Guido "Wugi" Wuyts @ Dilbeek, Belgium, Europe, World, Solar System, Milky Way, Local Cluster, ...

Faster than light?

> > History shows that:
Science considered speed of over 30 kph would be deadly for the human body...
Until the railroad was invented.

> That is not true.

> > Science considered moving faster than acoustic waves would be impossible for mankind...
Until the jet plane was invented.

> That is not true.

> > Science considered moving faster than light would be impossible for any matter...
Until ... I don't know either, but remember that although our race is gathering knowledge
faster than ever before, we don't know yet everything!

> Please, you will have to try harder than that.

OK, let's have a try.

If you visualise the world line of a faster than light particle in a Minkowsky diagram you will notice
that it would coincide with the space axis of some inertial system. This implies that in one particular
inertial system such a particle would actually have to appear as an instantaneous, one moment phenomenon.

I myself like to picture mass as a nest of light clocks, i.e. a web of radiation that goes to and fro within
a tiny volume that could be associated with this moving mass. Now, this description by itself seems to be
indifferent whether you evolve it along a time-like axis (where it would describe subluminal mass) or
along a space-like (where it would describe a tachyon).

There are some snags however. Firstly, as you leave a one-dimensional space-description for a two- or three-
dimensional one, you realise that there is no symmetry when reverting the roles between one time-axis
and three space axes. So the objects described would certainly differ highly in nature.

Secondly, there is the phenomenon of phase correlation, that would apply to any pair of e.g. photons having
been "together" somewhere "in the past". The light clock concatenation process that would go with ordinary
matter ensures conservation of this phase correlation between any succession of a "coming" and "going"
photon. But the process covering tachyons would have to cope with, from our point of view, the meeting
between two uncorrelated photons (or light cones), in alternation with the departure of two correlated ones. So
this concatenation would not transmit phase correlation. A good reason perhaps for considering tachyons as an
impossibility.

Faster than light, and space expansion?

Could somebody enlighten me in general terms on the following questions I have been brooding about.

1. How do they explain the possibility of ultraluminal (more than light speed) recession speed between remote
objects, or else : space expansion exceeding light speed?

> The light-speed restriction in relativity merely requires that one
(timelike) observer will not measure another (timelike) observer as having
a speed greater than 'c' with respect to it, at an event at which their
worldlines cross. Thus, it says nothing about the relative velocities
of distant objects. In fact, you can't even _define_ the relative
velocities of distant objects in a unique way in general relativity.

2. Couldn't it be that light speed precisely mirrors the (local) space expansion rate, so that matter is doomed to
"surfing" somewhere between these expansion limits?

> I don't understand this question.

3. How does one explain that space expansion has to occur between galaxies, but obviously is not doing so
within their boundaries, as they don't seem to be flying apart? Again, why does matter not fly apart with space
expansion (I would understand some surfing process as I mentioned above)?

> 3.1. Because space is only expanding _on the average_. On the average, the
matter in the universe is spaced out quite a bit. But there are little
clumps of denser-than-average material, called galaxies, which are dense
enough to locally halt or reverse the expansion due to gravitational
attraction -- within that extra-dense region, of course. But on the
large scale, things are expanding, and the extra-dense galactic regions
recede from each other.

>> 3.2. I suggested in a recent posting to this newgroup that one can picture the
expansion of the universe by pasting pennies on a balloon and inflating it--
the pennies uniformly move apart from each other but do not change their size.

This balloon model describes the "matter dominated closed universe Friedmann
solution" to the field equations of GTR. The basic assumption made in deriving
this model universe is that matter consists of "dust particles" which interact
ONLY via gravitation and which moreover at all times possesses a spherically
symmetric constant density (with respect to any one dust particle).

Small bodies like people and skyscrapers are held together by electrical
forces (gravitational forces are present but much weaker than the electrostatic
forces between atoms, so they are overwhelmed by the latter forces). Thus
at these scale levels, you clearly cannot treat matter as dust particles interacting
only via gravitational forces.

Gravity IS the dominant force even at galactic scales, so it's a little trickier
to explain why galaxies do not expand at the same rate as the Hubble expansion.
It is clear that explaining this is the same thing as explaining why the stars
in a galaxy cannot be modeled (even as a first approximation) as dust particles
even though they interact (essentially) only via gravitity. I would guess that
the explanation involves the fact that galaxies are ROTATING about definite
"center"; moreover, many galaxies are roughly disk shaped and all have a definite
stellar density gradient which increases as you move toward the center. So the
symmetry assumptions made in deriving the Hubble expansion in GTR are grossly
violated.

However, not that it is best to think of the expansion of the universe as occuring
everywhere all the time; this expansion is simply not observed at small scales.
But it would be incorrect to think of "more cubic meters" being inserted into
the universe only in between galaxies. Rather, think of the balloon expanding
and the pennies resisting that expansion just enough to maintain their size and
shape.

