Re: Orlow cardinality question

Martin Shobe said:
> On Wed, 15 Jun 2005 10:44:33 -0400, Tony Orlow (aeo6)
> <aeo6@xxxxxxxxxxx> wrote:
>
> >Martin Shobe said:
> >> On Mon, 13 Jun 2005 09:51:04 -0400, Tony Orlow (aeo6)
> >> <aeo6@xxxxxxxxxxx> wrote:
> >>
> >> >Martin Shobe said:
> >> >> On Wed, 1 Jun 2005 14:15:50 -0400, Tony Orlow (aeo6)
> >> >> <aeo6@xxxxxxxxxxx> wrote:
> >> >>
> >> >> >Martin Shobe said:
> >> >> >> On Tue, 31 May 2005 12:52:34 -0400, Tony Orlow (aeo6)
> >> >> >> <aeo6@xxxxxxxxxxx> wrote:
> >> >> >>
> >> >> >> >David Kastrup said:
> >> >> >> >> Uh, no. You string length is unlimited over the set, meaning that
> >> >> >> >> there can't be a fixed maximum for the complete set, but every single
> >> >> >> >> one of those lengths is finite.
> >> >> >> >Why?
> >> >> >>
> >> >> >> There can't be a fixed maximum for the complete set becuase if there
> >> >> >> was, we could append another member of S to the end of such a string
> >> >> >> and have a finite string with a greater length.
> >> >> >>
> >> >> >> Every single one of those lengths is finite because the set in
> >> >> >> question is the set of all finite strings.
> >> >> >And what is the upper bound on the length of those strings?
> >> >>
> >> >> 1) There isn't a unique upper bound. Any infinite ordinal is an
> >> >> upper bound (Since all the strings are finite.) There are no finite
> >> >> upper bounds for reasons given earlier. Therefore, the least upper
> >> >> bound is the smallest infinite ordinal. I.e. omega.
> >> >Oh wow, so omega-1 must be that magical greatest infinite.
> >>
> >> I'll assume you mean greatest finite here, otherwise I don't see any
> >> connection to what I said. And, no omega - 1 is not the "magical
> >> greatest infinite". omega - 1 doesn't exist.
> >>
> >> That's one for the
> >> >books. If you simply declare omega to be the smallest infinity, why not declare
> >> >alpha to be the largest finite?
> >>
> >> I don't simply declare omega to be the smallest infinity. In ZFC, it
> >> is a theorem that a smallest infinite ordinal exists. Once we have
> >> proven this, we name that smallest infinite ordinal, omega.
> >A theorem proven by what axioms?
>
> ZFC, NBG are both capable of proving it, and from my point of view as
> a dabbler, they seem to be far and away the most common axioms used
> when dealing with sets. (Especially ZFC).

That didn't answer the question. I'd like to saa anyone derive this "fact", and
I'll be happy to show you the false assumption upon which it rests.

>
> >The smallest infinity is as nonsensical as the
> >largest finite.
>
> Yet in most set theories actually used, there is a smallest infinite
> ordinal. But there isn't a largest finite ordinal.

Interesting. So? That rests on the nonsensical notion that subtracting froma
value doesn't always make that value smaller. Conveniently non-functional.

>
> >The same logic applies, that a smallest infinity would be
> >finite if 1 were subtracted, but that's simply false.
>
> Subtraction isn't defined on ordinals. In part becuase a + c = b + c
> does not necessarily imply a = b.

Yes, convenient, and nonsensical.

>
> >There is no such thing,
> >and if ZFC claims there is, that's just another glob o' junk for the bucket.
> >What axioms prove this? They need review.
>
> Then don't use ZFC. But, you should keep two things in mind.
>
> 1) Just because you don't like a result, it doesn't make it
> inconsistant. You have claimed that ZFC is inconsistant, but so far,
> you have only shown that it proves things you wish it wouldn't.

I have shown that it makes assumptions that are inconsistent with other
assumptions within set theory and outside of it.
>
> 2) You are going to have a hard time making what you want consistent
> without introducing even bigger problems (from the point of view of
> most mathematicians).

I have heard that opinion numerous times, along with admonishments about how
it's not that easy, etc. Yet, no one has pointed out, despite their claims to
the contrary, any inconsistencies in my understanding, either internally or
externally, with anything except cardinality, which doesn't agree with anything
else anyway. There are no bigger problems this introduces. Only ones it
addresses.

>
> >>
> >> > But then I guess you'd have to say alpha+1
> >> >=omega, and that would violate Peano. Smallest infinite ordinal? There ain't
> >> >none. That's unstable conjecture.
> >> >>
> >> >> >> Because you would be talking about a different set. You can certainly
> >> >> >> talk about sets with infinite strings. But there are no infinite
> >> >> >> strings in the set of all finite strings.
> >> >> >Yes, only a finite amount of finite strings in your set, unless of course
> >> >> >you're working with an infinite set of symbols.
> >> >>
> >> >> Nope. Proof. Let S be the set of symbols. let a be in S. Let A be
> >> >> all the finite strings that contain only the symbol a. (I.e. A = { a,
> >> >> aa, aaa, aaaa, ...}. A is obviously a subset of the set of finite
> >> >> strings for the all the symbols in S. Now definie as follows. f(x)
> >> >> is the strings xa. f is clearly a bijection. and there is no x in A
> >> >> such that f(x) = a. Therefore A is infinite. Therefore, S is
> >> >> infinite.
> >> >
> >> >> Martin
> >> >>
> >> >>
> >> >Woohoo! You struck gold! We all gonna be rich now!
> >> >
> >> >I don't even want to wade through your convoluted proof. I just want to point
> >> >out that, unless you made a mistake, you just proved what you just denied, that
> >> >S is infinite, meaning you require an infinite set of symbols to have an
> >> >infinite set of strings all of finite length. Very good, Martin. Verrrry good,
> >> >indeed. Nice job. Have a beer on me.
> >>
> >> Convoluted? If you think that's convoluted, you'd better stay away
> >> from the difficult stuff, 'cause your brain will explode. Anyway,
> >> there is a mistake there. The last sentence should be "Therefore, the
> >> set of finite strings for all the symbols in S is infinite." not
> >> "Therefore, S is infinite."
> >You better reexamine that some more. I am sorry, but I have no problem with
> >difficult logic if it makes sense, but watching people work with Cantor and ZFC
> >is like watching the keystone cops. You drew MY conclusion, and now you want to
> >throw the word "finite" in there, so as not to concede anything. It's looking
> >pretty clumsy, I hate to say.
>
> You obviously do have a problem with difficult logic. You call an
> introductory level proof, convoluted. When I admit I made a mistake
> and post a correction, instead of working with the corrected proof,
> you post an ad hominem. QED.

Excuse me, but you suggested I couldn't handle logic and my brain would
explode, but I don't suppose that's ad hominem? Gimme a break. All the proofs
in ZFC are convoluted in my mind, because so many of the assumptions are
unfounded. It was enough to see you derive my conclusion. Why should I bother
giving myself a headache analyzing yet another convoluted proof that's wrong,
when you gave me the answer I wanted anyway? I may be strange, but I'm not
stupid enough to waste my time milking a bull.

>
> Martin
>
>

--
Smiles,

Tony
.

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