Pearl Jam‘s second album, Vs, contains a song called Rats (along with several other masterpieces of their catalogue). The song features a beautiful bassline on E minor, as well as some interesting lyrics. It is those lyrics that came to mind recently.

The song essentially attempts to compare the behavior of rats with that of human beings. The first verse goes:

They don’t eat, don’t sleep
They don’t feed they don’t seethe,
Bare their gums when they moan and squeak
Lick the dirt off a larger one’s feet

From the get-go, the lyrics are weird. The first line appears to elevate a negative characteristic of rats and explains that humans are unlike that (the assumption here is that we sleep more regular hours and eat more regular meals than rats). But the second line is unclear. Humans clearly seethe, and so do rats. The third and fourth ones clearly describe behaviors of rats.

Second verse:

They don’t push, don’t crowd
Congregate until they’re much too loud
Fuck to procreate till they are dead
Drink the blood of their so-called best friend

Well clearly this stanza chastises humans, with the possible exception of the third line, since I have the feeling that rodents tend to reproduce much more often than primates.


They don’t scurry when something bigger comes their way
Don’t pack themselves together and run as one
Don’t shit where they’re not supposed to
Don’t take what’s not theirs, they don’t compare

This is definitely in favor of humans except…

Verse #3:

They don’t scam, don’t fight
Don’t opress an equal’s given rights
Starve the poor so they can be well fed
Line their holes with the dead one’s bread


Chorus again, twice this time:

They don’t scurry when something bigger comes their way
Don’t pack themselves together and run as one
Don’t shit where they’re not supposed to
Don’t take what’s not theirs, they don’t compare

Bridge to outro (not sure if this is even a correct term):

They don’t compare, rats
They don’t compare, rats
They don’t compare

And outro proper:

Ben, the two of us need look no more (x6)

In all likelihood a reference to the namesake character of this movie.

As one goes through this track, it’s hard not to “keep score”. Assuming that some of the more confusing lines are either pro-rat or pro-human, one starts making imaginary scoreboards in their mind.

– Maybe we need to split the first one 3/4 for humans and 1/4 for rats…

-Last line of chorus is a snide remark on humans thieving….

-Doesn’t the bridge favor rats since one would typically use the phrase “They don’t compare” in a positive way for whoever “they” is?


Until at some point, hopefully sooner rather than later, you realize that you are forced to count votes in order for humans to win a moral battle against rats.


On character

(And / or the inherently unquantifiable nature thereof.)

Of diamonds, flair, millions, bleach white teeth smiles, linen robes and endless miles of red carpets,

of 4 out of 4 GPAs and thousands of references, athletic prowess, spectacular extracurriculars,

of shining silver eyelids covering black eyes without irises, pink fonts on top of chopped moments describing what to see, and short sentences feeding you the need for more,

of fame for the sake of being famous, instead of for something that you did not for the sake of being famous, but because it was par for the course, whatever course that you have otherwise clearly set on yourself,

of comfortable people walking through life like a Markov process, no past queues, nothing in their past to draw pain from, and cry and learn, yet always a persistent avoidance of pain,

I’ve seen thousands.

Of pairs of piercing eyes, mouths that seldom speak, yet say so much,

of silence screaming, iterative self-doubt, pain-shaped cheeks,

of rebirth through disappointment, of mistake after mistake braved, regretted, braved again

of soccer cleats kicking dirt around on a dry May day, alone

of cigarette smoke emanating towards a purple August night sky, huffed from powerful, powerful lungs

of genius homeless dropouts with collections of pirated math texts

of cores that guide and give purpose, hearts that are armed and ready to get scraped

I see way too few.

Το δικό μου “χάρτινο”

Η πρώτη φορά που ασχολήθηκα με την ΑΕΚ σοβαρά ήταν τη χρονιά 2002-03 (η χρονιά της Ριζούπολης). Έξι ισοπαλίες στο Τσάμπιονς Λιγκ, πόλεμος στον Μπάγεβιτς, Ψωμιάδης με φουσκωτούς στο σπίτι του Ντέμη και σακούλες με λεφτά στις προπονήσεις, Νοέμβρης / Δεκέμβρης με 5 σερί ήττες που τελείωσαν τη χρονιά εντός συνόρων, 12 σερί νίκες στο δεύτερο γύρο με διπλά σε Τούμπα και Ριζούπολη, υπεροψία με Μάλαγα, μπαλάρα και τερματισμός δύο; Τρεις βαθμούς πίσω;

Ναι, τα θυμάμαι έντονα. Βλέπετε, τα χρόνια εκείνα μια νίκης της ομάδας σου σήμαινε μια καλύτερη Δευτέρα όταν ξεκίναγες με Μαθηματικά Κατεύθυνσης και μετά το πρώτο (μεγάλο) διάλειμμα, Αρχαία. Εσένα ήταν ο νους σου ακόμα στο γκολ στο ντέρμπι… Πώς να μη θυμάμαι τη χρονιά τέλεια; Πώς να μη θυμάμαι την επόμενη χρονιά επίσης, όταν σχεδόν διαλύθηκε η ΑΕΚ, και τερμάτισε οριακά πάνω από το Αιγάλεω εν μέσω αντιδράσεων για το αν έπρεπε να συμμετάσχει καν στο Ουέφα… Αντιδράσεων από το Αιγάλεω, παρακαλώ.

Πέρασαν πολλά χρόνια. Για μένα, για την ΑΕΚ, για την Ελλάδα. Το 2007-08, στο “ζενίθ” της προσπάθειας του Ντέμη, ως πρόεδρος πια, να μας φέρει πίσω στην κορυφή, τη χρονιά του Ριβάλντο και του Αρουαμπαρένα, τη χρονιά της υπόθεσης Βάλνερ και του “χάρτινου”, το πρωτάθλημα το χάσαμε, και ξεκίνησε η παρακμή. Σαν αυτούς τους στρατούς στους Παγκόσμιους Πολέμους που υπερέβαιναν τις γραμμές ανεφοδιασμού τους χωρίς, πάντως, να κατακτήσουν το στόχο. Δεν είχαν ελπίδα, δεν είχε και η ΑΕΚ.

