Re: Infinity
- From: Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx>
- Date: Tue, 10 Jan 2006 13:10:57 -0700
In article <MPG.1e2dbff9503e972698a95b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> MoeBlee said:
> > Tony Orlow wrote:
> > > MoeBlee said:
> > > > Tony Orlow wrote:
> > > >
> > > > > So, a Dedekind-infinite set is one where every member has a
> > > > > successive member, so they go on forever. It's not a bad
> > > > > definition, but there are some set theory conclusions
> > > > > regarding infinity which indicate some problems along the
> > > > > way. For instance, according to a proof by Banach and Tarski,
> > > > > one can cut a solid ball into 6 pieces and reassemble them
> > > > > into two solid balls, each the same volume as the original.
> > > > > Interesting mathematical fact, eh? :(
> > > >
> > > > That is misinformed and misleading.
> > > >
> > > > The definition of 'Dedekind infinite' is: x is Dedekind
> > > > infinite iff there exists a proper subset of x such that there
> > > > exists a 1-1 function between x and the proper subset.
> > >
> > > Okay, that's the proper definition, but is essentially equivalent
> > > anyway.
> >
> > If you claim that 'for every member of x there is a successive
> > member' is equivalent ("essentially equivlalent" has no import -
> > either the definitions are equivalent of they're not) with 'there
> > exists a y such that y is a proper subset of x and there exists an
> > f such that f is a 1-1 function from x onto y', then you need to
> > prove it.
> >
> > But that's silly for me to say, since proof requires a logistic
> > system and axioms, neither of which you have. So not only is
> > 'successive' not defined by you, NOTHING is defined by you, since
> > definitions must revert to primitives, of which you have none. But
> > since, when we talk about 'Dedekind infinite' we're talking about
> > set theory, if you're making claims about this, then you should be
> > making your proofs in set theory. But that you can't do, since you
> > don't know what set theory IS.
>
> While you're right that I misstated the Dedekind definition of an
> infinite set, and I apologize, the original question had nothing to
> do with sets persay, but defined "infinite" as larger than any real
> number. The op wanted a more concrete understanding, so I offered
> some geometrical images of infinity which would be immediately
> accessible, rather than talking about surjections. When it comes to
> denumerable sets, what I said was correct, and besides, I think it's
> a better definition at its root. Of course, that's just my opinion.
Since TO's opinions are of no value outside of the cloud cuckooland of
TOmatics, he would be etter served not to expose than in public.
>
> > But moving on:
> >
> > There are at least a few things we can start with:
> >
> > the successor of x = x u {x}.
> >
> > y is an R-successor of x <-> <x y> e R.
> >
> > E!y y is an R-successor of x -> (y = the R-successor of x <-> y is
> > an R-successor of x).
> >
> > y is an immediate R-successor of x <-> (y is an R-successor of x &
> > Az(<x z> e R -> (<y z> e R v y = z))).
> >
> > E!y y is an immediate R-successor of x -> (y = the immediate
> > R-successor of x <-> y is an immediate R-successor of x).
> >
> > R satisfies trichotomy on D <-> Axyz((xeD & yeD & zeD) -> exactly
> > one holds: <x y> e R; <y x> e R; x = y).
> >
> > R is transitive on D <-> Axyz((xeD & yeD & zeD & <x y> e R & <y z>
> > e R) -> <x z> e R).
> >
> > R is a linear ordering of D <-> R satisfies trichotomy on D & R is
> > transitive on D.
> >
> > To make sense of your claim, we'd have to first make it a formula.
> > To give you the benefit of the doubt, we should make the
> > formulation the easiest one to prove that's a reasonable
> > formulation of what you might have in mind. So perhaps you'd like
> > to show:
> >
> > D is Dedekind infinite <-> Er(r is a linear ordering of D & Ax(x e
> > D -> Ey(y e D & <x y> e r))).
> >
> > If that's not what you mean to say, then you need to say just what
> > it is you mean to say IN THE LANUGAGE OF SET THEORY, since that's
> > what's called for when you claim that your formulation is
> > equivalent with that of the definition of 'Dedekind infinite'.
>
> Okay, I agree, it's not exactly equivalent, but was more of a
> descrition of an inductively defined infinite set. I don't suppose
> all the above is for the op's benefit?
It is certain that TO's egocentricity will prevent TO from benefiting
from it.