(Your first two questions were too imprecisely formulated for me to understand
what you were asking.)

Relativistic mass.

> I have two quick questions.
My first question is this, does the E=m*C^2 equation say that the total
energy of an object (the rest energy plus the kinetic energy) is equal to
m*C^2 (for a given reference frame)? I am pretty sure that the answer to
this first question is yes, but I just want to verify this.

Talking about relativistic mass, it may be awarding to realise that the difference between rest mass and moving
mass is rather dim. There is no such simple thing as "objects" to attribute mass to. It is more realistic to view
relativistic -and otherwise- objects always as composite systems including endowed with internal movement. So
the "rest mass" of an atom comprises the relativistic masses of the electrons whirling about as well as of the
nucleons dancing within the nucleus.

My favourite example is light clocks, ie photons going to and fro. Photons have no rest mass, yet they do have
energy and so, relativistic mass. The light clock system itself behaves as an ordinary "object having mass",
including rest mass in the system where it is "at rest".

I have come myself, in fact, to view all manifestation of matter as a feature of light clock colonies, and mass as
a "trick" of photon energy to travel light-clockwise, surfing on the space-time tissue as it were, at infraluminal
speeds! ....

Time dilatation and twin paradox.

> Before I begin I would like to explain that I am actually a biology major,
and my understanding of physics is thus quite limited. I see what seems
to be an apparent contradiction with the relativity theory. I am sure I
am wrong, but I don't know why. This is my problem:

According to relativity if two people on different ships (we'll call them
A and B) are moving apart from eachother at a rate nearly the spead of
light, person A can say he/she is stationary and person B is moving away
at nearly the spead of light, while person B could say that they are
stationary and person A is moving away at nearly the spead of light.

Another theory, I believe it is special relativity, indicates that if a
rocket leaves the Earth and travels at nearly the spead of light for one
hour on the ships clock, then returns, the people on Earth will be much
older then the person on the ship who has only aged by one hour.

So in the case of the two rockets moving away from eachother, who is
moving at nearly the spead of light depends on the frame of reference,
thus who time is moving slower seems to depend on the frame of reference
as well. Who actually gets older? With the ship leaving the Earth,
couldn't the person on the ship say he remained stationary while the Earth
moved at close to the spead of light? In this case time would move much
more slowly for people on earth, and when the Earth returned to the rocket
the person on the rokect would be much older.

I hope I am making my confusion clear. I would guess that there is just
some key piece or pieces of information I don't know or have overlooked.
Any responses would be greatly apprecitated. I am sort of an idiot so I
would like to request that any responses be worded as though you are
speaking to a child so I have some chance of understanding it. Thank you
very much.

1. (Me) Your relativity problem can be solved by disentangling the two separate paradoxes it entails. It is not the place here to detail fully the theory, "relatively" simple though it is. Only special relativity is involved anyway.

The first paradox is: how can two observers in motion among each other, observe each other's time lagging behind "at the same time"? But you can picture the result by visualising a pair of scissors. Consider that the end points of both cutting edges represent two world points of the first observer, and that the two fingergrip ends represent likewise two world points of the second observer.

If you hold the scissors a little open along a horizontal line in front of your eyes, we will assume that both lower points (cutting and fingergrip) represent a clock tick for each observer, say "one second". Then the legs will link each observer's clock event with another event noted by the other observer, which represents the "lag" of the other one's clock with respect to "his own clock". The situation is symmetrical, as both legs of the scissors link a lower, say previous, point from observer 1, with a higher, say later, point of observer 2.

In fact, the scissor legs represent space axes, or lines of simultaneity, of each observer (the one whose "1 second" point lays on the leg). Whilst the cutting and, otherwise, the fingergrip points represent world lines, or time axes, of both observers. Once you get the picture you realise how two observers can see each other's time system lag behind "simultaneously".

Now for the second paradox, which is the so-called twin paradox. The difference here is that the situation is no longer symmetrical! The observer staying on earth, means that he never leaves his (supposedly inertial) frame of reference. The travelling partner, however, at least has to switch between two inertial frames of reference: a parting one and a returning one. Around the turning point his perception of the world changes abruptly, as his time axis suddenly tilts from a position where it used to point towards his partner's past, to a new one pointing towards his partner's future.

If the change is abrupt (without deceleration and acceleration processes which are even trickier as they leave the framework of special relativity...), a whole episode of the partner's lifetime is "lost" in the time system of the traveller. As he returns home he finds his partner older by the amount of time "lost" in the description by his own framework.

2. (> third party) I think this question is best handled by responding to two
sub-problems in turn.

First, how is it that two people can each
think the other's clock is ticking slowly. That's the two-clocks-
receeding-from-each-other case.
For a discussion of this, try
http://sheol.org/throopw/sr-ticks-n-bricks.html
or http://shoel.org/throopw/sr-superbowl.html
The first one compares clock ticks and time coordinates to
wall bricks and x/y coordinates, to illustrate the issue, and then
continues on to reference the second half of the problem.
The second one answers a specific question about clocks *approaching*
each other in some detail, and compares time coordinates to football yardage.