Αλλά ποιός από τους ΑΕΚτζήδες αλήθεια θυμάται το πόσο ΧΑΛΙΑ έπαιζε η ΑΕΚ τότε, ιδιαίτερα τηρουμένων των αναλογιών, ενός συνόλου με Ριβάλντο, Μπλάνκο, Εντίνιο, Λυμπερόπουλο, … Αρουαμπαρένα, ….  Ταμαντανί Ενσαλίβα… που πέρα από το γκολ με τη Μπόλεσλαβ δεν έκανε τίποτα άλλο. Ποιός θυμάται το ότι ο καλύτερος παίκτης της στον πρώτο γύρο ήταν ο Παπασταθόπουλος, που σκόραρε και με τον Άρη στη Θεσσαλονίκη; Ποιός θυμάται το παιχνίδι με τον Ηρακλή στο ΟΑΚΑ, όπου έχασες τα άχαστα, μαζί και τους πρώτους σου βαθμούς; Όταν φάνηκε το ότι ήσουν ακόμα μικρή ομάδα, γιατί οι μικρές ομάδες χάνουν ευκαιρίες, ενώ οι μεγάλες κερδίζουν όταν είναι χειρότερες;

Ποιός θυμάται μια φάση από το εντός έδρας παιχνίδι με τη Βιγιαρεάλ, με το σκορ είτε στο 0-0 είτε στο 0-1, πρώτο ημίχρονο, στην οποία ο τερματοφύλακας (Μορέτο;) δίνει με τα χέρια στον Τόζερ, αυτό το “μαγυάρικο άτι”, για τον οποίο μας είχαν πρήξει τα ούμπαλα οι δημοσιογράφοι; Μια φάση στην οποία ο Τόζερ κινείται προς τα μπρος με τη μπάλα, και σε κάποια φάση ένας Ισπανός (μάλλον επιθετικός, half field pressing των αδιάφορων Ισπανών γαρ) κινείται για να του πάρει τη μπάλα, και εκείνος αντί να δώσει πάσα πλάγια, ή τουλάχιστον να επιχειρήσει τρίπλα, ΑΠΛΑ ΣΚΟΥΝΤΗΣΕ ΤΟΝ ΕΠΙΘΕΤΙΚΟ ΤΗΣ ΒΙΓΙΑΡΕΑΛ ΚΑΙ ΤΗΝ ΕΧΑΣΕ;

Με εκείνη τη φάση κάτι μέσα μου “έσπασε”. Η ομάδα ήδη είχε δώσει τραγικά στοιχεία μην όντας καν ανταγωνιστική κόντρα στη Σεβίλλη, στα προκριματικά (απλά ανταγωνιστική, δε μπορούσε ποτέ καμία Ελληνική ομάδα να κοντράρει εκείνη τη Σεβίλλη), και κερδίζοντας αυχενικά στο πρωτάθλημα. Στη δε ρεβανς των προκριματικών του Ουέφα κόντρα στην Σάλτσμπουργκ, με τον πλαστικό χλοοτάπητα, μάλλον τυχερά φύγαμε με μόνο 1-0, καλά να ήταν το 3-0 του πρώτου. Και μόλις είδα εκείνη τη φάση, πραγματικά κάτι μέσα μου “έσπασε”, και πήρα διαζύγιο από την ΑΕΚ.

Μετά από ένα χρόνο (για την ΑΕΚ, Δώνης, Σκόκο, αρχή παρακμής) πήρα διαζύγιο και από την Ελλάδα. Τρεις μέρες μετά τη δολοφονία του Γρηγορόπουλου πήγα να διαμαρτυρηθώ στο Κέντρο με πολλούς συμφοιτητές μου. Η πορεία γρήγορα χαβαλεδιάστηκε, με καμένα αυτοκίνητα πέρα δώθε στις εισόδους των Εξαρχείων, σπασμένες βιτρίνες, βεβηλωμένες τράπεζες και εκκλησίες. Για να μπορέσουμε να αναπνεύσουμε, κάτσαμε στα Προπύλαια τα οποία είναι ελαφρώς υψωμένα σε σχέση με την Πανεπιστημίου.

Ένα διαμέρισμα στην Κοραή καιγόταν. Δε θα μου έκανε εντύπωση καμία αν ήταν κάποιο φροντιστήριο με παιδιά μέσα: υπήρχαν πολλά στην περιοχή και Δευτέρα βράδυ είναι ώρα λειτουργίας για τα φροντιστήρια. Φτάνει ένα πυροσβεστικό από την πλευρά της Ομόνοιας, και ένας κουκουλοφόρος βγαίνει στην Πανεπιστημίου, δείχνει την παλάμη του για να κάνει τον οδηγό να σταματήσει. Ο οδηγός σταματάει. Σε στυλ Grand Theft Auto, ο κουκουλοφόρος τον βγάζει έξω από το πυροσβεστικό και πετάει μολότωφ μέσα.

Πάντα ήθελα να κάνω το κάτι παραπάνω στη ζωή μου όσον αφορά Ακαδημαϊκά / Μαθηματικά (λόξα μου, γούστο μου, καπέλο μου), αλλά οποιοαδήποτε προοπτική με είχε να κάνω Μάστερ ή Διδακτορικό στη Θεσσαλονίκη, στο ΕΜΠ ή στα Χανιά χάθηκε εκείνη τη στιγμή. Η Ελλάδα, εκείνη τη στιγμή τελείωσε για μένα.

Ταυτόχρονα, η ΑΕΚ ακολουθούσε την αναπόφευκτη κατρακύλα της προς τον υποβιβασμό. Μια αστεία ομάδα που δεν άξιζε το 12-13 να μείνει στην κατηγορία. Και καλώς έπεσε, καθώς το να έχεις Κατίδηδες στην 11άδα δε σε κάνει ελκυστική ομάδα για μένα. Ο κυνισμός μου και για την ΑΕΚ και για την Ελλάδα εξελίχθηκε σε φιλοσοφία, φιλοσοφία την οποία ασπάζομαι ακόμα και σήμερα.

Η 2η Απρίλη του 18 είναι μια ιδιαίτερη όμως μέρα. Χθες η ΑΕΚ κέρδισε τον Παναθηναϊκό 3-0 σε στυλ μεγάλης ομάδας: μεγάλη απόκρουση του τερματοφύλακα όταν σε παίζουν για την πλάκα, και γκολάρα από το πουθενά. Το ότι δώθηκαν δύο κίτρινες κάρτες σε παίχτη του Παναθηναϊκού (Ινσούα; Ρε, οι Παναθηναϊκοί βρίζατε το Μπασινά, και τώρα έχετε Ινσούα;) στο πρώτο ημίχρονο σε κανένα άλλο πρωτάθλημα δε θα θεωρείτο “έλλειψη σεβασμού”. Έκανες μαρκαρίσματα για κίτρινη; Πάρε 2 και φεύγε αγόρι μου. Εκεί εστιάζουν οι ελάχιστοι “άλλοι” σήμερα, και όχι π.χ στο ότι ο Βράνιες έκανε φράγμα μέσα στην περιοχή όταν το σκορ ήταν στο 1-0 (οπότε πιστεύω ότι θα έπρεπε να δωθεί έμμεσο) και χέρι μέσα στην περιοχή ενώ το σκορ ήταν 2-0 (πέναλτι-μαρς). Ο Βράνιες, ο οποίος κατά τα άλλα είναι το καλύτερο σέντερ μπακ της χρονιάς για πλάκα.