>
> >
> > > The only exception to that is in continuous sets, such as sets of
> > > reals in any finite or infinite interval, where the elements are
> > > generally considered not to have distinct successors.
> >
> > Oh wait, now you've made it 'distinct successor'. I don't know what
> > your definition of 'distinct successor' is. Perhaps your 'distinct
> > successor' could be taken as 'the immediate successor'. But even
> > granting that, whatever 'generally considered' is supposed to mean,
> > you're already committing to a particular ordering. It is true
> > that, with the standard ordering of the reals, the numbers don't
> > have immediate successors. But what about other relations on the
> > reals? You don't know a priori that there are not relations on the
> > reals in which each real has an immediate successor. Also, when you
> > say "the only exceptions" are such sets (putting aside your use of
> > 'continuous set' where 'dense ordering' would be much better), you
> > haven't proven that those are the only exceptions. You just assert
> > it because it's something that SEEMS like it would be true to you.
> > You have no idea of the orderings that exist on all kinds of sets
> > other than sets of numbers. You have not a clue how you would prove
> > that Dedekind infinite is equivalent with 'every member has a
> > successive member'.
>
> True, and it's probably not generally true for all Dedekind-infinite
> sets. However, it may be. It is certainly possible to sequentially
> order the real numbers, or even the complex numbers
As it has never been done for the reals, what makes TO so sure?
Or does To have some well ordering of the reals that he has been hiding
from us?
>
> I haven't given a formal proof that the H-riffics cover all reals,
> but they do.
To hasn't given a formal proof of anything, and on present evidence is
incapable of doing so in even the simplest of cases.
> It's probably something you could do in an afternoon,
> with your experience using Dedekind cuts. I am a little more
> interested in implementing them on the computer
Dedekind cuts on a computer? Or any uncountable set on a computer?
C'est a rire.
> but I need to be
> able to take 2 to the power of any binary fraction, and haven't found
> an efficient way to do this yet. The gradal numbers are kind of
> intractable so far, like the Collatz conjecture, but I am looking for
> a method to do these exponents. Any suggestions?
>
> >
> > > There is some feeling among some that the set of real numbers
> > > does not really qualify as a set, because it lacks this discrete
> > > nature and elements cannot always be distinguished from their
> > > "neighbors".
> >
> > That's garbage. The question was about Dedekind infinite. That's
> > set theory. The set of real numbers exists in set theory no matter
> > whatever you mean by your undefined "distinguished".
>
> The question, originally, was about infinite being defined as greater
> than any real, and seeking a more concrete understanding. It said
> nothing about sets, but set theorists seem to think EVERYTHING is
> about sets.
When one starts talking about numbers, one either has some set theory
basis in mind or one is talking non-mathematically, and in TO's case
non-logically into the bargain. There are no /number systems/ in
mathematics that are not, at least naively, set based.
>
> >
> > > It's my feeling that they are indeed sets, but of a different
> > > nature from discrete sets, until a discrete enumeration of them
> > > is derived, such as the H-riffic real number sequence.
> >
> > Your FEELING indeed. And of course the standard ordering is dense,
> > not discrete.
>
> True, but discrete orderings are indeed possible, just not
> "countable" in the Dedekind sense.
Does TO have any examples of uncountable but discretely ordered sets?
>
> >
> > > > However, another definition of infinite is: x is finite iff
> > > > there exists a natural number such that there exists a 1-1
> > > > function between x and the natural number. x is infinite iff x
> > > > is not finite.
> > > >
> > > > It turns out that if x is Dedekind infinite then x is infinite;
> > > > but the axiom of choice is used to show that if x is infinite
> > > > then x is Dedekind infinite.
> > >
> > > Personally, I take exception to that. While the set of finite
> > > naturals is Dedekind infinite in the sense that it can be
> > > bijected with a proper subset, the identity function between
> > > element count and value, coupled with the fact that all values
> > > are finite and that the value range is therefore finite, since
> > > the difference between two finites is always finite, means that
> > > there cannot be an infinite number of naturals in that finite
> > > range.
> >
> > You started out not so bad: "the set of finite naturals is Dedekind
> > infinite in the sense that it can be bijected with a proper
> > subset". Now take out the redundant 'finite' in 'finite naturals'
> > and take out the silly, gratuitious 'in the sense that' and replace
> > with 'since' and you have a true statement: the set of naturals is
> > Dedekind infinite since it can be bijected with a proper subset.
