Re: abundance of irrationals!)


mueck...@xxxxxxxxxxxxxxxxx wrote:
> Randy Poe wrote:
>
>
> > If the set size were finite, you would reach a point where
> > there is no next.
>
> If the magnitude were finite, you would reach that maximum.

Is this a counter to my statement?

> If by definition the magnitude is always finite, and that works, why
> can't, by definition, also the cardinality be always finite?

Because the definition includes the property that every
n has a successor, and a finite cardinality set can't have
that property.

"Defining" the cardinality of {1,2,3,...} to be finite
would contradict the definition of 1, 2, 3, ...

> > Do you think there exists a finite integer n such that n+1
> > is not an integer, or is not finite?
>
> Do you think there exists a finite number of elements such that
adding
> another one the result gives an infinite set?

No, I do not think that any finite set is infinite.

> Remember: Every set of even numbers contains numbers which are larger
> tan the cardinality of that set.

So does the set {15}.

So what?

> This is valid for *every* set of finite numbers.

This is valid for every finite set of even numbers.

> There is no reasonable arguing claiming that this
> situation should change in case of an "infinite" set.

No reasonable arguing except that you have just introduced
a new axiom into the system: "Everything that is true of
finite sets must be true for infinite sets". And that
axiom is (a) not part of our logic because (b) it will
lead to inconsistencies.

There is VERY reasonable arguing, namely that there is
no such axiom and creating such an axiom would cause
insurmountable problems.

"I don't see why not" is not a proof that you can
extend an argument from the finite to the infinite.

- Randy

.

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