# User:Dan Gluck

I am an Israeli algorithm designer, working on image analysis. I've done my PhD in physics, in string theory (or is it mathematics?), and did post-doc in biology.[1] Also have studied math and philosophy.

So please don't argue with me about physics unless you're a physicist yourself.

(Sorry for the arrogance but you may understand the need after reading this)
Wikipedia:Babel
he ‏משתמש זה דובר עברית כשפת אם.‏
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## My contributions

### Major contributions

(pages I've started or was significantly involved in their writing):

## A proof sketch that $2^{\sqrt{2}}$ is irrational

We first consider the following matrix: $M_{ij} = (s_i + t_i\sqrt{2})^{u_j}2^{(s_i + t_i\sqrt{2})v_j}$

where i runs over all (2S-1)2 pairs (si, ti) with -S < si, ti < S, and j runs over all (L0+1)(L1+1) pairs (uj, vj) with 0 ≤ uj ≤ L0, 0 ≤ vj ≤ L1.

S is an arbitrarily large integer, L0 = S [Log(S)], L1 = [S /Log(S)]. (here [x] denotes the integer value closest to x from below).

Mij is a (2S-1)2 by L matrix, and since (2S-1)2 > L for large enough S, we may look at any determinant of an LXL sub-matrix of M, which we denote Δ.

|Δ| can be shown to satisfy an upper bound, |Δ|≤x-L2/2 for some x>1, as we later discuss.

Now let us assume that $2^{\sqrt{2}}$ is rational. Then for some integers a, b with b>1, $2^{\sqrt{2}} = a/b$. Therefore:

$M_{ij} = (s_i + t_i\sqrt{2})^{u_j}2^{s_i}a^{t_i v_j}b^{-t_i v_j}$,

and hence:

$\Delta = (A+\sqrt{2}B)p a^{t_i v_j}b^{-t_i v_j}$, for some integers A, B, p.

Then one uses Liouville theorem on diophantine approximation to show that |Δ| satisfies a lower bound, |Δ| ≥ bSL1L.

For every x,b>1 there exists an S large enough such that: bSL1L< x-L2/2. We thus arrive at a contradiction, and therefore $2^{\sqrt{2}}$ must be irrational.

We turn now to the upper bound of Δ. As S (and hence L) grow larger, the determinant Δ contains multiplication of more factors (which tend to increase its size), but it also contains more lines and columns which become closer and closer to being linearly dependent (which tend to reduce its size), and the second effect is shown to be more prominent .

To show this, we see that every entry in the matrix M satisfies: $M_{ij} = W^{u_j} 2^{W v_j} f_j(z_i)$, with $f_j (x) = x^{u_j}2^{W v_j (x-1)}$, $z_i = (s_i + t_i \sqrt{2})/W$ and some $W > S(1+\sqrt{2})$. Note that $|z_i|, |f_j(z_i)|<1$ for all i, j. Therefore:

$\Delta = L! W^{L\sum_j u_j} 2^{W L \sum_j v_j} \Delta_1 L! W^{L\sum_j u_j} 2^{W L \sum_j v_j} \Delta_1$ where $\Delta_1$ is the determinant of the LXL sub-matrice fj(zi) (instead of Mij).

## An example for a Gödel number in Peano arithmetic

Let's try to write a concrete example for the Godel number required here, with Peano arithmetics and the decimal system. We will have to use abbrevations for lengthy expressions.

(under work)

## A list of words similar in Hebrew and Greek

     α-            alpha privativum, expressing want or absence.
αγείρω   (root ΑΓΕΡ-)   to bring together, gather, assemble
δωμα      a house, a chief room, a hall,
αδρός      strong, great.  Of persons: large, fine, well-grown.
αγαπάω   to treat with affection, love, be fond of
έχω         to have or to hold    (Ionic imperative:  εχεσκον) Iliad: to grip
αυριον    dawn, aurora,     ουρανός    the sky [perhaps source of light?],
ελλος       a young deer, fawn
ελλοψ      Hesiod = mute (for fish)
ύλη          a wood, forest, woodland.  Homerus and Herodotus:  forest trees.
άνθρωπος         man, or woman. (latin: homo, not vir) a human being
οπτάω     to cook  by means of fire or dry heat.
όφις         a serpent, snake
αράχνη    a spider,  a spider’s web
αρά          I.  a prayer. Especially a prayer for evil, a curse, imprecation.  II. The
effect of the curse, bane.
δόλιχος     long
έρα            ground
ερώ           say, speak
αίθω          (1) to light up, kindle. (2) intransitive: to burn or blaze.
αιθάλη       soot.  Αιθίοψ =   Ethiopian = פני אש.  αίθος  = fire
αστράπτω  hurl lightnings (of Zeus)  αστραπή = lightning,  αστήρ =star
σοφός        skilledin any handicraft or art, cunning in his craft. Clever, wise.
βαίνω (ΒΑ-)      to walk, to step, to go, to come, to arrive, to depart, etc.
βάραθρον            a pit, (at Athens a left behing the Acropoliw, into which
criminals were thrown

