Re: abundance of irrationals!)

Randy Poe said:
>
> 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.
Then why is cardinality based on counting, and yet claiming to say ANYTHING
about infinity? Do you use a microscope to take someone's pulse? Don't you see
it's THE WRONG TOOL????

>
> > 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.
Apparently not. Please prove that assertion.
>
> > 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.
That's your opinion, and mathematics has suffered from the same timidity and
condusion for long enough.
>
> > 2) Seems logical when thinking about numbers.
>
> Provably so, not just for numbers but for anything with a
> complete order relationship.
Prove it.
>
> > If we specify all the members in
> > our set, we can compare them quantitatively
>
> Does compare them quantitatively mean determining <, =,
> or > for each pair?
No duh.
>
> > 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.
Can you specify every natural number? No, because they go on forever? That's
what I thought. You do know what "specify" means, don't you?
>
> > then it may not be possible to discern any largest element.
>
> Can't guess what you're thinking here.
I seem to recall a chorus of repeated mantra "there is no largest natural".
Weren't you in the chorus?
>
> > If the set is defined recursively,
>
> As we all know N is.
Yes it is, by counting. That's naturals, a specific enumerated subset of the
reals....
>
> > implying an infinite process of creating elements
>
> No, no infinite process. You've just left our planet
> again.
Then why, again, did you claim the set is infinite?
>
> > 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.
Isn't that why you say the set is infinite?
>
> > 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.
Only if you do it an infinite number of times. Are there an infinite number of
naturals or not?
>
> So, no contradiction.
Big-time contradictions.
>
> 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.
Apparently, one can have a value they know somehow is finite, even though they
can't specify what it is, and even though it's infinitely far from zero. That's
the position of the vast majority here, and it's the floppiest hook I've ever
seen anyone hang their hat on, who wasn't living on the street.
>
> > 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.
My case is solid, despite all the squirming around me.
>
> - Randy
>
>

--
Smiles,

Tony
.

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