Re: infinity

Virgil said:
<snip>
> > Only given your assumptions. Don't you ever play with modifying your
> > axioms?
>
> Not while discussing the consequences of the axioms as they actually are!
>
> And as these various axiom systems are the assumptions on which all
> standard artithmetics are based, there is o point in changing them in
> order to discuss the properties of standard arithmetic.
Axioms should be reviewed when the need arises. If the axiom set does not cover
certain recognized aspects of its subject matter, then it should adjusted. If
you don't like people adjusting pre-=existing axioms, then you must dislike
Hilbert, eh?
<snip>
> > You never commented on my adjustment of the Peano axioms, did you?
>
> Yes! I commented that they were irrelevant to, and contradictory of,
> standard arithmetic. And they are.
Not contradictory at all, or at least you didn't point out any contradictions,
and not irrelevant to the subject at hand, which is a non-standard approach
including infinite whole numbers, so your standard system makes some sense.
> > >
> > > 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?
>
> Not precisely. I am applying only finitely many successor operations at
> any one time, but with no finite least upper bound on the number of
> operations.
Finite but with no LUB? What was your definition of finite again? I thought it
was that one could find a number larger. If you are applying only finitely many
successors, then you are generating only finitely many natural numbers.
>
>
> Can you explain this?
What?
> > >
> > > 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.
>
> I have proved that it does. If TO choses to ignore that proof, as he
> ignores all the other proofs of his errors, he is merely proving himself
> to be invincibly ignorant.
Someone else "proved" it by assuming the contrapositive. That is not not a
proof based on anything but the conslusion desired. You, as always, generally
do not respond constructively.
>
>
> Besides, you and your compadres DO claim to
> > have this smallest infinite, omega,
>
> But whatever any of us says omega may be, we do not ever say that it is
> a natural number, in fact, we say quite definitely that it is NOT an
> natural number.
It is a whole number in the infinite spectrum of numbers. To claim that you can
subtract 1 from a whole number and not change the value is contradictory to the
definition of subtraction. The number circle is symmetrical, and treating the
two ends of it incosistently is a mistake.
>
>
>
> > that the smallest infinite should have a finite predecessor, you
> > simply invent an axiom stating that omega-1=omega.
>
> Since we are no longer working in the set of naturals, what holds in the
> set of naturals need not be required to hold.
By my adjusted definition, the infinite whole numbers ARE in the set. Let's
call them the "orlovian" numbers, as someone has called them. It doesn't
matter. This set of axioms resolves some of the problems with your system.
>
> If finite cardinalities and infinite cardinalities were in all respects
> the same, there would be no point in distinguishing between them
The fewer distinctions and the more general the rules, the better. Occam's
Razor rules.
>
> > 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
>
> Only if one ignores the fact that a boundary has been crossed and that
> things differ on opposite sides.
Things are mirrored on each side. They are not as different as people claim.
The finites are infintie compared to the infinitesimals, and infinitesimal
compared to the infinites.
> > >
> > > 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.
>
> This "largest natural" only exists if one accepts TO's assumptins, and
> is totally impossible under the standard assumptions of standard
> arithmetic. So It is not my "mantra", it is TO's assumptions added to
> standard arithmetic that bring this beast into existence.
I don't believe in ANY largest whole number. Even the infinities are
infinitesimal compared to meta-infinities, and there is no end to the nesting
of infinities any more than there is an end to the digits in finite numbers
which lead to the infinities.
>
> > 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.
>
> In the sense that there are rules that we must agree to follow in order
> to make things work properly, it is precisely like a game.
>
> But TO keeps wanting to change the rules in the middle whenever he
> doesn't like the way things is going.
I am trying to suggest rules that make the system actually consistent and
meaningful.
>
> We have said many times that TO is free to set up the rules of his own
> game, but until he has a much more complete and contradiction-free set
> of rules than he has so far produced, he will not find anyone to play
> his game with him.
Are Peano's axioms more complete and contradiction-free? What contradictions do
you detect WITHIN my set of axioms?
> >
> > 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?
>
> Let us see how it affects the of whole arithmetic, including that for
> the rationals and reals and all those areas of mathematics dependent on
> those arithmetics.
That is an area worth exploring, and some operational differences are bound to
occur, at least at first. The hope is to derive a system that treats finites,
infinitesimals and infinties consistently.
>
> And since "infinity" is already much overused, try some other name for
> your extra "number".
No, thank you. Infinity is what we're talking about, and is a fine term for
systems and quantities that don't end.
>

--
Smiles,

Tony
.

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