oo.
>
> >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.
But then, the largest of that set would be one less than its size, which is
aleph_0. Does that mean aleph_0 is one greater than the largest finite? If not,
why not?
>
> 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.
Even though the size is al;ways equal to, or one greater, than some element of
the set? Nope, it stinks.
>
> >> 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.
Try "number of elements". Stick to basics. Remember Occam's Razor.

If you can simply say it's the definition, then I can simply say the definition
of an infinite set is a recursively defined set such that every element has at
least one successor, and where an infinite number of successor operations is
allowed. Then, by definition, the set of Peano numbers is infinite, if an
infinite number of iterations is allowed to occur. I don't need to "prove" it.
It's "the definition".
>
> >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.
Call it cardinality. Transfinite cardinality does no correspond to infinite set
size.
>
> Go through the definitions. Where is there any
> contradiction, or ambiguity, or undefinedness?
In treatment of induction, for one. In the postulation of a smallest infinity
with special prop-up rules.
> 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.
Your definitions work that way, but not very well.

>
> 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.
I am developing a whole other system, where balls don't magically disappear,
where set size changes when elements are added or removed, and where the size
of a set of points correlates with the size of the object that contains those
points. it's not just terminology. A piece of crap by any other name would
smell as rank. :D
>
> >> 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.
It is valid for infinite n as long as it is not proving some inequality where
the difference has a limit of 0 at oo.
>
> >What a great system. It stinks.
>
> You have trouble with it because you are basically
> an idiot.

Yes, I know, and if I disagree with you politically then I'm an ***, and if
I have a different religious outlook, then I'm evil. Whatever floats your
stool.

> 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 don't like the conclusions, examine the premises. That's what I'm doing.
>
> 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.

I am less interested in trying to teach old dogs new tricks than developing new
trick for the next generation of puppies. Of course, one can get an idea of
what the puppies can learn by playing with old dogs.

>
> What you are doing is complete bull***. You are
> trying to impose new and different assumptions on
> *top* of structures defined in classical mathematics.
Actually I am trying to supplant them with better, more fruitful assumptions
and axioms.
> 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.
I can examine the definitions used to derive wrong results, and induce better
definitions for the same concepts. there is no rule preventing me.

> They are different objects, and any proof you make
> about them has no relevance to the objects of classical
> mathematics.
Whole numbers are whole numbers. Sets are sets. You keep cardinality. I don't
need to talk about that.
>
> >> 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.
Oh, immediate predecessor. And yet, there are ordinals before it. You just
can't identify the one directly before it. Sounds like a whole other set of
numbers, and yet, you call them both ordinals. Hmmm.....
>
> >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.
That's what I've been doing, but whenever I get you folks up against the wall,
the hands start waving, and suddenly a set of values has no range, and
everything boild down to the largest finite mantra. it's wuite the monkey's
fist knot you have yourselves tied up in.
>
> >You can keep it, but I see no use for it or justification for it.
>
> That's because you are basically an idiot.
Hmph. Funny that IQ tests rate me as a junior genius, and I got a perfect score
on my Math level II achievement test, and all A's in math and science,
including logic. I suppose, given my primitive writing style and inability to
present argument after argument poking holes in your impenetrable theory, it
seems natural to come to that conclusion. But, if I am an idiot, what does that
make you? Do you think an idiot could formulate an alternative to standard set
theory? That's what's going on here, whether you understand that or not.
>
> >> 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.
I refer you to http://paias.org/Mathematics/Paradoxes/paradox.htm#Set%20Theory
>
> >> 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.
Thank you. Coming from you, that is a compliment.
>
> You pick and choose axioms from classical mathematics when
> you like them, and reject them when you don't like them.
I have that right.

> 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.
I reject misapplication of axioms. That is my duty.
>
> 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.
If you looked that them with any visual understanding, you would see the sense
in them, and feel compelled to reject the standard system as I do.
Unfortunately, just like an evangelical christian who thinks evolution is
bull***, or a businessman that thinks socialism is bull***, or a general that
thinks peace is bull***, you are probably simply incapable of seeing what I
see. So be it. Don't look.
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

Tony
.