       φαείνω    Skt. *bha-   to shine, give light           בהר, בהק
βύσσος     a fine flax and the linen made from it     (a later borrowing?)
βωμός      any raised platform, or stand, araised place for sacrificing, altar.
πυρός       wheat
flee, flight  Skt. palay-                    ברח      (Gothic: pliuhan, etc.)
πέσσω      I : of the sun: to ripen.  II.  By the action of fire: to cook, bake.
όχθη                                                       גדה
γίγας       mostly plural: Γίγαντες = Giants, a savage race destroyed by the Gods
κυλίω      to roll,         גל    (see,  γογγύλος – עגול)
γλάφω     to scrape up (the ground). γλαφυρος =   polished, finished.
γλυπ-, γλυφ-   (akin to γλάφω)  to carve, cut out with a knife
γοάω, γόος       to wail, groan, weep
Αίγυπτος         Egypt                      (αια-γυπ)     ארץ גושן
γεφυρόω         to bridge over, make passable by a bridge,   γέφυρα = a bridge
τιτθός         a teat, nipple
γύρος           a ring, circle
δύη             a woe, misery, anguish, pain.
θολός        mud, dirt, the dark juice of the cuttle-fish.  θολόω   to make turbid
τρέχω         to run, move quickly
έρχομαι  (fut. ελθω)    to come, to go
ολολύζω,  αλαλάζω     (= crying aloud to the gods in prayer or thanksgiving)
όρος          a mountain, a hill (Cf. C.D. Buck, A Dictionary of Selected Synonyms in the Prinicipal Indo- European  Languages:  “Gr. όρος ..…  from  IE *er-  ‘move, raise, rise’     seen in Skt. ŗ- , G r. όρνυμι , Lat. oriri.”.    Heb.  רם= high, also connected?
αράω (old verb: to harm)
σφάζω     to slay, slaughter, properly by cutting the throat.    σφαγή   slaughter,
σεύω (έσσευσα)   to put in quick motion.  σείω  to move to and fro.                                                                                                   σεισμός = earthquake
σκώληξ          a worm.
μέλος  (for the  σ  see μικρ-, σμικρ)   a song, tune
[ζέω ( =  to boil)                                      זעם  (= anger)]
ρέω (ροή)    to flow, stream, gush
σπείρω    ( Lat. sero)    to sow seed                     Akk. zāru      זרע
κόπτω       to strike, smite, knock down, cut off chop down
χωμα       earth thrown up, a bank, a mound,   a dike
Lat.  forare ,                 Akk. harāru     חרר      ( hole, Old Eng.: hol,   חור )
άλσος       a glade or grove            (see ύλη,  אלה, etc.)
γάλα   (root: ΓΛΑΚ-, ΓΛΑΓ-)   milk                                                  חלב
connected to γλαύκος, γλυκύς, glow, etc.   ?  and to λευκός,     --   לבן  ?
γάμος, γαμέω)      a wedding, wedding-feast, marriage, wedlock.
ξένος    In Homer: a guest friend, i.e. any citizen of a foreign state, with whom one
has a treaty of hospitality for self and heirs,  a refugee.
σκεπάω         to cover, shelter.
σκάπ-τω        to dig
σκότος          darkness, gloom
χήρα (-ος)     a widow, bereaved               Akk. Һarābu        חרב

and κάρφω  to dry, destroy
ορρωδέω       to fear, dread, shrink from. (formed so as to express the shuddering
of fear)
χαράττω (ΧΑΡΑΚ-)     to cut into, to sharpen, whet,   scratch.   (χαράσσω)
αρόω         to plow
δείδω   (epic aor. έδδεισα)    to fear, dread   ω
εδνόω   to betroth, to marry
σφαγ- σφακ-      slaughter (see above:  טבח (זבח)
δρόσος  =  dew,    υδωρ = water,   θάλασσα = sea, τέλμα = standing water, marsh,
pool, pond.
τρέφω (έθρεψα)       to rear, feed.   τροφή   food, nourishment
μογέω, μόγος       to toil, to suffer
*δάω (anc. root =learn, perf.: to know)      Skt. vid-  = to know    Akk. idu     ידע
οίδα (perf. of to see= know)
οίνος        wine
οιωνός    = a bird of omen, and also omen.  Aesch. οιωνός γόος,   but the only
bird  that makes a wailing sound – γόος --  is the dove!
Ιάπετος       one of the Titans (sons of Uranus and Gaea), father of Atlas,
Prometheus, etc. According to a tradition – rather late—he formed
the body  of the first man with earth and water (perhaps his own
tears) into which Athena breathed soul and life.
לא במקרה,הסופר בחר שם זה לאבא של פינוקיו!  :Japetto