>
> That's fine for Dedekind, but given the fact that the naturals have
> exactly one element per unit of value, such that any 1-unit interval
> on the real line contains precisely 1 natural, an actually infinite
> number of them covers an infinite number of unit intervals, and
> therefore an infinite domain on the real line, such that there will
> be two points infinitely distant from each other. Both of these
> points cannot be in finite positions, if they are infinitely far from
> each other.
No two naturals are, or need to be, infinitely far apart, any more than
any two points on the standard real line (which has no endpoints) need
to be infinitely far apart for the line to be "infinitely long".
> One of them must be infinite. That's my quantitative
> reasoning
Only a TO would have the gall to call that reasoning.
> >
> > > There is only one natural per unit of element value, and the
> > > values only have a finite range. So, Dedekind-infinite does not
> > > always equate to truly infinite in my book.
> >
> > Your book is not a book of mathematics. No one needs to convince
> > you that Dedekind infinite is equivalent with your personal visions
> > of infinity. The definition of 'Dedekind infinite' sets a 1-place
> > predicate symbol as an abbreviation for a longer string of symbols.
> > It's not a question of whether that captures your intuitions of
> > what infinity is.
>
> For me, it is.
Then what To is doing is NOT mathematics.
> If a mathematical description of a problem does not
> give results that are expected, then it's not a good mathemtical
> description for the problem.
WRONG!
If a mathematical description of a problem does not
give results that are CORRECT, then it's not a good mathemtical
description for the problem.
Consider the problem of the precession of the perihelion of Mercury,
for example. The Newonian model gave what was expected, but not what was
correct!
It sometimes happens that the correct result is unexpected. Relativity
and quantum theory abound with instances.
> Transfinite set theory is described as
> "highly countintuitive", and the assumption is that no more intuitive
> theory of infinity can be devised, because infinity is simply
> counterintuitive. Well, for someone who has developed his intutitions
> regarding infinity by looking at a number of different approaches
> over the years, that statement is unsatisfactory.
It is nevertheless correct.
> Yes, for me, it is
> a matter of concern if a system of axioms produces conclusions that
> appear patently false, and reason to try to formulate a better
> system.
Better men (and women) than TO will ever be have struggled with this,
and their best conclusions are either (1) that any infinity is
counterintuitive, or (2) there is no such thing as an infinity.
>
> >
> > > The solution is to allow infinite natural numbers, which are NOT
> > > prohibited by the Peano axioms.
> >
> > The Peano axioms don't define the natural numbers.
> Combined with incrementation between successive elements, they do.
The set of natural numbers, as delimited by the 5th Peano axiom,
excludes anything other than naturals finitely generated from the first
natural.
> >
> > > > Moreover, the proof of the Banach-Tarski theorem does not use
> > > > consequences of definitions of infinity (in the way Orlow would
> > > > have us believe), but the proof of the Banach-Tarski theorem
> > > > uses the axiom of choice. The theorem stands in the primitive
> > > > language of set theory irrespective of ANY definitions. Indeed,
> > > > mathematical definitions do not have consequences other than to
> > > > establish abbreviations. Definitions do not affect the class of
> > > > models for a first order theory in its primitive language, so
> > > > definitions do not have the kinds of implications Orlow thinks
> > > > they have (he is still unfamiliar with the basic concepts and
> > > > the particulars of the axiomatic method and should not be
> > > > giving misinformation about it). Any structure remains a model
> > > > of a theory in its primitive language or not a model of the
> > > > theory in its primitive language irrespective of any
> > > > definitions that extend the language of the theory. That Orlow
> > > > thinks the definition of 'infinity' has any material bearing on
> > > > the Banach-Tarski theorem is a function of Orlow's lack of
> > > > familiarity with the basics of set theory, and even more
> > > > fundamentally, of the axiomatic method.
> > > >
> > > > Moreover, the Banach-Tarski theorem has to do with specifics
> > > > that are glossed over by Orlow's use of 'solid' and 'volume',
> > > > as these are UNdefined by him.
> >
> >
> > > There are perfectly well-understood notions of "solid" and
> > > "volume", such that it is not my responsibility to define them.
> >
> > That's really intellectually dishonorable. YOU claimed that the
> > Banach-Tarski theorem says something about solidity and volume. So
> > YOU are responsible for definining 'solidity' and 'volume' in
> > whatever sense YOU claim the Banach-Tarski theorem talks about
> > them.