Η ΑΕΚ είναι φέτος η καλύτερη ομάδα στο πρωτάθλημα. Ο ΠΑΟΚ είναι 8 πίσω από την ΑΕΚ αλλά 9 πίσω από τον τίτλο. Οπότε η ΑΕΚ, με 4 παιχνίδια να απομένουν, με Πλατανιά, Λεβαδειακό, Κέρκυρα και Απόλλωνα Σμύρνης, στην παλιά γειτονιά, εκεί που έζησα τα φοιτητικά μου χρόνια και εκεί που ακόμα μένει ο πατέρας μου, πρέπει να πάρει 4 βαθμούς. Ούτε η μικρή ΑΕΚ που έχουμε συνηθίσει τα πολλά τελευταία χρόνια μπορεί να το χάσει αυτό το πρωτάθλημα.

Ένα πρωτάθλημα το οποίο όλοι σωστά αναφέρουν ως  γελοιοδέστατο, από το ΠΑΟΚ-Ολυμπιακός και έπειτα, όμως. Καθώς μέχρι τότε ήταν “το καλύτερο των πολλών τελευταίων ετών” και “με ανταγωνιστικό ΠΑΟΚ, Ολυμπιακό που δεν το καθαρίζει από το Νοέμβριο” κτλ. Καθώς για πλάκα ξεχνάμε. Ένα πρωτάθλημα στο οποίο όσο ήταν υποφερτό, πάλι η ΑΕΚ ήταν η καλύτερη ομάδα. Ένα πρωτάθλημα το οποίο η ΑΕΚ το αξίζει.

Για μένα η ΑΕΚ ήταν πάντοτε ανώτερη από την Ελλάδα. Τα πολλά τελευταία χρόνια είμαι έξω, και δε νομίζω ότι θα γυρίσω σύντομα. Δεν έχω ευκαιρίες να μιλήσω Ελληνικά. Το να φωνάζω “ΓΚΟΛ” και “ΑΝΤΕ ΓΑΜΗΣΟΥ” σε μια τηλεόραση στην οποία προβάλλω παράνομο στριμ της πλάκας που βρίσκω στο Reddit είναι οι μόνες μου ευκαιρίες, πέρα από κάποιους ελάχιστους Έλληνες φίλους στην περιοχή. Έχω χαθεί με την Ελλάδα, μέρα με τη μέρα μου φαίνεται όλο και περισσότερο σαν ξένη χώρα.

Αλλά αυτή η κιτρινόμαυρη φανέλα μου προξενεί ακόμα ρίγος, χαρά, λύπη. Πιο πολλή λύπη, έχουμε συνηθίσει στις κατραπακιές, αυτές που προξενούσε η παράγκα και αυτές που προξενούσε η ανικανότητα της ομάδας για πολλά χρόνια. Ακόμα και σήμερα με κάνει να σκέφτομαι τα Μικρασιατικά εδάφη, το 1921 που, 100 χρόνια αφού ξεκινήσαμε να πάρουμε την Πελοπόννησο, χάσαμε για πάντα το Αϊβαλί, τη Νίκαια, την Προύσα, τη Σμύρνη, την Κωνσταντινούπολη. Σε εποχές τέτοιες, σκέψεις σαν αυτές λέγονται σκέψεις ακροδεξιές. Δε μπορώ να πείσω τον αναγνώστη περί του ότι είναι διαφορετικό το να είσαι Τραμπ / Μιχαλολιάκος / Πετέν και το να εύχεσαι τα πράγματα να ήταν διαφορετικά, ώστε τώρα η χώρα σου να είχε διπλάσιο μέγεθος, να είχε το Βόσπορο, να είχε τις πρώτες ύλες, τα λεφτά, την υποδομή για να μην είναι εκ των φτωχών συγγενών της Ευρώπης σήμερα. Για να μην έχει υπάρξει απαραίτητη ανάγκη να φύγουν όλα τα νεαρά παιδιά έξω, για να γλιτώσουν από τους βλαχογκιαούρηδες που διοικούν αυτή τη χώρα.

ΑΕΚ > Ελλάδα. Είναι ακόμα μικρή ομάδα και θέλει μεγάλη απογοήτευση και ρίσκο για να ξαναγίνει μεγάλη. Δεν το πήρες με Τσάρτα και Νικολαΐδη, αλλά το παίρνεις με Λιβάγια και Αραούχο. Παίκτες ανύπαρκτους που απλά “κόλλησαν” και πήραν το πρωτάθλημα. Ένα πρωτάθλημα το οποίο είναι της πλάκας και μάλλον θα συνεχίσει να είναι της πλάκας. Αλλά για μένα μικρή σημασία έχει: γιατί κάπου στο θυμικό μου υπάρχει ακόμα ο μικρόκοσμος της Κουντουριώτου και της Βεΐκου, του σχολείου μου, του φροντιστηρίου μου, του πιο δύσκολου ξυπνήματος της ζωής μου την επομένη του Γκρασχόπερς – ΑΕΚ 1 – 0 (συντρίμ τιμ, 2003-04).

Το δικό μου “χάρτινο” είναι ένα πρωτάθλημα που η ΑΕΚ έχασε γιατί έπαιζε χάλια. Δέκα χρόνια μετά το χάρτινο και με έναν υποβιβασμό ενδιάμεσα, παίρνει στα ίσα το πρωτάθλημα με Μάνταλο και Γιόχανσον τραυματίες όλο το χρόνο, αήττητη στο Γιουρόπα και με απίστευτη εμφάνιση στο Κίεβο. Τι άλλο να θέλει κανείς;

A better number sequence for teaching Cantor’s diagonalization.

I’ve been teaching CMSC250: Discrete Structures in CS UMD for a while now. Essentially a more CS-friendly version of Discete Mathematics, with more logic, set theory, structural induction, countability. Countability, a rather esoteric subject, is challenging for our students. Naturally, a student that doesn’t get countability because they don’t get bijections cannot be helped until they understand bijections, but when it  comes to the Cantorian diagonalizing proof that the reals are uncountable (to be precise, that the interval [0, 1]) is uncountable), one can lose many more students.

