Re: Orlow cardinality question

In article <MPG.1d20ca236aac34cc989e49@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:

> > 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 "number"
> > 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 need not be the case, as in the first example above. But in any
case, it is the sets which exist first and the numbers, as
representations of the cardinality of those sets, which are derived from
the sets,


>
> I have talked about the two unit infinities

Is that anything like finite infinities?


> > 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.

As a representative of set size.

Meaning that there is a standard set, sometimes named by its last
element, which is the standard set for finite cardinality.
.

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