Re: infinity


Tony Orlow (aeo6) wrote:
> Randy Poe said:
> > No, the definition of natural numbers does not require
> > that the collection be built one at a time. There is
> > nothing in the definition that makes such a requirement
> > on the set.
> The set is recusively defined,

No, the ELEMENTS are recursively defined. Your dyslexis is
acting up again. Let's look at the axioms (as specified at Mathworld,
http://mathworld.wolfram.com/PeanosAxioms.html)

1. Zero is a number.

Axiom about one particular element.

2. If a is a number, the successor of a is a number.

Axiom about each element: the successor of every element is
an element.

3. zero is not the successor of a number.

Axiom about the special element zero.

4. Two numbers of which the successors are equal are themselves equal

Axiom about elements.

5. If a set S of numbers contains zero and also the successor of
every number in S, then every number is in S.

Only axiom that is about the *set* of naturals, and it is not
recursive. It defines the set as the collection of all numbers obeying
axioms 1-4.

So, not surprisingly, your repeated assertion that the set is
"defined recursively" or has to be "created" by growing from finite
sets one element at a time, is not found anywhere among the Peano
Axioms.

> each member being derived from the previous
> through successor().

Yes, each member is defined by the successor operation. Note the
subject of your sentence: "each member". That is not talking
about sets but, as I said, the elements. The set is merely
defined as the collection of members which obey these axioms.

> If each element is a finite number of steps from 1, then no element is an
> infinite number of steps from 1,

Correct. We all have been saying that.

> and the set does not have an infinite number
> of elements, since each element is a step.

Non sequitur and incorrect. The set is the collection of all numbers
which can be reached in finite steps. At best it is in dispute
(in this thread, not in any discussion where the rules of logic
are followed) as to whether that collection is finite or infinite.
Again you are getting confused, and simply declaring by faith
that the collection of things at finite distance must, must,
MUST be infinite (stomp, whine, gnash teeth).

> > This isn't your usual quantor dyslexia, it's some other
> > sort of element-set dyslexia.
> It's Bigulosity.

No, it's element-set confusion. Bigulosity is your wacky theory
of a particular property of sets. But again, it's a set property,
not an element property. Another manifestation of the element-
set dyslexia.

> > Be more specific. Tell me what part of "infinite series
> > for god's sake" implies that one can reach an produce
> > value by a process of incrementing.
> Increment means add one.

Yes. And adding one to a finite thing, as we all agree, NEVER
gets you an infinite thing. Never. Not ever.

> If you do this an infinite number of times

That involves going to the end of a process which you have agreed
many times has no end.

> you get sum (x=1->oo: 1)

No, you do not. This sum is defined only in terms of the behavior
of the finite partial sums SUM (x=1:n, 1) as n takes on increasing
FINITE values. Nobody ever talks about "adding infinite terms"
or "getting to infinity".

> Infinite series says that this sum diverges

Indeed it does. And diverge has a very precise meaning, none
of which has anything to do with "getting to infinity" or
"adding an infinite number of terms".

> Therefore, if you have an infinite number
> of elements, each one greater than the last, then you will have infinite values
> as the result of the infinite increments.

More precisely, if you'd been awake in the class where this was
taught, you would realize that you don't "have infinite values".
Rather, you have finite values which are not bounded.

> If you do not allow such infinite sums,

Infinite "sums" are allowed but do not have the meaning you think
they do. In particular...

> then you must not allow infinite increments,

.... this is not true. There is nothing in the definition of an
"infinite sum" which actually implies that an infinite number
of terms are being added.

> > Perhaps you should "look up infinite series for god's
> > sake" and familiarize yourself with what it means for
> > a series to diverge. One thing it doesn't mean is that
> > anything ever reaches an infinite value.
> What??? YOU look it up again.

I'm quite clear on the definitions, but I'd be happy to cite
them from a text or two. Perhaps tomorrow. Are you willing
to find a text bolstering your position?

> There are three possibilities for an infinite
> series. It can converge,

Which does not mean that it ever necessarily actually reaches
the value which it converges to.

> such that the sum of all terms is a finite number.

No, such that the limit of the sequence of partial sums is a
finite number. This does not imply that that number is ever
reached or that the operation "sum of all terms" ever actually
happens.

> It can diverge in the sense that the sum over all terms is
> infinite.

No, not in that sense. It means that there is no finite upper
bound to the finite sequence of partial sums. And what THAT
means, more precisely, is that for any FINITE value you pick,
there will be at least one FINITE partial sum, a sum over
a FINITE number of terms, which has a larger but still FINITE
value.

Nothing as ill-defined as "the sum over all terms" ever actually
crops up in these definitions.

> Or it can diverge in the sense that the sum is never infinite,
> but has no specific limit

Again this is defined by what happens with the sequence of
FINITE partial sums over FINITE numbers of terms, at which they
take FINITE values. Nothing like "the sum of all terms" ever
occurs. Indeed, in your mixed-up recollection, you have bits
of the actual definition since you talk about "the sum is never
infinite". There is of course at most one sum of the whole series
(defined as a limit). If you talk about more than one value for
a sum, what you are recalling in your muddled way is the
behavior of the FINITE partial sums.

> at n=oo. This third form is generally an oscillating sequence.

The PARTIAL sums form a sequence which does not converge.
Oscillation is one possibility, not the only one. The
convergence criterion is quite precise, and not converge
merely means the sequence doesn't match that criterion.

> In this case, we
> are looking at a sum which is clearly infinite.

We are looking at a monotonically increasing sequence of
finite values. The LIMIT is infinite because we can't find
a finite upper bound, even though every element of the
sequence is finite. But there's no such thing as "the sum
of all the terms". That terminology never actually crops up
in formal discussion of series.

- Randy

.

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