Then second, if each can see the other ticking slower, how is it decided
which one "really" loses time. The major point here is, moving
"inertially" in space (that is, not accelerating) is modeled in SR by
the same math that models straight lines in geometry. Thus, just as we
know that the bricklayer that lays the crooked wall will have a longer
wall than the one who lays a straight wall, even though at any point
either bricklayer can say their own bricks span more distance (per point
1 above), we also know that the person who accelerates (between
separating and reuniting) will have a shorter elapsed time than the
person who does not. The core of the matter is this:

The person on the ship cannot say he *remained* stationary. Because
Einstein only claimed that *uniform* motion (motion with no acceleration,
no "curves" in spacetime) is relative. So, we can decide that the earth
is motionless throughout, or we can decide that the earth moves near
lightspeed throughout. But we can't say that the rocket moved at a
single, constant rate throughout, whether that rate is zero or
near lightspeed.

The second half of http://sheol.org/throopw/sr-ticks-n-bricks.html
illustrates this diagramatically. There is also a long (and very good)
second of the relativity FAQ dealing specifically with the twin paradox at
http://www.public.iastate.edu/~physics/sci.physics/faq/twin_paradox.html

By the way, if you see a way to improve or simplify the presentation
on the mutual dilation puzzlement in the first half of the ticks 'n bricks
page, or the comparison of the bricklayer's paradox in the second half,
let me know. I'd be glad to improve it if I can.

In general, the comparison of SR to simple plane geometry is an
excellent "intuition pump" to use when considering SR "paradoxes".
Usually what's going on in such "paradoxes" is that the relativity of
simultaneity is overlooked, which is like doing a rotation of one axis
in geometry and forgetting the other one. When your axes are no longer
at right angles, you get weird behaviors from the usual geometrical
formulae; in SR, when your space and time axes no longer follow the "law
of propogation of light", you get weird behaviors from the usual time
dilation / length contraction formulae.

Galaxy time and universal time; time dilatation and expansion rate.

> Can someone answer the following as it has been puzzeling me for quite
a while:-
The galaxies are moving away from the centre of the big bang at
different rates therefore according to the time dilation theory they
must be existing and evolving at different time rates.

(no input of mine. Just interested)

>> It all depends on how you think of it.
*** sigh ***
Not sure I feel up to trying to explain this in detail for the tenth time.

The simple cosmological model you have in mind is no doubt the matter
dominated Friedmann solution. In fact, because these solutions (4 dimensional
curved spacetimes) happen to posses a family of "spatial" (3-dimensional)
slices which are respectively 3-spheres, euclidean planes, or hyperbolic planes,
there is a notion of "universal time" defined for bodies (idealized galaxies)
following world lines with constant "spatial" coordinates. That is, the
"universal time" is defined by the proper time of the galaxies.

The point is that each galaxy observes the same effects as any other galaxy.

It is impossible to overestimate the extreme idealization of the Friedmann
models, in which the universe is imagined to consist of a uniformly expanding
"dust" of point masses, each of which represents a galaxy. A "dust" is
a perfect fluid in which the fluid elements exert no pressure on each other;
that is, the only forces present in this idealized universe are gravitional.

> Now, as the big bang theory suggests there must come a time when this
expansion slows down and, prior to contraction, universal matter must
come to absolute rest.,ie zero velocity.

>> In all of the Friedmann models, the expansion is continuously slowing.
Only in the "closed" Friedmann models (spherical spatial slices) does the
expansion halt after finite galactic proper time.

The "closed" model can be visualized as a football (the circles of latitude
represent the spherical spatial slices; the "equator" of the football
reprsents "space" at the moment where the galaxies are momentarily motionless
to respect to each other before the contract phase begins). The paths
taken by light rays spiral up and around the football in such a way that
their projection along the axis of the football are cardiods ("heart-shaped"
curves).

But again, re Klaus Kassner's recent post, the 3 dimensional euclidean space
in which this model football universe is embedded is only an artifact with
no physical significance.

> Does this mean there is an infinite increase in the rate of time flow, If time dilates with
velocity it must surely expand with a decrease in velocity or am I
missing something?

>> You are confused. During the expansion phase, each galaxy observes light
emitted by other galaxies to be red shifted. During the contraction phase,
such light appears blue shifted.

Distances and age of the universe.

> I read recently that scientists have discovered a galaxy(ies) as far away
as 14 billion light years. That means the light left the galaxy (and that
point in space) 14 billion years ago and travelled a distance of 14 billion
light years to reach us here today. Assuming (as most believe) that the
universe is no older than 10-15 billion years and assuming the correctness
of big bang theory (all matter exploding outward from an initially
centralized mass), how is this possible?