I conjecture that one possible reason for this is that the sequence of numbers that is used to create  the 2D matrix is usually a bit arbitrary in both texts and slides. For example, in my own slides, I have this following sequence of numbers:

Screenshot 2018-02-28 12.50.38

With such a sequence of numbers, I have observed that even medium-to-strong students oftentimes have trouble. So let’s try to improve our example. Here’s a nice trick: Write down only the diagonal portion of the listing of reals:

r_1 = 0. \mathbf{2}xxxxxx\dots

r_2 = 0.x\mathbf{4}xxxx\dots

r_3 = 0.xx\mathbf{2}xxx\dots

r_4 = 0.xxx\mathbf{9}xx\dots

r_5 = 0.xxxx\mathbf{0}x\dots

r_6 = 0.xxxxx\mathbf{8}\dots

\ \vdots \qquad \vdots \qquad \vdots \qquad \vdots

Since the diagonal values are the only ones that you need to construct the number of interest, go ahead and construct it. I call it r' (r “prime”) only because I would like to use r later.

r' = 0.353019\dots

And now we will go replace all of those x‘s with the same-index digits of the number itself. For example, we will replace the 2nd, 3rd, 4th, 5th and 6th digit of r_1 with 5, 3, 0, 1 and 9 respectively. We will also replace the 1st, 3rd, 4th, 5th and 6th digits of r_2 with the relevant digits of r', and so on and so forth:

r_1 = 0. \mathbf{2}53019\dots

r_2 = 0.3\mathbf{4}3019\dots

r_3 = 0.35\mathbf{2}019\dots

r_4 = 0.353\mathbf{9}19\dots

r_5 = 0.3530\mathbf{0}9\dots

r_6 = 0.35301\mathbf{8}\dots

\ \vdots \qquad \vdots \qquad \vdots \qquad \vdots


The benefit of carefully constructing the number sequence in this way is that now the student can see that those numbers, no matter how hard they try to look like our own constructed number r', they fail in at least one decimal digit. In fact, with this visualization of the first 6 decimal digits for every real r_j, there is a difference of exactly one decimal digit. We have essentially pushed the argument to its limits: even if we had a real r_k whose 1st, 2nd, 3rd, …. (k-1)th, (k+1)th, (k+2)th, …. digits are the same, r_{k_k} is guaranteed, by construction of r', to be different from a_k.

The final piece of the argument can perhaps be shown as follows: The statement “[0, 1] is countable”, can be re-worded as: “For every real r in [0, 1], there is some positive integer j such that r = r_j.

The way I like to proceed from that point onward is the following: Since there is some positive integer j such that every single r \in [0, 1] can be written as r_j, then this is also the case for our constructed real number r' since, clearly, by the way that I have constructed r', it is a number between 0 and 1!

Maybe this j is equal to 1. But this can’t be, since we notice that the numbers differ in the first decimal digit, again by construction. And this is where the construction comes to help: all the other (visible) digits are the same. However, it suffices for there to be a difference in a single digit in order for us to say that, for a given j,\ r \neq r_j.

Maybe j = 2. The first digit is ok, it is 3 in both r_2 and r', But the second digit gives us a difference, despite the fact that all other digits are the same!

Maybe j = 3, and so on and so forth.

So the point is that this enumeration actually provides a Discrete Mathematics student with more intuition about why the proof works.





Spiderman: Homecoming is entertaining, but morally bankrupt.

I have recently began watching the entire Marvel Cinematic Universe, mostly out of curiosity. I’ve been enjoying it! Ridiculous C.G.I, good humor, some good acting, hilarious plot holes, and, perhaps surprisingly, some interesting moral arguments here and there. For example (obviously, spoilers follow):

To say that I am wasting my time trying to find morality in Marvel movies because their focus is on the CGI and action is an all-or-nothing statement, and experience has shown that all-or-nothing statements tend to be null. Unfortunately, watching Spiderman: Homecoming, one of the latest additions to the series, has left me troubled.

This movie essentially reboots the character of Spiderman and casts him within the modern-day framework of the Avengers.

About ten years after the Battle of New York, featured in the end of the first Avengers movie, Peter Parker (Tom Holland) is a “typical” white teenager who has found himself “blessed” with extreme agility and physical prowess. During the day, he is a sophomore in a STEM-oriented highschool in his home NY burrough of Queens. During the night, he is neighborhood-friendly vigilante Spiderman. Using a make-shift spider suit, he shoots webs from his hands as he hops from building to building, thwarting petty crime and being the all-around neighborhood good guy. He is being put up by his aunt Mae (Marissa Tomei) in Queens, whom he views essentially as his mother. It is implied by the movie that Peter’s parents are either dead or unavailable.

During the airport confrontation featured in Captain America: Civil War, Spiderman was on Tony Stark’s side. For his services, he is given an advanced suit and a “line” to Tony Stark’s personal assistant, Happy. Excited, Peter manages his daily routine in such a way that he finishes school duties as fast as he can and then he joins the so-called “Stark internship”, which is essentially a misnomer for “I’m fighting petty Queens crime until Happy calls me for an actual mission.”

The movie is entertaining. There’s short, biting humor, some funny highschool stereotyping based on a teenager’s changing body and psychology, Holland is a very good choice, and so is Keaton (Vulture, the main antagonist). With Keaton’s character, however, begin some problems.

Statement #1: “Evil government” here to steal the fruits of your labor

The movie actually begins with a shot right after the Battle of New York, so there’s a flashback to 10 years before “present day”. The Vulture leads a team of scavengers who are collecting and sorting the alien invasion’s fallout (exotic minerals, weapons, fuel). This “business” is lucrative, since it was made obvious from the invasion proper that the materials the aliens use have significant energy and weaponization parameters that could revolutionize human industrial endeavor, at both a small scale (small / medium businesses) and large scale (factories). Of course, peddling the weapons themselves in the black market is the most straightforward way for a quick, huge cash flow. The Vulture is obviously interested in this latter venture (he is the arch-villain, after all).

The Vulture has assembled this team by using his own money, arranging for trucks to bring the workers from other areas of the country, set up tents, equipment, arranged buyers, etc. Very few minutes into the movie, an unnamed federal government entity swoops in and informs the crew that as per a certain Executive Order, the area is now under the jurisdiction of the Feds. The people making up this entity are portrayed very coldly, in sharp black suits, led by a seemingly emotionless and detached woman, who responds to The Vulture’s protests with “sorry, there’s nothing we can do”.

And it is these protests that are the problem. Perhaps in an effort to offer some kind of logical basis about how Keaton’s character actually ends up being the Vulture, the movie shows what this governmental overtaking means for Keaton and his crew; Keaton has “overextended” (verbatim), assembling this crew using his own finances and bargaining with significant income that can support the families of his workers. This he makes quite evident to the officials arriving on the scene, even punching one out of sheer classic white man working-class frustration.

Dude, no. Alien shit has fallen from the sky. The CDC and DHS MUST weigh in on how much of this stuff is radio-active and all-around dangerous. Just because you went out on a limb to help people with what is a most definitely illegal operation doesn’t mean that the government is the bad guy here. You’re the idiot. Not to mention that it’s quite clear what you’re planning on doing with the weapons. 

In times such as this that  an inbred anti-authoritarian is head of State and his Neanderthal – level supporters equate government with spending and spending is bad and taxes and don’t take my freedoms and other such BS, this is not something that you want to popularize.

