Re: infinity

Virgil said:
> In article <MPG.1d62941df41df45b989ff9@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1d618c01dce78c27989fea@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > >
> > >
> > > What Virgil is interseted in is the source of the drivel that TO keeps
> > > cooking up. Since TO's positions are so many and so mutually and self
> > > contradictory, no one with any sense would try to keep track of them.
>
> > My positions are non-standard, but consistent.
>
> Only in TO's mind are they consistent. Nowhere else.

Apparently not in your mind, yet.

> > >
> > > What number system does TO propose that has, as he now claims, a
> > > "largest" number in it, so large that that number plus one is not a
> > > larger number?
>
> > I never claimed to have any such system.
>
> But what TO does claim requires it.
Only given your assumptions. Don't you ever play with modifying your axioms?

>
>
> > > The Peano axioms prohibit any complete Peano set from having a largest.
>
> > No kidding. "largest finite...largest finite" I see we're back to irrelevant
> > mantra. Can it.
>
> That "mantra" is the death knell to TO's delusionary theories. TO's
> theories contradict the Peano postulates, as that mantra shows.

No, that mantra is a last-ditch effort to make what I am suggesting sound
meaningless, but it's really irrelvant to what I am putting forth, and really
seems to be a major hangup and the source of much confusion in this area.

> > >
> > > The Archimedean property of the standard reals prohibits the reals from
> > > having a largest.
> > >
> > > Even the non-standard reals of A. Robinson do not have largest members.
> > >
> > I never claimed there was any such thing, and Virgil knows that, but repeats
> > his lies in an attempt to discredit those that point out his flaws. Sad.
>
> It is not that TO directly claims it directly but that his assumptions,
> added to the Peano postulates, imply it.
You never commented on my adjustment of the Peano axioms, did you?
>
> TO claims that there exists at least one natural number which cannot be
> reached by a "finite number" of successor operations.
Well, after all, you ARE applying an INFINITE number of successor operations,
IF you claim to have generated an infinite set with your stepwise difninition,
now HAVEN'T you? Sheesh! How can you have an infinite number of counting
numbers, without counting and infinite number of times? Can you explain this?
>
> This , together with the Peano properties, requires that there be a
> smallest such "infinite" natural, which in turn implies the existence of
> a largest "finite" natural.
No, it really doesn't. Besides, you and your compadres DO claim to have this
smallest infinite, omega, but to get around the implication that the smallest
infinite should have a finite predecessor, you simply invent an axiom stating
that omega-1=omega. Now, if that isn't a kludge, and a contradiction to what
you see at the top end of the finite end of the number line, then the moon IS
made of cheese, and germs ARE spontaneously generate in the ether.
>
> So it is TO's "largest natural" not mine. Unless he wishes to revoke one
> or more of the Peano postulates.
I already suggested that, if you want to cling to your omega, you might as well
declare alpha to be your largest finite natural, invent another nonsensical
axiom that states that alpha+1=alpha, and put your "largest finite" mantra to
rest. Of course, that sounded silly to you. It is. But, it's already what
you're doing anyway on the infinite end, so why not? It's all just a game to
you fellows anyway. I call flying kings! :(

Since you don't seem to recall my adjustment of the Peano axioms, and since you
are bringing them up as "proof" that I am wrong, let's review our axioms, and
see if they can't be adjusted to accomodate the requirements of our infinite
set:

Peano Axioms (from MathWorld):

1. Zero is a number.

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

3. zero is not the successor of a number.

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

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


Orlow's Peano Axioms:

1. Zero and Infinity are numbers.

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

3. Zero and Infinity are not the successors of any number.

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

5. (induction axiom.) If a set S of numbers contains Zero and Infinity and also
the successor of every number in S, then every number is in S.

Notice that what we have done is essentially set up two counting sequences, and
while it is not specified, they count in opposite directions, but can never
meet in any finite number of steps. Similarly unspecified in the original
axioms is the idea that each number is a unit quantity greater than its
predecessor. Of course, this would be the opposite for the successor of
infinity.

Personally, I would probably define the numbers thus:

1. Zero and Infinity are numbers.

2. If a is a number, the successor and predecessor of a are numbers.

3. If a and b are numbers, a is the predecessor of b iff b is the successor of
a.

4. If a set S of numbers contains any number, and also the successor and
predecessor of every number in S, then every number is in S.

5. The successor of every number is 1 greater in value than that number and the
predecessor of any number is 1 less in value than that number.


This way, we include not only the infinite end of the number circle, but also
the negative half of it, for a definition of the true infinite set of whole
numbers.

Does anyone object to such a set of axioms? Does it allow for infinite whole
numbers? Does it work?



--
Smiles,

Tony
.

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