>
> Perhaps, if mathemticians want to claim such things, THEY need to
> define their special use of terms
All the terms used in the BT theorem are quite adequately defined in
mathematics. But those definitions do not usually conform to TO's
misunderstanding and misusage of them.
, or make up entirely new terms, so
> as not to seem to be claiming magic. But, that IS what is claimed,
> and I'M dishonorable? Perhaps honor is relative.
Not to those who know what it means.
>
> >
> > > If you claim that Banach-Tarski has nothing to do with infinite
> > > sets, then why does the paradox only apply for an ideal
> > > infinitely-divisible solid?
> >
> > I don't claim that the theorem has nothing to do with infinite
> > sets. Since you didn't think about what I wrote, you have no idea
> > what I'm trying to get across to you. The Banach-Tarski theorem
> > depends on the axiom of choice. We don't need an axiom of choice
> > for finite sets, since provably each finite set has a choice
> > function. So of course the Banach-Tarski theorem wouldn't use the
> > axiom of choice if only finite sets were involved. What I was
> > trying to get across to you is that the DEFINITION of 'infinite' is
> > not what enables the Banach-Tarski theorem.
>
> No, it's the axioms regarding infinite sets that enable it.
What axioms are those?
> The
> definition either satisfies our notions of the subject or doesn't,
> and the Dedekind definition has problems for me this way.
Since mathematicians have no problem with that definition, the problem
with TO's understaanding and not in the definition itself.
>
> > If set theory NEVER definied 'infinite', then the Banach-Tarski
> > theorem would still stand except everywhere 'infinite' appears,
> > instead would appear 'not bijectible with a natural number' or
> > 'bijectible with a proper subset', depending on the context, or
> > some equivalent of those. The WORD 'infinite' is irrelevent except
> > as a heuristic. If set theory never uttered the word 'infinite',
> > then we'd still have every single theorem, except in more primitive
> > formulation. This is one of the many things that you just cannot
> > get through your obtuse skull: Definitions are NOT CREATIVE. That
> > means that a defintion does NOT enable new theorems that aren't
> > already theorems in the notation before the definition is
> > introduced. NOT ONE, NONE, NO THEOREM depends on definitions except
> > in the sense that the definitions allow the formulas to be shorter.
> > If we had to write the formulas in full primitive notation, with NO
> > definitions AT ALL, then we'd have EXACTLY an EQUIVALENT set of
> > theorems, including the Banach-Tarski theorem.
>
> Sure. So the set of finite naturals is Orlow-finite, as per my
> definitions. That doesn't affect your set theory at all does it?
As long as Orlow makes clear in every usage of "Orlow-finite" that Orlow
is using Orlow's own definition, there is no problem. But when Orlow
uses in a mathematical context a word that has an agreed upon meaning in
such contests in a way conflicting with that meaning, Orlow is being
anti-mathematical.
> >
> > > The proof is not possible with a collection of finite-size and
> > > therefore finitely many particles, but rests on notions of
> > > infinity within set theory.
> >
> > No it does NOT. Just as I explained above. The proof rests on first
> > order logic and the axioms of set theory with the axiom of choice.
> > No "notions" of infinity are invoked. The proof is a sequence of
> > symbols. If we had NO IDEA IN THE WORLD what those chicken
> > scratches are about, they'd still be a proof. You still don't
> > understand this after all these thousands and thousands and
> > thousands of posts.
>
> Every axiom states some kind of fact or rule which is based on
> "notions" or ideas regarding the subject. On the symbolic level, it
> doesn't matter what they "mean", as long as we know what implies
> what. How one decides what axioms to include depends greatly on their
> ideas regarding the problem.
If TO thinks he can produce a better set of axioms, he should spend his
energies in that direction and quit wasting them by polluting this NG
with his nonsense.
>
> >
> > > I didn't get into some of the roots of the most egregious errors
> > > in the theory.
> >
> > Because there are no "errors" that anyone knows of. The only thing
> > that can be an error is an inconsistency. No one has shown one.
>
> I agree there is no identifiable internal inconsistency in set
> theory
So the various set theories will stand as is until someone does find
some internal inconsistency.
> and so cannot be called "wrong" in the pure mathemtical
> sense.
As set theory is about as "pure" as mathematics ever gets, TO has just
confessed that set theory is all right!
> > Your point whatever it was, led you to provide misinformation as to
> > what the definition of 'Dedekind infinite' is and to blather,
> > completely uninformed and misinforminlgy about set theory including
> > Banach-Tarski.