ύπνος         sleep, slumber
ιασπις                                Akk. (j)ašpu   ישפה  (but not a basic, primitive word…)
ιθύς           straight, direct    ιθύνω:  make straight, guide in a straight line
καίω          to light kindle, Passive: to get burned,   καύμα : burning heat.
κικυς   (in Homer)    strength, vigour.
(h)όλος    whole,  entire,              Akk. kullatu    כל
κλείω    to shut, close, bar, to confine.
σκύλαξ (OE hwelp)  a young dog, puppy, a whelp, generally: a dog --  Κέρβερος
κλωβός      a bird cage      (El Amarna:  kilubi)
κάμνω (root: KAM-)     Iliad: to work, of smith’s work.
κύπελλον    a big-bellied drinking vessel, a beaker, goblet, cup
κυφός   (r.: ΚΥΠ-)     bent forward, bent, stooping
κριός        a ram
χράω     (1) to be eager to do  (2) to furnish what is needed.     χρη = must, need
κείρω    (I) to cut the hair    (II) to cut, to hew  out (of wood), destroy
γρύψ , Eng. Griffin
κρόκος, IE crocus             Akk. kurkanu         כרכם
χιτών  (κιθών)   the garment worn next the skin, in early times: a man’s frock.
Later worn by women as well.
νάσσω, νάττω      to press, squeeze,  ναστός: a well kneaded cake
λανθάνω (root: ΛΗΘ-  ΛΑΘ)       to escape notice, be unseen, etc.
λέων        a lion                           Akk. nesu                ליש
μανθάνω (r.:  ΜΑΘ-)   Iliad: to learn, to know, to understand.
λέγω        to say, to speak                          larynx =  לע       ( whence Arab.
λάρυγξ  = לע + ρύζω(=to growl) ?
λωβάομαι    to treat despitefully, to outrage,    λώβη    an outrage, dishonour
λείχω           to lick up
γλω̃σσα       the tongue, the organ of speech.  Homerus:  language
μω̃̃̃μος          a blame, riducule, disgrace.     μωμεύω (Hom.) to find fault.
μίσγω         to mix one thing with another, join together, to mingle.
μυελός =  marrow, μύχιος = inward, inmost,  μυχός = the innermost place.
Connected to μυέω, μυστήρια?    סתר – מוסתר?)
Lat. amarus    French:  amer, = bitter
μετρέω     to measure
πίμπλημι (r.:ΠΛΕ- or ΠΛΑ-)    to fill full of a thing .  Passive: to be filled.
μέγαρον       Homer:  a large room or chamber, a hall.
βασιλεύς      a king, chief
μάσσω         to handle, touch
εγγύς = near, near at hand
ναίω, ναιετάω  = dwell
νέω                  to go,         νέομαι      to go or come
έχις, έχιδνα      an adder, viper   (Skt.  ahi)
πίπτω        to fall
δάκνω (root: ΔΑΚ-)   to bite
δίδωμι (root: ΔΟ-)       to give, to grant
σηκός, σακός         a pen for lambs)
σής               a moth (which eats woolen stuff)
αγαπάω        to treat with affection, to caress, love, be fond of..
γογγύλος      round
γερανός         a crane
κόραξ            a raven, crow
αιξ                a goat
αετός             an eagle
αχλύς             a mist, a mist over the eyes of one dying
άναξ               a lord, master, being applied to the gods, esp. Apollo and Zeus.
A title given to all men of rank and note.

