Re: infinity

In article <MPG.1d8a5d065cfe0aad98a241@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Randy Poe said:
> >
> > 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.
> Starting condition
> >
> > 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.
> f(n)->f(n+1). Recursive infinite chain of logical implication
> >
> > 3. zero is not the successor of a number.
> >
> > Axiom about the special element zero.
> Semi-redundant statement of the starting condition.
> >
> > 4. Two numbers of which the successors are equal are themselves
> > equal
> >
> > Axiom about elements.
> Statement of strict linear order.
> >
> > 5. If a set S of numbers contains zero and also the successor of
> > every number in S, then every number is in S.
> Defines the set based on the element definition, as is always the
> case for infinite sets.
> >
> > 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.
> Of which 2) is the recursive statement.
> >
> > 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.
> It is in axiom 2. Each natural is defined based on the previous one.
> >
> > > 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.
> As all infinite sets are defined by the properties of their elements.
> >
> > > 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.
> You have been saying you have an infinite set, which implies elements
> that are an infinite number of steps from 1.
> >
> > > 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).
> Oh, bull. By the way, you mean "finite". If you don't understand any
> of my reasoning, then I am not repeating it.

Good! Stop repeating your garbage.

> You claim I am throwing
> a tantrum, but you are projecting your own silliness on me.

TO has quite enough of is own silliness so that no additins are needed
to make him the ass he is. Failing at so many pons.

> I am
> making logical statements

To can't even tell logic from illogic, much less produce logic.

> Your claims that my proof that the largest element is always the set
> size for any initial segment of the naturals doesn't apply to the
> entire set is bull.

Then it is TO bringin in a "largest element" argument of his own, and
one a good deal more phony that the valid one he keeps objecting to.

For every finite natural, n, in he set of all finite naturasl, N,
n < n+1 = size({1,2,2,...,n+1}) <= Size(N), so that, despite all TO's
arguments, there is no member of the set of all finite naturals that is
the size of the set of all finite naturals.




> The entire set IS an initial segment of itself. Do you
> also claim the set is not a subset of itself?

No only that its size is not a member of it, as every member is too
small.
> >
> > > > This isn't your usual quantor dyslexia, it's some other sort of
> > > > element-set dyslexia.
> > > It's Bigulosity.

Which is how TO spells "Garbage".
> >
> > 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.
> If you think it's element-set confusion, then explain where this
> confusion is. I am making no such mistake, and you know it.

You are, and most of us know it!

> > 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".

> Bull***.

That is what TO is producing a lot of, certainly.

> Bull***.

TO describing his next esay, which I have snipped for the benefit of the
queasy.

> >
> > > > 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?
> What exactly is in question here?
> >
> > > 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.
> The limit is that value. As n goes to oo, the sum goes to the
> convergent value.
> >
> > > 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.

> Gimme a break.

Why should anyone encourage TO's idiocies?

Sum(x=1->oo:1/2^x)=1. It EQUALS 1 at x EQUALS
> infinity.

It CONVERGES to 1 as the number of terms increases without limit.

> Do you claim the infinite series 1/2^x does no have a sum
> of 1?

No sum but a limiting value of 1.



> This way of speaking is another manifestation of the political
> garbage that surrounded the explorations of infinity in the 19th
> century. We can talk about "approaching" infinity, but we all agree
> never to "reach" it. It's a bunch of cow pie.

Apparently TO's favorite snack.

> The sum of an infintie
> series of 1's is infinite, and if you want to argue against that,
> then you're just being stupid on purpose.

Since TO is so often stupid on purpose, what is his objection to it?
> >
> > > 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.


> Ho hum.

TO finds truth boring again!
.

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