Statement #2: “Ultra (???)-liberals” are annoying

One of the protagonist’s friends is a young girl with a strong political stance. She is not comfortable with the status quo and displays a very distinct way of thinking and a general aura of not being comfortable in her own shoes, since other people have had, have, and will very likely still have it worse than her. She is forced to read a piece of dialogue where she tells the protagonist and his best friends that it’s somehow completely natural that they have no friends (see statement #4) and that she doesn’t either. Of course. Politically involved people are outcasts that should have no friends.

Statement #3: Thighs over brains, yet again

The protagonist’s erotic interest is an oversexualised senior. The girl playing is very beautiful, no doubt. Tall, beautiful skin and lips, deep eyes, beautiful hair.

And that is pretty much her role in the movie. In a STEM-oriented highschool, the directors have the opportunity to elevate the position of women by clearly depicting desirable role models: a beautiful woman who can also enter a lab and make the same mistakes and the same breakthroughs as everybody else. Instead, she’s leading the debate team, and her own contributions to the debate team are from the managerial standpoint. Nowhere in the sequence of the movie is there a single implication that this woman, in addition to visually striking, can be intelligent.

Statement #4: Geeks are unpopular and unattractive

This hits home a bit, since I’m a computer geek. The protagonist’s best friend is portrayed as an overweight, unpopular kid who, when not in the lab or not hacking passwords, spends his day eating Cheese Puffs and watching Netflix / getting high (the last activities are not actually portrayed in the movie, but anybody with half a brain -in short supply these days, I know – can tell that this is  exactly the mental picture that we are fed with). No, Jon Watts. Geeks can wear nice clothes, or not. They can be sexually active, or not. They can eat Cheese Puffs, or Quinoa soup and salads. They can be not stereotypes. It’s their choice. Show them this. Give me something original. Give me a geek that is actually grounded in the real world. Tell our undergraduate students who are struggling to find their identity that it’s ok to be socially awesome, sexually active, politically involved. Show them as I see them in reality, people who are struggling emotionally yet are growing in all sorts of weird ways. Give us an original example of what is really going on. Challenge my worldview. You have the power; your audience is millions that are watching the show absent-minded or high: tickle their subconscious by passing the right message.

Some might say that I’m being over-sensitive and over-feminist. The latter word makes no sense in Jasonland and it’s quite dangerous, but this is a discussion for a different time. But I very much want to apply emphasis on the use of the preposition “over”: No. I am not being over-whatever. I’m just challenging your world view. I’m challenging dangerous stereotypes. There exist stereotypes that are either not dangerous at all, or, if we want to ground ourselves in the real world where absolutes don’t exist, infinitesimally dangerous. For example, stormtroopers in Star Wars always missing their target, which extends towards pretty much every action movie there’s ever been: bad guys are terrible shots. I guess this stereotype could be harmful if people want to go out there and shoot criminals, since in the real life, even a bad shot is better than a movie bad guy shot….

In a time when we hear all kinds of degrading comments about minorities and women, with the very president having suggested groping women’s genitals, we have to look deeply into our own behaviors and our inputs and thing: “how did this make me more into a Trump today?” Or, we can choose not to and continue the proliferation of male-only jokes, and “no girls in guys’ nights”. Because we want to be sexist and call women “bitches” and we don’t want girls around because they will be offended. Yeah, of course they will. You’re calling them bitches. Would you like it if women called you or the entirety of male-hood a bastard or an asshole during ladies’ night? Do you not get aggravated when you hear statements such as “all men are the same”?

It’s very, very easy to get offended without trying to rationally think why the other person is thinking the way they are thinking. I have to put myself in the equation because lately I’ve been hearing so many things that defy my logical pattern of thinking that all I get to is contradictions. But it’s much harder, and a sign of a better person, a person I want to become, to be able to understand how the person reached the conclusion, and whether they are “over” – sensitive or “over” – not sensitive. Maybe then I wouldn’t be calling Trump supporters Neanderthals.

What Trump means for a white, privileged male from a different country

I was born and raised in Athens, Greece, leaving the country for a PhD in Computer Science in UMD back in 2012. While it is true that Greece has been hit hard by the recession since 2009, what most people won’t tell you is that even in times of financial prosperity, it’s always been a corrupt, stagnant, sexist, racist and all-around horrible place to live in if you have any kind of empathy for your fellow human being and you are entertaining thoughts such as actual, compassionate gender equality and gay people not being devil’s spawn. To quote a certain video game character, most of modern Greeks would not be able to tell right from wrong if one of them was helping the poor and the other one was banging their sister.

I didn’t take Trump lightly at all. I slept 2 hours that night and woke up with the taste of a bunch of stale cigarettes in my mouth. Held my ordinary office hours during a day that half the workplace was absent without even bothering to call in; who can blame them? Depressed students were seeing me about their quizzes and homeworks and stuff. Begging the question of whether anything of that sort actually mattered.

It’s futile to rehash what’s happened in the political arena since. Suffices to say that not even Greeks would be dumb enough to vote for somebody like that, Russian probe notwithstanding. What’s not futile to bring up is a fantastic question that I was asked by a friend of mine who wanted to challenge my thinking and make me feel better in the months that followed the election, months during which my line of thinking was completely binarized: I’m either Greek or American, I’m either lazy or hard-working, I’m either liberal or conservative. Nothing in between. The question was:

“Why are you so affected by Trump? As a white male invested into mutual funds of companies who can only benefit from deregulation, as a non-constituent who could leave at any point and go to Canada (I have Greek and Canadian citizenship), as a non-Muslim white person with a car and a credit card, why is it that you are having that hard of a time with this?

It was a fantastic question, and it’s taken me some time to find an adequate response. I’ve also had to take time off of the news to avoid aggravation. Since August, there’s been no Facebook, no Washington Post, no front-page news, nothing. Home->Work->Home. It’s been going nicely. Lets my mind clear of the stupid shit you are likely hearing every day and leaves me time for my music.

Trump affects me because his presidency makes me understand that I am a person without a country, despite having two passports. I speak Greek, my family is in Greece, but I don’t feel Greek. I haven’t for many years. I take pride in my work and I hold Science as the ultimate ideal. I can’t live in an environment like Modern Greece, where the path of least resistance is glorified. Where higher education and healthcare are both constitutional rights for citizens, but have stopped actually working since the 70s. Where women are passively accepting guys disregarding their complaints because “it’s probably that time of the month”. Where black people are considered subhumans and are the subject of very unsettling jokes, even for somebody with a very dirty mouth like myself.