>> In cosmology, there are several different types of distances, the
distance from A to B is not necessarily the same as the distance from B
to A, distances can DECREASE with INcreasing redshift, and the distance
from A to C is not necessarily the sum of the distances from A to B and
from B to C even if B is `right between' A and C. Also bear in mind
that the actual distance is, for high redshift objects, calculated from
the redshift and the cosmological parameters. Since we don't know the
cosmological parameters very accurately, distances at large redshift are
uncertain by a factor of 2 or so, even if you know what distance you're
talking about. So it's a quite complicated issue but there is not
necessarily something wrong with the statement above (though I would
have phrased it quite differently).

> If nothing travels faster than
the speed of light (and presumably galaxies travel and the universe expands
at a much slower rate),

>> The speed of light is an upper limit in the Minkowski space of special
relativity. In cosmology, where Minkowski space is not necessarily a
good global approximation, this simply does not apply.

A good introduction to both of these issues (and many more interesting
aspects of cosmology, with an emphasis on things which are often
misunderstood and misconstrued in the popular press) is the wonderful book by

Edward R. Harrison: COSMOLOGY, THE SCIENCE OF THE UNIVERSE
(Cambridge University Press, 1981).

If you read only one book about cosmology, this should be it.

At a slightly more technical level, a couple of colleagues and I have
recently published a paper which has two goals:

1) gather together, in one place and with a uniform, understandable and nice notation, all
definitions of cosmological distances and discuss the interrelationships
among them and

2) present an efficient numerical method (and publicly
available code:) to calculate distances in a very general manner (i.e.
not limited to certain special cases).

Both goals also consider the effect of inhomogeneity on distances. Although it is a good
approximation at large and universal scales to assume the universe is
homogeneous, at smaller (angular) scales, comparable to those at which
measurements of high redshift objects are made, this is, depending on
what you're doing, not necessarily a good approximation.

Our paper appeared in Astronomy and Astrophysics Vol. 318, No. 3, pages
680-686 (issue February III 1997). See also the URL
http://hsvax1.hs.uni-hamburg.de:8000/helbig/research/publications/info/angsiz.html

A new english spelling.

>
=======================================
PONDER THIS.....HUMOR........EUROENGLISH
=======================================

The European Commission has just announced an agreement whereby
English will be the official language of the EU rather than German,
which was the other possibility. As part of the negotiations Her
Majesty's Government conceded that English spelling had some room
for improvement and has accepted a 5 year phase-in plan that would be
known as 'Euro-English'.

In the first year 's' will replace the soft 'c'. Sertainly, this
will make the sivil servants jump with joy. The hard 'C' will be dropped
in favour of the 'k'. This should klear up konfusion and keyboards kan
have one less letter. There will be growing publik enthusiasm in the
sekond year when the troublesome 'ph' will be replased with the 'f'.
This will make words like 'fotograf' 20% shorter.

In the 3rd year, publik akseptanse of the new spelling kan be
expected to reach the stage where more komplikated changes are possible.
Governments will enkorage the removal of double leters which have
always ben a deterent to akurate speling. Also al wil agre that the
horible mes of the silent 'e' in the languag is disgraseful and it
should go away. By the 4th yer peopl wil be reseptiv to steps such
as replasing 'th' with 'z' and 'w' with 'v'. During ze fifz yer ze
unesesary 'o' kan be dropd from vords kontaining 'ou' and similar
changes vud of kors be aplid to ozer kombinations of leters. After
ziz fifz yer ve vil hav a rali sensibl riten styl. Zer vil be no mor
truble or difikultis and evrivun vil find it ezi tu understand ech
ozer.

ZE DREM VIL FINALI KUM TRU!!

Very funny indeed. Still, it is also possible to consider seriously the possibility
of a corrected English spelling. I did so, and below you'll find some samples
of English writing following "my" rules.

Inglish saampls.

Dhe cheif difranses bitwein Eithieupian and Reuman Caetholic Churches.

Dhe Jhezueits thoaht dhaet sym elimants ov Eithieupian Cristiaeniti caim from dhe
Jhous e.g. rouls abaut meit, dhe Saebath, dhe jouz ov Taebots etc. Dhis woz partli
trou.
Dhe Jhezueits disaprouvd strongli ov dhe Emprar being hed ov dhe Church.
Meust important wer difranses euvr dhe naitjr ov Jheizus Cryist's bodi- hau mych
woz hjouman and hau mych divyin. Dhe Ei. Church iz monophizyit.

Importans ov dhe mishons.
Dhe Jhezueits hindrd dhe wurk ov rikyvri aaftr Graen, and dhe wurk ov
consolidaiting dhe Empyir against Turkish and Gaela ataecs.
Dhiz coazd rivolts and lesnd rispect for dhe Emprars.
Faesilidas gaind grait presteizh.
Meust important, it led tu yisolaishonizm. Faesilidas aaskd dhe Turks and Muslim
roulrs tu cleuz dheir ports tu Portjugeiz or Jhezueits and dhei did seu.

