Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> Dave Rusin said:

> > > Now add one to it. What do you get? A contradiction.
> >
> > Right, that's it.
> >
> >
> This logic depends on two assumptions which are unfounded:
>
> 1) That counting is the ONLY way to get to infinity, and

Um, I don't know how you get that from a chorus of
people who have been saying, unanimously, that counting
will NEVER get you to infinity.

That is the heart of the critiques of your "induction to N",
that counting is a very bad way to get "to infinity",
because you will never get there.

> 2) That ANY finite set must have a largest member which is known.

Any finite set of things with a complete order relationship
(all pairs of items are either <, =, or >) has a largest
member. That is true.

> 1) is simply not true.

Certainly not, for the reasons I state.

> In fact we can look at the infinities created when we
> divide by zero, and we can look at infinities as sums of things OTHER
than
> natural numbers, and infinities created as limits of functions.

In mathematics, we never get "to infinity". ISTB. We never
do any of those things. We don't divide by zero, we don't
really add up divergent sums, and we don't take limits
"to infinity". When we observe that something is taking
increasingly large finite values, that's the end of the story.
You leave the world of logical deduction every time you
try to make claims about things that happen "at" infinity.

> 2) Seems logical when thinking about numbers.

Provably so, not just for numbers but for anything with a
complete order relationship.

> If we specify all the members in
> our set, we can compare them quantitatively

Does compare them quantitatively mean determining <, =,
or > for each pair?

> to see which is largest. But, if we
> can't specify all our elements quantitatively,

Oh, well if some of them don't have a <, =, or >
relationship, then you have a "partially ordered set"
rather than a "complete order". But aren't we talking
about numbers here? Numbers have a complete order.

> then it may not be possible to discern any largest element.

Can't guess what you're thinking here.

> If the set is defined recursively,

As we all know N is.

> implying an infinite process of creating elements

No, no infinite process. You've just left our planet
again.

> with a constant finite difference, but
> restricted to finite values, then we are forced to ask where
> the counting ends,

It doesn't of course.

> and this endpoint cannot be specified.

There isn't one.

> This is because there is a contradiction
> between the recursive specification of the set, implying infinite
range of
> values,

There's no upper bound on the range. You can always add
another digit.

> and the restrictive definition of natural numbers as finite,

Being able to add another digit does not contradict "finite".
Adding another digit doesn't make a number infinite.

So, no contradiction.

Weren't you going to say something about FINITE sets not
necessarily having a largest value? A finite set is
one where you can count the elements from 1 to n,
stopping at some finite natural number n. You don't
seem to be talking about such a set.

> So, you can't specify the size because it is not
> specified in the definitions being used, and no way is supplied to
figure it
> out. This is also certainly not a general rule about sets.

No, just sets with ordering.

> Is there a largest element in a set of 10 marbles?

Not unless we define "<", "=", and ">", which could be
done.

But we're talking about finite sets of NATURAL NUMBERS,
so appealing to unordered objects won't help your case.

- Randy

.

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