Re: Infinity
- From: Tony Orlow <aeo6@xxxxxxxxxxx>
- Date: Tue, 10 Jan 2006 12:59:12 -0500
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.
> 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?
>
> > 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, and so, I am not sure there is any non-
sequentially orderable set that is really a set of comparable values.
>
> > However, there are at least two different approaches
> > whereby the contnuum can be viewed as sequentially ordered points.
>
> With the axiom of choice, since there is a well ordering of the reals,
> a fortiori, there is a linear ordering of the reals. I haven't thought
> about whether, without the axiom of choice, we can prove that there is
> a linear ordering of the reals. But I am not aware that you've proven
> it.
I haven't given a formal proof that the H-riffics cover all reals, but they do.
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, 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.
>
> > 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.
>
> > > 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. One of them must be infinite. That's my quantitative
reasoning, as opposed to Dedekind's set-theoretic approach.
>
> > 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. 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. 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. 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.
>
> > 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. Perhaps you
would like to expound on that?
>
> > > 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, 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.
>
> > 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. The definition
either satisfies our notions of the subject or doesn't, and the Dedekind
definition has problems for me this way.
> 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?
>
> > 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.
>
> > 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, and so
cannot be called "wrong" in the pure mathemtical sense. But, it appears to me
to contradict conclusions drawn from other areas of math and general intuition,
to an alarming degree. For me, it's beautiful when two different mathemtical
approaches reach the same goal and agree. It makes the world seem universally
consistent, and that's good. This theory is a big hole in that fabric, in my
eyes. Externally, it doesn't play well with others, even if it's internally
healthy.
>
> > My point was to direct the naive inquirer to
> > the geometric concepts of infinity,
>
> You have the gall to use the word 'naive' not self-referentially.
Sure, I'm naive. So are you in ways. Everyone has their expertise and naivete.
Oh, and I also have gall too. Don't you? :)
>
> 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 ;)
>
> > before statements of set-theoretic "fact"
> > were delivered as gospel, so that they would have a more well-rounded answer
> > than the standard creed.
>
> If you have a different definition of 'infinite', then you're welcome
> to give it, but you are intellectually dishonest to negligently
> misrepresent a common definition such as that of 'Dedekind infinite'
> and to continue to try to weasel out of just admitting that you
> mistated the definition, and then to misrepresent the import of
> Banach-Tarski by getting it all mixed up with your own UNdefined use of
> 'solid' and 'volume' and to incorrectly represent that the theorem
> depends on the definition of 'infinity' (see comments above).
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. If transfinite set theory took
into account set densities and such measures, I wouldn't have a problem with
it.
>
> > 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. But, aren't axioms stated for the defined objects, and don't the
kind of axioms one could form depend on what objects are defined? In fact,
aren't many definitions actually axioms? In Peano, is 0 defined? it would apear
to be the natural that is not a successor. So, I am not sure I get the
distinction, probably because I am thinking on a more concrete level than you
are regarding this.
>
> > 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? Isn't that what you have just
gone on and on about? ;)
The unit interval is the real line divided by N.
>
> MoeBlee
>
> P.S. If subsequent replies by me seem to ignore intervening relevent
> posts, it's probably due to the fact that, for some reason, some of my
> posts are not posted for many hours after I've submitted them.
That must be annoying. I would just have assumed you didn't care any more, and
wept. :(
Anyway, have a good one.
>
>
--
Smiles,
Tony
.
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