Dhe Gondar Empyir. Introdycshon.
Gondar iz important bikyz for dhe first tyim for 900 jiars dhear woz a setld
caepital in Eithieupia.
It woz faundid byi dhe Emprar Faesilidas aez a dilibrat part ov hiz polici tu
rei-junyit church and stait, hwich haed bein torn apart in dhe relighos strygls ov
dhe preivios 30 jiars.
Dhe haenging ov hiz eun yncl, Se'aala Kristos (for conspiraci widhdhe Caetholic
paitriarc Mendez ixpeld in 1663), sheud hau necisari it woz tu heil dhe divizhons
left byi dhe convershon ov hiz faadhr tu Caetholicizm.

Drugs.

Apart from dheiz scyiantificalli efectiv poizns, hwich dhei pripear eupnli widhaut
eni ov dhe maeghical pricoashons and complicaishons hwich syraund dhe maiking ov
cjuraarei a litl furdhr north, dhe Nambikwaara haev ydhr, mor mistiarios poizns. In
tjoubs yidentical tu dheuz containing dhe trou poizns, dhei keip particls ov rezin
exjoudid byi a trei ov dhe bombaex speishi, dhe trync ov hwich iz sweuln in dhe
midl; dhei bileiv dhaet byi threuing a particl ov dhis rezin on tu an opeunant,
dhei wil coaz a phizical condishon similar tu dhaet ov dhe trei- in ydhr wurds,
dhaet dhe victim wil swel yp and dyi. Dhe Naembikwaara haev eunli wyn wurd, nandé,
hwich dhei jouz beuth for trou poizns and for maeghical sybstansies. Conseiqwantli,
dhe term haez a wyidr raingh ov meinings dhaen aur wurd poizn. It dineutz eni kyind
ov thretning aecshon, az wel az prodycts or objhects lyikli tu be jouzd in sych
aecshons.

Dheiz preliminari explanaishons ar necisari for dhe yndrstaending ov dhe rest ov
myi stoari. Yi haed broaht widh me a fjou ov dhe mylticolrd balouns maid ov
tisjou-paipr, hwich ar fild widh hot ear byi meins ov a litl torch fixd yndrneith,
and hwich dhe Brazilians rileis in dheir hyndrids on Midsymr's Dai. Wyn eivning Yi
haed dhe ynfortjunat yidia ov sheuing dhe Indians hau dhei wurkd. Dhe first baloun
coaht fyir on dhe graund, and dhis coazd grait hilaeriti, az if dhe naitivs haed
sym yidia ov hwot oaht to haev haepnd. Dhe secnd, hauevr, sycceidid eunli tou wel:
it reuz qwicli intu dhe ear, reichd sych a hyiht dhaet its flaim merghd widh dhe
stars, driftid abaut abyv ys for a long tyim and dhen disapiard. Byt dhe inishal
mirth haed givn wai tu ydhr imeushons; dhe men wochd intentli and aengrili, wyil
dhe wimin hid dheir faisies in dhe cruk ov dheir arms and hydld against eich ydhr
in fiar.

Jheivs.

It'z rymi hau sleiping on a thing oftn maikz jou feil qwyit difrant abaut it. It'z
haepnd tu me euvr and euvr again. Symhau or ydhr, wen yi weuk next morning dhe euld
hart did'nt feil haaf seu breukn az it haed dyn. It woz a perfectli toping dai, and
dhear woz symthing abaut dhe wai dhe syn caim in aet dhe windeu and dhe rau dhe
birds wer kiking yp in dhe yivi dhaet maid me haaf wyndr hwedhr Jheivs woz'nt
ryiht. Aaftr oal, dheuh she haed a wyndrful preofyil, woz it sych a caech being
engaighd tu Florans Nyihtingail az dhe caezhual observr myiht imaeghin? Woz'nt
dhear symthing in wot Jheivs haed sed abaut her caeractr? Yi bigaen tu rialyiz
dhaet myi yidial wyif woz symthing qwyit difrant, symthing a lot mor clinging and
drouping and praetling, and wot not.
Yi haed got az far az dhis in thinking dhe thing aut wen dhaet "Tyips ov ethical
thiori" coaht myi yih. Yi eupnd it, and yi giv jou myi honist wurd dhis woz wot hit
me:
Ov dhe twou aentithetic terms in dhe Greik philosophi wyn eunli woz rial
and self-sybsisting; and dhaet wyn woz yidial thoaht az opeuzd tu dhaet hwich it
haez tu penetrait and meuld. Dhe ydhr, corisponding tu aur Naitjr, woz in itself
phenominal, ynrial, widhaut eni permanant futing, haeving neu predicats dhaet held
trou for twou meumants tugedhr; in short, rideimd from negaishon eunli byi
inclouding in-dweling riaelities apiaring throuh.
Wel -yi mein tu sai- wot? And Nietzsche, from oal acaunts, a lot wurs dhaen dhaet!
"Jheivs", yi sed, wen he caim in widh myi morning tei, "yi'v bein thinking it euvr.
Jou'r ingaighd again."
"Thaenk jou, Sir".
Yi sykd daun a chiarful mauthful. A grait rispect for dhis bleuk's jhyghmant bigaen
tu seuk throuh me.