So where is home then? Is it the US? Where we’ve had an African-American published neurosurgeon declare, on camera, that slaves were just “immigrants who worked very hard”? Where cheering Trump NYC voters yell “back to the reservation, faggot” to gay people after the election was over? Where there’s still debates over whether it’s good to actually give people assault rifles? Still? In 2017? Where every 20-30 year old spends their free time drinking at bars again and again and again? Where the social inequality is so obvious that the best students in our major come from three elite highschools? Where everybody is courteous and professional but nobody actually cares about anybody but themselves? Because, no matter how many “How are you doing today”s you tell me, I know that if I were in need, you wouldn’t give two shits. In Greece, if you say “hi” on the phone, you would get -at best- a welcoming grunt. But if you were hurt in a car accident, or if you were so sad you couldn’t lift yourself from bed, or if you were hurt by a loved one, you would immediately get strangers, actual strangers, hold your hand, buy you a beer and dance with you. Because Greeks, no matter how corrupt, backwards, smelly, creepy, sweaty they might seem, are actually in touch with the only thing that matters: people. It’s all about people, and going out with people, laughing with them. Loving them with all that you got at that point in time, at that location in space.

I was the proudest man in the world to come to the US. My previously documented health troubles were the first blow. An understanding that the healthcare system of the most powerful country in the world can’t figure out how to take a blood test. Avoiding giving a patient medications that will actually push them closer to death appears to be an area of active research. I was tempted to stay back during my recovery in late 2015. My PhD in the USA was in shambles, researchers I knew in Athens reached out, and I was even formally accepted into the PhD program of my alma matter. But I changed my mind and in February 2016 I came back to the States with less than what I had in 2012; a hostile advisor who couldn’t accept the fact that I fell badly sick, no guarantees of funding or Visa status 3 months after my plane landed, no signed lease. Just a suitcase and a plan. Because my will to actually excel in Computer Science and be able to do something was still irresistible. Because that has always been what the USA meant for me: Unbounded potential for somebody skilled and motivated. And so I was, letting go of the cancerous doctorate and working as an instructor of 500+ students every semester and loving every single one of them and myself more, day by day.

And then Trump.

So all this is the answer to my friend’s question, which I believe is shared by many foreign over-achievers who wanted to come here out of pure will, pure fire to excel in their trade. Unless you are one of us, you cannot hope to understand our extreme mental dichotomy and our quiet, overwhelming despair about where exactly on this goddamn planet we belong. Because even if we are in the upper echelons of Maslow’s Pyramid, this doesn’t make us immune to pain, mental health issues and existential crises. We still hurt because we have compassion, because we are logical people who can’t accept the dissemination and glorification of idiocy. Because we have invested time, money and tears in our education and expect the decision makers to be educated people who understand Science and think with equal parts numbers and compassion. And, in the US of all countries, it is stupidity that now rules.

Let me know when all this ends, please. I won’t be able to tell, because I no longer follow the news.

A deep take on countability, cardinality and ordering

I’ve been teaching CMSC250, Discrete Mathematics, over the past year in CS UMD. Last semester, I typed a more philosophical than mathematical post on Countability, Cardinality and Ordering, which I’m repeating here for the community’s sake.

After our ordinality lecture last Tuesday, I had a student come to me and tell me that they were not sure how to think about ordinality: they were understanding the relationship between cardinality and size, since it is somewhat intuitive even for infinite sets (at least to them!), but ordinality still appeared esoteric. That’s 100% natural, and in this post I will I’ll try to stray away from math and try to explain how I think about countability, cardinality and ordinality intuitivelyThis post has exactly zero things to do with the final, so if you want to limit your interactions with this website to the exam-specific, you may stop reading now.

Before we begin, I would like to remind you of a definition that we had presented much earlier in the semester, I believe during an online quiz: A set S is dense if between any two elements of it, one can find another element. Note something interesting: only ordered sets can be qualified as dense or not! Technically, we had not presented the notion of an ordered set when we discussed dense sets, but it is intuitive enough that people can understand it.


We say that any enumerable set is countable. Enumerable, mathematically, means that we can find a bijection from the non-zero naturals to the set. Intuitively, it means “you start from somewhere, and by sequentially making one step, no matter how long it takes, you are guaranteed to reach every single element of the set in finite time”. Whether this finite time will happen in one’s lifetime, in one’s last name’s lifetime, or before the heat death of the universe, is inconsequential to both the math and the intuition. Clearly, this is trivial to do for either the non-zero naturals or the full set of naturals: you start from either 1 or 0, and then you make one step “forward”.

However, we also saw in class that this is possible to also generalize for the full set of integers: we start from 0 and then start hopping around and about zero, making bigger hops every time. Those hops are our steps “forward”.

Those results are probably quite intuitive to you by now, and I feel that the reason for this might that both LaTeX: \mathbb{N} and LaTeX: \mathbb{Z} are non-dense sets.There are no naturals or integers between LaTeX: n and LaTeX: n+1 (LaTeX: n \in \mathbb{N}  or LaTeX: n \in \mathbb{Z} ).

Let’s stray away from LaTeX: \mathbb{Q}  for now and fast-forward to LaTeX: \mathbb{R} . We have already shown the mathematical reason, Cantor’s diagonalization, for which the set of reals is uncountable. But what’s the intuition? Well, to each their own, but here’s how I used to think about it as a student: Suppose that I start from zero just to make things easier with respect to my intuitive understanding of the real number line (I could’ve just as well started with LaTeX: -e^8). 

Then, how do I decide to make my step forward? Which is my second number? Is it 0.1? Is it -0.05? But, no matter which I pick as my second number, am I not leaving infinitely many choices in between, rendering it necessary that I recursively look into this infinite interval? Note that I have not qualified “infinite” with “countably infinite” or “uncountably infinite” yet. This was my personal intuition as a Discrete Math student about 11 years ago about why LaTeX: \mathbb{R}  is uncountable: Even if you assume that you can start from 0, there is no valid ordering for you to reach the second element in the sequence of reals! Therefore, such a sequence cannot possibly exist!

But hold on a minute; is it not the case that this argument can be repeated for LaTeX: \mathbb{Q} ? Sure it can, in the sense that between, say, LaTeX: 0 and LaTeX: \frac{1}{2}, there are still infinitely many rationals. It is only after we formalize the math behind it all that we can say that this is a countable infinity and not an uncountable one, as is the case of the reals. But still, we have to convince ourselves: why in the world is it that the fact that every one of these infinite numbers can be expressed as a ratio of integers make that infinity smaller than that of the reals?

