Re: Orlow cardinality question

In article <MPG.1d28b9ff6a083bc0989ee0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Jan de Vos said:
> > In sci.math, Tony Orlow wrote:
> > > I have never argued with Peano, but with interpretations of his
> > > definitions. I think inductive proof needs the caveat I have
> > > suggested, that it be used for proving constant equalities, or care
> > > be taken to check the results with infinite series. I do not reject
> > > the pofnats as a set, but I do reject, on solid mathematical
> > > grounds, that this set can be infinite while only containing finite
> > > values as you claim. It is a finite but indeterminate set, the way
> > > you define it. it has no determined maximal element, and has no
> > > determined size.
> >
> > What, your Bigulosity can't even determine the size of a very simple
> > set of /numbers/?
> >
> > That's not very usefull :-)
> >
> >
> > Jan
> >
> >
> It's not a simple set by definition if the elements are defined as "less than
> any infinity". That's not a definite upper bound.

A set is well-ordered if every nonempty subset has a first or smallest
member.

Any set satisfying the Peano axioms is well-ordered.

A well-ordered set is finite if every subset (including the set itself)
has a last or largest member.

For every natural, the set of naturals no greater that it is finite.

There is no set of naturals with a maximum member that is not finite.
.

Relevant Pages

  • Re: Orlow cardinality question
    ... > to be finite if the set of naturals up to and including it is a finite ... > set by the Cantor definition, is, by the Cantor definition, an infinite ... > member of the set. ... Even worse, I posit that the first may come after the last, and the last before ...
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  • Re: Orlow cardinality question
    ... to be finite if the set of naturals up to and including it is a finite ... set by the Cantor definition, is, by the Cantor definition, an infinite ... Then the size of the set is not a member of the set because TO agrees ... finite objects. ...
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  • Re: Orlow cardinality question
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