Infinity and countability.

I have this problem with the distinction between countable-noncountable infinities.
What does a countable series mean anyway? And yet, some theorems such as Gödel's eagerly call for countable sets, eg of theorems. In my opinion countability suffers at least two big restrictions of applicability:

1. it just is a the result of a particular way of ordering a set, albeit a happy one, amidst a sea of less such happy ways;

2. it postpones the enumeration of elements with ever more complicated description, ever more ahead (and in the end scrambles the border of distinction with related domains of noncountable elements).

The set of natural numbers is the basic model, as a matter of fact the criterion for comparison of countable sets. However, it takes a minimal mix-up to transform them in kind of an unoverseeable, uncountable desert. Take as a rule of enumeration: first the evens, then the odds. The "then" implies a never ending enumeration that has to precede a next one.

The rational numbers are proven countable thanks to a cunning way of diagonal counting. But taken in the natural order, any "finitely describable" number has no such neighbour, so there is no enumeration in finite terms along this order.

In the case of natural and rational numbers we don't worry about that, since we know some ways "that work". But what with establishing a supposedly countable set of theorems as Gödel would like?

Let us try a "clever" way of ordering theorems, by grouping for instance theorems that would, given a number A, perform an operation with A+n. Each number n (natural or, worse, real) would determine a theorem Tn. In doing so, we obtain a series T0, T1, T2, ... , Tn, ... from which there is no escape before being allowed to continue the enumeration with theorems of a "next" type.

OK then, let's try a "dumb" method. We accept all ASCII codes, ordered in a definable way and then proceed by eliminating the ones which do not represent theorems. First, how definable may that ordering be, eg how long may the lines become, and how many...? Second, we now have to establish whether a code is a theorem, which seems to counter our intention to produce theorems mechanically. Well, at least we can establish whether it respects syntax. My point is that this way we will doubtlessly stumble upon endless subsets of meaningless code before encountering a "next" code-with-a-theorem.

My argument leads to the assertion that there is no such thing as a well-ordered set of theorems, supposedly representing the human mind about some theory, and out of which Gödel would then playfully pick a non-provable theorem to prove incompleteness. The end result of his statement remains the same of course, but it is looked upon in another way.

The noncountability of real numbers raises its own problems. The "power of infinity" (I don't know the proper English term) of R defined as a set of all decimals, or of R defined as a straight line in algbraic geometry, is fundamentally arbitrary, in that you make it as great or as small as you wish to, and moreover both representations of R are mutually independent. The power of the "real", physical line in space is what it is, probably unprobable, and independent of our mathematical representations of R (however neatly they seem to settle down on it).

My last problem is with limits, one of the tricks to approach the reals. If we state eg, that the limit of a series 1/2, 1/3, 1/4, ..., 1/n, ... equals zero, then I say to myself, no, it is at best a nest of real numbers that couldn't be disentangled, as crowded as you like, and amongst which zero is the only member describable in finite terms.

Discussions:

1. > (it postpones the enumeration of elements with ever more complicated description, ever more ahead)
If there is one way to enumerate, there are many, yes.
(and in the end scrambles the border of distinction with related domains of noncountable elements)
Ah!-- call the countable anything else, ... but not "uncountable"!

>> (My reply) But any "successful" enumeration has to push ahead in its tasklist the elements whose description becomes increasingly complex, eg the rationals that become too "real"-like, the elements bordering the non-rational domain...

>>> (I) The zig-zag through m/n's ? The higher the m, n the later you get there; why is that a problem?

If every x in A shows up as an f(k), f is an enumeration of A; if not, it isn't. There aren't any shades of grey in between! (Maybe you have in mind an example where it is not easy to tell -- but it still is a yes or no issue).

> >The set of natural numbers is the basic model, as a matter of fact the criterion for comparison of countable sets. However, it takes a minimal mix-up to transform them in kind of an unoverseeable, uncountable...

>>> Hey... you said the "U" word again...

>> ... desert. Take as a rule of enumeration: first the evens, then the odds. The "then" implies a never ending enumeration that has to precede a next one.

>>> One mentally goes over the first part, then the next... -- like you just did; you didn't stumble or get confused either.

>>>> But you cannot mechanize this letting your program "mentally" jump over those tasks...

>>>>> There are highly complicated wellorderings of w ( = {0, 1, ...} ). Of course if you _know_ X is a w.o. of w then you know it is countable without performing any tasks to "read" it! If not, then sure, it may take some effort to recognize what it is. "Mechanize" is a separate matter, though. If you only use computable functions on w, you can recognize/describe some of the w.o.'s of w, but not all; countable w.o.'s beyond a certain level of complexity are out of reach.

