Re: Orlow cardinality question

Virgil said:
> In article <MPG.1d1ccb0af4b7d541989e1b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1d1b9b0c5dea7d6c989e09@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > >
> > >
> > > > > Who says that number of elements is the one universal measure of
> > > > > sets? That is something you just made up. You have to define
> > > > > "number of elements" before the question even makes sense.
> > >
> > > > Which term do you not understand, "number" or "element"?
> > >
> > > How do you, TO, define "number of elements" of a set?
> > > It is not that we do not understand in general, it is that we do not
> > > know what your understanding about that phrase is.
> > How about the integral of the density over the domain? Does that satisfy your
> > need for mathematical definition? There are several ways to state this, but
> > "number of elements" is the most basic, and the way we intuitively think
> > about
> > sets. A set is a number of elements, members, units, or whatever.
>
> Such vagueness is mathematically unsatisfactory. For finite sets,
> bijectability with a standard set can be used to define the "number" of
> members in an arbitrary finite set. Such standard sets are commonly
> taken to be initial sequences (a set of all elements before or up to
> some member of the sequence) of, for decimal numbers, either the
> ordered set {0,1,2,3,...} or the ordered set {1,2,3,...}, either of
> which can be defined inductively so that they consist of only finitely
> long strings of digits..
>
> If one takes {0,1,2,3,...} as the reference sequence then the initial
> sequence is the set of all elememnts before the given element.
>
> If one takes {1,2,3,...} as the reference, then the initial sequence is
> the set of all elements up to and including the given element, with the
> empty set as a special case.
>
> In either case, one then takes the element in question as the "numb er"
> of objects in the set in question based upon a bijection.

That's very interestingly stated, given your repeated objections to my
statement that a set of naturals beginning with 1 always has the set size as
its maximal element, which is what you have just stated. You are just arguing
for the sake of arguing. How boring!

I have talked about the two unit infinities and the ability to classify all
infinities in terms of them. Aren't those the "standard" infinities you are
talking about? If I dig deep enough, I am sure I can show you how your
definitions are as vague as you claim mine are, but I don't really feel like
playing the nit-picking game when it is pretty clear what both of us is saying.
Is math like a court of law? Is that what it's degraded into? Or, are there
still some out there that actively seek truth in numbers without regard to
political nonsense?
>
>
> > >
> > >
> > >
> > > > > You can use induction to prove that all finite sets have a certain
> > > > > property. You cannot use it to prove that an all infinite sets
> > > > > have a certain property.
> > >
> > > > Induction is supposed to prove something true for all members of the
> > > > infinite set of naturals.
> > >
> > > Only those members that can be generated by adding one once to some
> > > other member. There is no provision in induction for anything other than
> > > one-at-a-time.
> > Yeah like the entire set of naturals.
>
> Induction only describes what is true for MEMBERS of the set of all
> naturals, it says nothing about the set itself.
It is true that, for every n in N, the set of naturals from 1 to n has n as a
maximal element and also as a set size.
>
>
> Adding one an infinite number of times
> > is
> > the same as adding infinity once, even if done one-at-a-time. There is not
> > "time" involved" That is a figure of speech.
> > >
>

--
Smiles,

Tony
.

Relevant Pages

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