>
> Banach-Tarski is proof by contradiction that set theory is messed up.
> QED ;)
Not at all, it is only proof that the combination of set theory and
Cartesian geometry used to prove BT do not entirely correspond to
physical reality.
But only a fool expects mathematics to be a perfect mirror of the
physical world.
> I admitted in my first response to you that it was misstated but
> essentially equivalent, which it is for many sets, though perhaps not
> all. Apologies again. Sheesh! And, what is the import of
> Banach-Tarski, and why is it a paradox, if we have simply fluffed up
> the material so it's half the density? Is it a paradox if I make a
> marshmallow twice the size by throwing it in the microwave oven for
> 10 seconds? No. The paradox claims the balls are solid, and does not,
> unless I am mistaken, take into account any change in density any
> more than any other result of transfinite set theory does.
The "density" of the real line is in the same way infinite, in that
number_of_points_per_unit_of_length is infinite. This means that half as
dense is equally dense. This only can happen when density (number of
points divided by units of space (length, area or volume, as
appropriate) is infinite so that dividing by a finite number does not
decrease it.
Discrete ses have finite density, dense sets have infinite density.
> If
> transfinite set theory took into account set densities and such
> measures, I wouldn't have a problem with it.
It does! Half of an infinite density is still an infintie density!
>
> >
> > > Why did you snip that part? Did you have any thoughts on that?
> > > Apparently not.
> >
> > I don't always quote and respond to everything in every post. I'm
> > certainly not going to commit to responding to every one of your
> > ruminations. But since you're brought it up, YOU did not respond to
> > this:
> >
> > > > mathematical definitions do not have consequences other than to
> > > > establish abbreviations. Definitions do not affect the class of
> > > > models for a first order theory in its primitive language, so
> > > > definitions do not have the kinds of implications Orlow thinks
> > > > they have (he is still unfamiliar with the basic concepts and
> > > > the particulars of the axiomatic method and should not be
> > > > giving misinformation about it). Any structure remains a model
> > > > of a theory in its primitive language or not a model of the
> > > > theory in its primitive language irrespective of any
> > > > definitions that extend the language of the theory.
>
> I don't think I really got your gist at the time. You seem to be
> distinguishing definitions from axioms, and saying definitions
> themselves don't change conclusions.
They don't.
> But, aren't axioms stated for
> the defined objects
The "objects" of the Peano axioms, which we usually come to call
naturals numbers, are undefined by those axioms even though there are
un=mbued with ceratin properties by those axioms. Sometimes they are
given concrete set representations as in the von Neumann theory, but
that is not inherent in the axioms themselves.
> and don't the kind of axioms one could form
> depend on what objects are defined?
You have it backwards. The kind of objects that will be allowed depends
on whether the objects have the properties required by the axioms.
> In fact, aren't many definitions
> actually axioms?
No!
> In Peano, is 0 defined?
No! At least not beyond being something that exists mathematically (but
not necessarily physically) and is not the "successor" of anything in a
particular set of objects.
> it would apear to be the
> natural that is not a successor.
At least not of some things.
> So, I am not sure I get the
> distinction, probably because I am thinking on a more concrete level
> than you are regarding this.
Numbers are about as unconcrete as things get.
>
> >
> > > I just saw an article where someone tried to make Banach-Tarski
> > > intuitive, by suggesting that the remaining balls somehow had
> > > half the density of the original. If set theory took into account
> > > this effect, then I would consider it perhaps a fair answer, but
> > > it does not, now, does it? So, perhaps I didn't represent set
> > > theory as you would like, but I still don't think anything I said
> > > was a gross mischaracterization either.
> >
> > It's not a question of how I like it. Your definition of 'Dedekind
> > infinite' is incorrect. And your comments about Banach-Tarski are
> > egregiously misinformative too.
> >
> > Please get a good book on mathematical logic so that at least you
> > can START to know at least SOMETHING about what you have been
> > spending thousands and thousands of post talking about while
> > knowing virtually NOTHING about it.
>
> I may not be steeped in set theory as such, but that is not the only
> approach to the concept of infinity. Sorry to have misstated a
> definition, but it doesn't really matter, now, does it, since
> definitions don't change the conclusions of an axiomatic system
> anyway, right?
They do if you get them wrong! If one has the wrong definitions of 5
then 2 + 2 = 5 could be a theorem.
.
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