Here we have strayed off cardinality, and gotten to ordinals... and effectiveness.

> (Well, at least we can establish whether it respects syntax. My point is that this way we will doubtlessly stumble upon endless subsets of meaningless code before encountering a "next" code-with-a-theorem.)
"Dovetailing". Alternate between work on the T's and work on the next part of the search.

>> Double-clever way. Still remains, how to order the theorems according to their operations...

>>> You don't really _need_ dovetailing for that; you have an algorithm that produces any theorem at some finite time... so as they come up you classify them. You cannot do better; no algorithm can tell you whether an arbitrary formula is or isn't a theorem.
Again, this is not just cardinality.

> It might be centuries before the first one shows up; the search never ends; but every proof+theorem will appear at some finite time.

>> When I said "endless" I meant infinite of the kind "all the evens", and without a mechanical way of jumping out of this routine...

>>> Dovetailing does show you don't have to be stuck. There may be infinitely many tasks t1, t2, ... to perform; carry out 1 step of t1, then 1 of t1 and 1 of t2, then 1 of t1, 1 of t2, 1 of t3...

> The reals, whether defined by decimals (R1), or Dedekind cuts (R2), or Cauchy sequences (R3), or integer sequences (R4)... etc have the exact same cardinality; there is a 1-1 onto mapping between any two R's.

>> I can "mentally crowd" my geometrical line with many more dots than needed for R1 or her sisters, or on the contrary think it a bit scarce for them.

>>> No doubt, but your line wouldn't then be the reals.

> Eh... isn't 1/2, or any 1/n finitely describable?! And 0 is not one of them anyway; it just happens to satisfy the definition of limit. I don't know what you are objecting to.

>>>> (a third party) What he probably means is that the *limit* of the sequence 1/2, 1/3, ... is a «nest of real numbers that couldn't be disentangled». The number 0 is the only number in the «nest» describable in finite terms.
However, I challange the original poster to describe this nest a little bit further.

>> The original poster thanks you for almost exactly producing the correction as he put it in a previous reply, which he/I repeat hereafter:
>> Sorry, but I do not mean members from the series 1/n, I do mean the creatures waiting at the end of that. They crowd a nest out of which we "observe" only zero. Or could you give me a series whose limit is "the closest neighbour of zero"? No, that last one hides in the nest...

>>> Not in R; 0 is the only limit here (all limits in R are unique). You can build nonstandard models extending R, where every x in R has a "cloud" around it, x + o, for o in I, the Infinitesimals. Maybe that is the 'nest'! But you are not in R then. And there still isn't a "closest neighbor of 0" ...
One might find wilder constructions yet... and might like such things and enjoy studying them. This doesn't change the properties of R, however, or make them problematic.

>> So the limit is the nest, not the series. What I mean is that the "real real" numbers, which are not describable in finite terms, do seem to me neither describable in terms of a series, infinite though it be, of elements with finite description each.

>>> Since 0.a1, 0.a1 a2, ... does describe x = 0.a1 a2 a3 ..., you must be thinking of sequences of a's longer than x.

>> You could reply, oh well, but take a look at, say, a series defining pi. So what, pi is describable in finite geometrical terms, but not as a decimal number: it is no less a nest...

>>> Pi is 2q, 0 < q < 2, 1 - (q^2)/2! + (q^4)/4! - ... = 0 (cos(q) = 0) and its decimal digits can of course be found.

>> ... than my "zero" example... My point is that the decisive distinction between (two "nearest") real numbers starts only after the aleph0-th decimal position, that is at the very point where the tentative series comes to an end!

>>> Well, there is no aleph-0 position; aleph-0 means "all a_n, n in N". But you may consider aleph-0 "plus 1": W = 0.a1 a2 ...; b1 -- or more b's. What intuition leads you to this? (if in fact this is what you are saying)
For one thing, are W and V = 0.a1 a2 ...; c1 "nearest" reals?
It is not clear how such things work. What is 0. ...; 6 + 0. ...; 7 say?
Sequences longer than aleph-0 (especially binary ones) have their place; but why should _they_ be "the reals"?

>>>> (me again) The problem which I argued occurs "after" aleph0, as a matter of fact dooms whilest the decimal series unrolls: it raises there the question whether two series like, say:
.1 .01 .001 .0001 .... and
.9 .09 .009 .0009 ...
converge "as a matter of fact" to the same number, zero;
statement which I allow myself to put in question, as doing so seems to me the only way in the decimal technique to keep a perspective for numbers that are irrational and yet keep their hold in an immediate, ever so tiny as you wish, vicinity around that very zero.

>>>>> (third party) You keep talking about this cloud of real numbers clustering around 0, in spite of the fact that everybody keeps telling you that there simply are no such (real) numbers. What makes you so certain that these numbers exist????