Here’s another intuitive reason why we will be able to scan every single one of these numbers in finite time: everybody open the slide where we prove to you that LaTeX: \mathbb{Q}^{>0}  is countable using the snaking pattern. Make the crucial observation that every one of the diagonals scans fractions where the sum of the denominator and the numerator is static! The first diagonal scans the single fraction (LaTeX: \frac{1}{1}) where the sum is 2. The second one scans the fractions whose denominator and numerator sum is 3 (LaTeX: \frac{1}{2},\ \frac{2}{1}). In effect, the LaTeX: i^{th} diagonal scans the following fractions:

LaTeX: \{ \frac{a}{b} \mid (a,b \in \mathbb{N}^{\geq 1}) \land (a + b=i+1)\}

For those of you that know what equivalence classes are, we can then define LaTeX: \mathbb{Q}^{>0}  as follows:

LaTeX: \mathbb{Q}^{>0} = \bigcup_{i \in \mathbb{N}}\{ \frac{a}{b} \mid (a,b \in \mathbb{N^{\geq 1}}) \land (a + b=i+1)\}

Let’s see this in action…

LaTeX: \mathbb{Q}^{>0} = \{ \color{red}{\underbrace{ \frac{1}{1}}_{i=1}}, \color{blue}{ \underbrace{\frac{1}{2}, \frac{2}{1}}_{i=2}} , \color{brown}{\underbrace{\frac{1}{3}, \frac{2}{2}, \frac{3}{1}}_{i=3}}, \dots \}

Note that essentially, with this definition, we have defined a bijection from LaTeX: \mathbb{N^{\geq 1}} \times \mathbb{N^{\geq 1}} to LaTeX: \mathbb{Q}. We know that LaTeX: \mathbb{N}^{\geq 1} \times  \mathbb{N}^{\geq 1} is countable, so we now know that LaTeX: \mathbb{Q}^{> 0}  is also countable! 🙂

Let’s constrain ourselves now to the original challenge that we (I?) are faced with: we have selected 0 as our first element in the enumeration of both LaTeX: \mathbb{Q}  and LaTeX: \mathbb{R}  (the latter is assumed to exist), and no matter which our second element is (say it’s LaTeX: \frac{1}{2}), we have infinitely many elements in both sets between 0 and LaTeX: \frac{1}{2}But now we know that those infinites are different: in the case of LaTeX: \mathbb{Q} . we know for a fact that we will reach all of those fractions whose decimal values are in LaTeX: (0, 0.5). In the case of LaTeX: \mathbb{R} , there is no such enumeration: any enumeration we define will still leave an… uncountably infinite gap between any two elements in “sequence”.

Remember how in our lecture on Algebraic and Transcendental numbers, we gave only three examples of numbers in LaTeX: TN, yet the fact that LaTeX: TN is uncountable when LaTeX: ALG is countable guarantees that there are “many more” Transcendental numbers than Algebraic? Same thing applies here with the rationals and irrationals: given any interval of real numbers LaTeX: (r_1, r_2), there are many more irrationals than rationals inside that interval... If you define a system of whole numbers (integers), there are many more quantities that you will not be able to express as a ratio of integers. That’s why back in the day (300 B.C) when Euclid proved that LaTeX: \sqrt 2 is not expressible as such a ratio LaTeX: \frac{a}{b} (or, more accurately, that LaTeX: 2 cannot be expressed as the square LaTeX: \frac{a^2}{b^2}) his result was so unintuitive; those Hellenistic people did not have rulers. They did not have centimeters or other accepted forms of measurement. The only thing they had were shoestrings, or planks of wood which they put in line and “saw” that they were the same length, and then they measured everything else as the ratio of such “whole” lengths.


Recall something that we said when we were discussing the factorial function and its combinatorial interpretations when applied on positive integers. Bill’s explanation of why LaTeX: 0!=1 was purely algebraic: If it were LaTeX: 0, then, given the recursive definition LaTeX: n!\:=\:n\:\cdot\left(n-1\right)! for LaTeX: n\:\ge1, every LaTeX: n! would be LaTeX: 0rendering it a pretty useless operation. My explanation was combinatorial: we know that if we have a row of, say, LaTeX: n marbles, there are LaTeX: n! different ways to permute them, or LaTeX: n! different orderings of those marbles. When there are no marbles, so LaTeX: n=0there is only one way to order them: do nothing, and go watch Netflix. 

Let’s stick with Bill’s interpretation for a moment: the fact that some things need to be defined in order to make an observation about the real world work. In this case, the real world is defined as “algebra that makes some goddamn sense”. My explanation is more esoteric. You could say: “What do you mean there’s only one way to arrange zero things? I don’t understand, if there are zero things and there’s nothing to do, shouldn’t there be, like, 0 ways to arrange them?”. So, let’s stick with Bill’s interpretation to explain something that I attempted to explain to a group of students after our first lecture this semester: Why do negative numbers even exist?

Here’s one such utilitarian explanation: Because without negative numbers, Newtonian Physics, with their tremendous application in the real world, would not work. That is, the model of Newtonian Kinematics with its three basic laws, which has been empirically proven to describe very well things that we observe in the real world, needs the framework of negative numbers in order to, well, work. So, if you’re not ok with the existence of negative numbers, you had better also be able to describe to me a framework that explains a bunch of observations on the real world in some way that doesn’t use them. For example, you probably all remember the third law of Newtonian motion: For every action LaTeX: \color{red}{\vec{F}}, there exists an equal and opposite reaction LaTeX: \color{red}{-\vec{F}}:

Recall that force is a vectoral quantity since it is the case that LaTeX: \vec{F} = m \cdot \vec{a}, and acceleration LaTeX: \vec{a} is clearly vectoral, as the second derivative of transposition LaTeX: \vec{x}

The only way for Newton’s third law of motion can work is if LaTeX: \vec{F} + (-\vec{F}) = \vec{0}. This is only achievable if the two vectors have the same magnitude but exactly opposite directions. No other way. Hence the need to define the magnitudes as follows:

LaTeX: | |\vec{F}|| = \frac{1}{2} \cdot m \cdot a^2,\ | |\vec{\color{red}{-}F}|| = \color{red}{-}\frac{1}{2} \cdot m \cdot a^2

and the necessity for negative numbers becomes clear. Do you guys think the ancient Greeks or Egyptians cared much for negative numbers? They were building their theories in terms of things they could touch, and things that you can touch have positive mass, length, height…

Mathematics is not science. It is an agglomeration of models that try to axiomatize things that occur in the real world. For another example, ZFC Theory was developed in place of Cantorian Set Theory because Cantorian Set Theory can lead to crazy things such as Russel’s Paradox. Therefore, ZFC had to add more things to Set Theory to make sure that people can’t do crazy stuff like this. If we discover contradictions with the real world given our mathematical model, we have to refine our model by adding more constraints to it. Less constraints, more generality, potential for more contradictions. More constraints, less generality, less contradictions, but also more complexity.

So when discussing the cardinality of LaTeX: \mathbb{N}  and LaTeX: \mathbb{Z}  and finding it equal to LaTeX: \aleph_0, we are faced with a problem with our model: the fact that LaTeX: \color{magenta}{\mathbb{N} \subset \mathbb{Z}} (I have used the notation of proper subset here deliberately). Now, I just had a look at our cardinality slides, and it is with joy that I noticed that we don’t use the subset / superset notation anywhere. That’s gonna prove a point for us.