        άγχω         to strangle, throttle,       αγχόνη       a throttling, strangling, hanging
ενίπ-τω     to reprove, upbraid          ( connected to όνειδος )
ασκέω       to work, to form by art, to fashion, to practice and art.
ίδη            in plural:  a timber tree,  in singular: a wood
ακίς                                                       עקצ, עֺקץ
σκορπίος                        Akk. aqrabu  עקרב
Έρεβος                          Akk. erebu           ערב
εγείρω                               Akk. êru                ער  (root: ,עור  Aram. עיר ) IE *ger-  Skt. jagr-, Av. gar-
είς                                      Akk. iśten          עשתי-  (found only in composites)
[Καφτωρ                        Akk. Ištar           עשתרת    Was the island named in honour of the goddess   (as Athens for Athena) and later renamed Crete under the           influence of Ugarit (as St. Petersburg –Leningrad)?
πεδίον                         Akk. šadu                 פדן     [a different form of שדה ?]
πω̃; όποι,  etc.  Lat. qu-                           פה
πέρδομαι,      Skt. pŗd-    Akk. balātu      פלט     (fart, peter, etc.)
πέλας, πελάζω                                         פלש
πατέω                                                      פסע      (and in the Bible: [פשע
βοάω, βοή,                                            פעה
πόρτις                         Akk. parru        פר, פרה
flight,     Skt. palay-                   ברח  ,פרח
οπώρα, fruit   Skt. phala-                            פרי     (connected to root פרה = grow)
Lat.flos,  IE *bhle-, bhel-   Akk.pirhu   פרח
πείθω                                                פתה, פֽתה    (later borrowing?)
Πύθων                               Akk. bašmu  פתן
σχελίς                                Akk. şelu      צלע
σκοπέω                              Akk. šafu        צפה       (and ; שקפ    (Latιn: aspicere)
κάπρος = wild boar        Lat. caper                        Akk. şapir       צפיר   ]
Lat. strictus                                          צרר   [ = bound together, whence narrow: Skt : avi-stīrņa- and avistŗa           צר           (It. stretto,   Fr : etroit ; etc )
καλέω     to call, to summon, to call by name, to name,
κλαίω    to weep, lament, wail, cry.
κράζω, κραυγή, cry out,

καίνω  ( = kill, in Iliad)                                קין  (the name)   ( Hitt.: kwen-)
εύ-κολος    easy
δύσ-κολος  difficult
κέλυφος     a sheath, case, pod, shell
καλύπτω  (r. KΑΛΥΒ-)    to cover with a thing
κάννα    a reed
κρύος      icy cold, chill, frost
κυρέω, κύρω         meet with, light upon, fall in with,
κέρας      the horn of an animal, horn as a material, anything made of horn

  οράω   (and  λάω)      to see, look at, behold
οργή      Ι.  natural impulse, temperament,   ΙΙ.  passion, anger, wrath
οργίζω       to provoke anger
ορμάω           to set in motion, spur on, push on,
ορμή          a violent movement onwards, onset, an attack
έρπω             to creep, crawl, to move slowly
ρύπος            dirt, filth, dirtiness
ορχέομαι       to dance in a row, and generally:  to dance
εθέλω or  θέλω  (Doric δήλομαι)   to wish, will, purpose
θήρ, φήρ       awild beast, a beast of prey
θηράω = to hunt or chase wild beasts         θήραμα = the hunted animal
ταυ̃ρος          a bull
σκηνή      a covered place, a tent, a tented cover
στέλλω      I.  to set in order, equip, furnish with   II.  to dispatch, send.
In Medium:  to  to send  for one.

συλάω      to strip off, esp. to strip off the arms of a slain enemy, to pillage, plunder,
rob .
λεηλατω̃  to drive away cattle as booty, to make booty, plunder, despoil
λεία     booty, plunder.      ληστής      robber, plundered.
πλήρης     full of a thing, complete.
οδούς  (gen. oδόντος)  a tooth
σινέομαι, σίνομαι        to do one harm or mischief,
πνίγω         to choke, throttle, strangle
θεάομαι      to look on, gaze, behold.
θύω            to rush on or along, of a rushing wind. To storm, rage.
θύελλα       a furious storm, hurricane.
θύρα          a door. In Homer mostly of double doors
πύλη          one wing of a pair of double gates. In plural (in Iliad): the gates of a city
έθειρα        hair. In Homer in plural: either a horse;s mane, or horsehair crest on
helmets.   Εθειράζω = to have long hair.
πηγάζω        gush forth with water.   πηγή  = mostly in plural: running waters, streams
σακίον, σάκκος       a sack, bag.          Akk. šaqqu       שׂק
έρπω           to creep, crawl
ρίζα            a root                        (Lesb.: βρίσδα )
πορφύρα     the purple-fish, purple dye, purple
ίστημι  (r.  ΣΤΑ-)   to stand.  To make to stand.  To set up.  To raise, rouse, stir up.
τίθημι    (r. ΘΕ-)  to set up, put, place
πίνω       (ρ.  ΠΟ-)  to drink.   πόσις  = a drinking, a drink, beverage.
θαυμάζω      to wonder, to marvel, be astonished.
θαυ̃μα          whatever one regards with wonder, a wonder, a marvel
τρέπω       to turn or direct towards a thing.