>>>>>> (me) Well, nobody seems to be able to accept the fact that there are, rational as well as irrational, numbers that you can pinpoint by some description (geometric or decimal, or as a converging series...), but that there are (to my taste many more) others that you can't.

Even for the rationals, whatever the order of counting down you may invent, some numbers will always hide beyond the "...etcetera" you find yourself at last putting behind your list. And what about the irrationals in their immediate vicinity...?

Sentence structure in languages

These are the gatherings I got from a request to forward translations of a (rather compound) sentence into various languages, with a view to comparing differences and subtleties between them. The sentence was first put by me in Dutch, the English word-to-word translation follows in each example.

Dutch:
------
De man (aan) wie ik gisteren dit boek gegeven heb /(of: heb gegeven)/, vroeg me het te mogen houden om het volgende maand als een verjaardagsgeschenk aan zijn moeder te kunnen aanbieden.
-----
The man (to) whom I yesterday this book given have /(or: have given)/, asked me it to may keep in-order-to it next month as a birthday-present to his mother to be-able-to offer.

German:
-------
Der Mann, dem ich gestern diesen Buch lieh, bat mich es behalten zu dürfen, um es naechsten Monat als Gerburtstagsgeschenk seiner Mutter anbieten zu können.
-----
The man, whom I yesterday this book lent, asked me it to-keep to may, for it next month as birthday-present to-his mother to-offer to be-able-to.

Problems:
You picked a heavily idiomatic sentence. For example, "asked me to be
allowed" is a problem in German as well as Spanish, French and other
Romance languages, as far as I know. The passive in the request is the
problem. If you had said "asked me to let him" it would be OK.
However, this experiment seems to be quite interesting. I'd be
interested in seeing the final results once you're finished compiling
the list.

Westerlauwer Frisian (Frysk):
-----------------------------
De man dêr't ik juster dit boek oan jûn haw, frege my om it hâlde te meien om it takomme moanne oan syn mem as jierdeispresintsje jaan te kinnen.

The man there /('t = clitic marking relative clause)/ I yesterday this book (him-)to given have, asked me for it keep /inf/ to may /"gerund" = infinitive + n/ for it next month to his mother as birthday-present give to be-able-to.

(The "gerund" is a sort of second infinitive, used after "te", among others. BTW not a very "idiomatic" way to express this sentence in Frisian, but perfectly grammatical.)

Norwegian:
----------
"Mannen (som) jeg gav denne boken til i går, bad meg om å få beholde den for å forære den til sin mor som fødselsdagspresang i neste måned"
-----
Man-the (to-whom) I gave this book-the to in yesterday, asked me for to may
keep it for to offer it to his mother as birthday-present in next month.

(right?)

(BTW I know my example is clumsy but it is in heavy-laden sentences that
language structure shows distinctive marks...)

Spanish:
--------
El hombre a quien le presté este libro ayer me pidió que pudiera
quedarse con él para poder dárselo a su mamá como regalo de cumpleaños el
próximo mes.
-----
The man to whom to-him I-lent this book yesterday me asked that he-might stay-himself with it /(= keep it)/ for-to may give-her-it to his mother as present of birthday the next month.

French:
-------
L'homme à qui j'avais prêté ce livre hier m'a demandé
de pouvoir le garder afin de pouvoir l'offrir à sa mère
comme cadeau d'anniversaire le mois prochain.

The man to hom I had lent this book yesterday me has asked
to may it keep in-order to may it offer to his mother
as present of birthday the month next.

Tagalog:
--------
Yung mama na pinahiraman ko
ng libro kahapon,
ay sinabi niya sa akin
na itago ko upang maipresentang regalo
sa susunod na buwan
sa kaarawan ng kanyang ina.

Yonder man which/that lending I
of book yesterday,
is speaking he/she at mine
which/that hiding I so_that presenting gift
at follow which moon/month
at birthday of his/her mother.

(I find it extremely difficult to give a word for word translation of the above. Note that in Philippine languages, there are no noun stems distinct from verb stems. Stems are neutral. The neutral stems are marked for usage. Obviously, there are words that I find extremely difficult to translate exactly into English. But I gave my best approximation of the English words for each word used in the above translation (although it may not even make sense because the true meaning of the original Tagalog words are lost entirely in the translation))

(I also know a bit of Chavacano (Creole Spanish spoken in very few places in the Philippines; Zamboanga, Cavite, Ternate). Here goes:)

Chavacano:
----------
El ombre que ya presta yo el libro ayer, ya dice elle conmigo que ta guarda yo aquel (libro) para pwede di presenta elle aquel (libro) na mes proximo na compleanyos de mama de elle.

The man which/that past_marker loan I the book yesterday, past_marker say
he/she to_me which/that present_marker keep I yonder (book) so_that possible future_marker present he/she yonder (book) at month next at birthday of mother of he/she.

(This is easier to to translate word for word into English. Although the grammar is very Filipino, the words are European which makes it a lot easier to translate for me into English)