So, back to the original problem: intuitively understanding why the hell LaTeX: \mathbb{N}   and LaTeX: \mathbb{Z}  have the same cardinality when, if I think of them on the real number line, I clearly have LaTeX: \mathbb{N} \subset \mathbb{Z}:

LaTeX: \underbrace{\dots, -4, -3 , -2, -1, \underbrace{0, 1, 2, 3, 4, \dots}_{\mathbb{N}} }_{\mathbb{Z}}

The trouble here is that we have all been conditioned from childhood to think about the negative integers as “minus the corresponding natural”. This conditioning is not something bad: it makes a ton of sense when modeling the real world, but when comparing cardinalities between infinite sets, that is, sets that will never be counted entirely in finite time, we distance ourselves from the real world a bit, so we need a different mathematical model. To that end, let’s build a new model for the naturals. Here are the naturals under our original model:

LaTeX: 0, 1, 2, 3, \dots

This digits that we have all agreed to be using have not been around forever. The ancient Greeks used lowercase versions of their alphabet: LaTeX: \alpha, \beta, \gamma, \delta , \epsilon, \sigma \tau ', \zeta,\ \dots\ \omega  to name a total of 25 “digits”, while the Romans used a subset of their alphabet “stacked” in a certain way: LaTeX: I, II, III, IV, V, VI, \dots, X, XI\dots . These “stacked” symbols cannot be really called digits the way that we understand them, especially since new symbols appear long down the line (LaTeX: C, M) etc. These symbols we actually owe to the Arabic Renaissance of the early Middle Ages.

The point is that I can rename every single one these numbers in a unique way and still end up with a set that has the exact same properties (e.g closure of operations, cardinality, ordinality) as LaTeX: \color{red}{\mathbb{N}}. This is formally defined as the Axiom of Replacement. So, let’s go ahead and describe LaTeX: \mathbb{N}  by assigning a random string for every single number, assuming that no string is inserted twice:

LaTeX: foo, bar, otra, zing, tum, ghi,\dots

Which corresponds to our earlier

LaTeX: 0, 1, 2, 3, 4, 5,\dots

Cool! Now the axiom of replacement clearly applies to LaTeX: \mathbb{Z}  as well, so I will rewrite

LaTeX: \dots, \color{blue}{-5, -4, -3, -2, -1,}\ \color{magenta}{0, 1, 2, 3, 4, 5,}\dots


LaTeX: \dots, \color{blue}{qwerty, forg, vri, zaq,  nit,}\ \color{magenta}{bot, ware, yio, bunkm, ute, kue,}\dots

Call these “transformed” sets LaTeX: \mathbb{N}_{new} and LaTeX: \mathbb{Z}_{new} respectively. Under this encoding, guys, I believe it’s a lot more obvious that LaTeX: \mathbb{N}_{new} \not\subset \mathbb{Z}_{new} in the general case. LaTeX: \mathbb{N}_{new} \subset \mathbb{Z}_{new} under these random encodings is so not-gonna-happenish that its probability is not even axiomatically defined. Therefore, now we can view LaTeX: \mathbb{N}  and LaTeX: \mathbb{Z}  as infinite lines floating around space, lines that we have to somehow put next to each other and see whether we can line them up exactly. If you tell me that even under this visualization, the line that represents LaTeX: \mathbb{Z} _{new} is infinite in both directions, whereas that of LaTeX: \mathbb{N}_{new}  has a starting point (0), then I would tell you that I can effectively “break” the line that represents LaTeX: Z_{new} in the middle (0) and then mix the two lines together according to the mapping that corresponds to:

LaTeX: 0, 1, -1, 2, -2, 3, -3, \dots

Now we no longer have the pesky notation of the minus sign, which pulls us to scream “But the naturals are a subset of the integers! Look! If we just take a copy of the naturals and put a minus in front of them, we have the integers!”. We only have two infinite lines, that start from somewhere, extend infinitely, and it is up to us to find a 1-1 and onto mapping between them. That is, it is up to us find a 1-1 mapping between:

LaTeX: foo, bar, otra, zing, tum, ghi,\dots


LaTeX: bot, ware, nit, yio, zaq, bunkm\dots

(Note that I re-ordered the previous encoding LaTeX: \dots, \color{blue}{qwerty, forg, vri, zaq,  nit,}\ \color{magenta}{bot, ware, yio, bunkm, ute, kue,}\dots according to the “hopping” map into  LaTeX: \color{magenta}{bot}, \color{magenta}{ware}, \color{blue}{nit,} \color{magenta}{yio}, \color{blue}{zaq}, \color{magenta}{bunkm},\dots .)

Under this “visual”, you guys, it makes a lot of sense to try to estimate if the two sets have the same cardinality and, guess what, they do 🙂

Not much else to say on this topic everyone. We can have a bunch of applications of the axiom of replacement to prove, for example, that the cardinality of the integers, LaTeX: \aleph_0, is also the cardinality of LaTeX: \mathbb{N} \times \mathbb{N}LaTeX: \mathbb{Q} , etc. It is only when we start considering sets such as LaTeX: \mathbb{R} , \mathcal{P}(\mathbb{N}) and LaTeX: \{0, 1 \}^\omega  that this idea that we can be holding two infinite lines in space fails.


There’s not much to say here except that the easiest way to understand how an order differs from a set is to consider an ordering exactly as such: an order of elements! Think in terms of “first element less than second less than third less than …. “. The simplest way possible. It is then that we can prove rather easily that LaTeX: \omega \prec y \prec \zeta .

Things only become a bit more complicated when considering the ordering LaTeX: \omega + \omega:

LaTeX: 0 < \frac{1}{2} < \frac{3}{4} < \frac{5}{6} < \dots <1 < \frac{3}{2} < \frac{4}{3} <\dots <2 <\dots

Please note that this ordering is clearly not the same as LaTeX: \eta, the ordering of LaTeX: \mathbb{Q} . Between the first and the second element, for instance, there are countably many infinite rationals: LaTeX: \frac{1}{100}, \frac{2}{5}, \dots , \frac{3}{7}\dots  which are not included in the ordering. 

Finally, realize the meaning of “incomparable” orderings: a pair of orderings LaTeX: \alpha, \beta  will be called incomparable if, and only if:

LaTeX: (\alpha \npreceq \beta) \wedge (\beta \npreceq \alpha).

So please realize that this is not the same as saying, for instance, LaTeX: \beta \nprec\alpha .

I think this is all, I am bothered when I can’t explain something well to a student so I thought I’d share my views on countability in case the subject becomes easier to grasp.