Everything Except Most Things

"All right, but apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, a fresh water system, and public health, what have the Romans ever done for us?"

— Reggie, the local Rebel Leader, Monty Python's Life of Brian

An informal Logical Fallacy where a generalization is made that is technically accurate, but has so many exceptions that the group which is generalized is significantly smaller than the arguer implies, and so small as to make the statement essentially meaningless.

For example, Alice says X is great and claims nobody disagrees. Except maybe every member of the crew who isn't Alice. And every student of the school who isn't Alice. And maybe the entire global population except Alice.

Or Bob asks when have his cunning plans ever failed — other than on almost every single occasion he's come up with a cunning plan.

Exceptions to the generalization can be provided when someone challenges someone else to Name One.

In its most extreme form, this trope borders on By "No", I Mean "Yes". Compare Overly Narrow Superlative, Hypocritical Humor, and Immediate Self-Contradiction.

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Examples:

Fan Works

• Jaune Arc in the RWBY fanfic What Happens in Vacuo Stays in Vacuo tries to make a Badass Boast, but it falls a little flat.
  Jaune: The only people who get to boss me around is my boss, my mother, my father, my seven sisters, the man who holds extensive blackmail over my head, the woman who is maybe trying to kill me, Glynda... and this psycho bitch here.
• In Walking In The Shadows, Robin tries to explain to Starfire that humans don't accidentally set their beds on fire, only for everyone else present minus Beast Boy, who doesn't have supernatural powers that would allow it, but including Xander from Buffy the Vampire Slayer and Sirius from Harry Potter, to mention they've done it before:
  "You have never gotten excited and accidentally ignited your bedding?" Star asked curiously.
  "That's not a thing humans do," Robin told her.
  "Not since I was thirteen," Raven said, thinking of her training.
  "I wear a mouth-guard to bed so I don't accidentally mutter in Latin while I sleep," Xander offered. "I lost a couple of blankets and my favorite pillow before I figured that out."
  "Ok, it's not a problem for most humans," Robin corrected himself, as Cyborg, Beastboy, and Sirius came in.
  "What's not a problem for most humans?" Cyborg asked as they joined them at the table.
  "Igniting their bedding," Starfire answered.
  "Haven't had that problem in close to five years," Cyborg said thoughtfully. "I had to put in a couple of power regulators so I didn't activate things in my sleep."
  Sirius laughed. "Now that takes me back. My family was actually considering disowning me as a changeling before I hit puberty and exhibited the Black Fire. I was taught fire suppression spells years before I learned contraceptive charms."
  Everyone turned to Beastboy.
  "I've never set a bed on fire," Beastboy said.
  "You poor dear," Starfire said giving him a hug. "I'm sure any day now you'll reach your manhood."
  "I've already reached my manhood!" Beastboy exclaimed.
  "Without fire?" Starfire asked.
  "While I will admit we aren't most people, the vast majority of humanity does not manifest fire upon reaching puberty," Robin assured her.
• My Hero Playthrough: All Might tells the class they are to offer each other constructive criticism. Izuku thinks the emphasis isn't needed. Except:
  Izuku: I mean, Todoroki-san was a bit cold. Iida-san was strict and somewhat hasty. Ashido-san seemed to sometimes speak without thinking. Mikoto could be a bit harsh. Okay, maybe the reminder wasn't the worst idea.
• In Departure From The Diary, Harry tries to argue against Tamelyn's insistence that all Muggles hate anyone who's "different", saying that just because she had bad experiences doesn't mean they're all like that. Yet when he starts thinking it over, Harry realizes that the Dursleys, his teachers, the neighbors, and every other Muggle he's ever met has been exactly the same.
• Freedom and Not Peace:
  Not you, too, said Harry, switching to silent speech as a Hufflepuff first-year came up the stairs and slipped over to a barn owl on a perch. She kept giving him awed glances. Harry stared out the window and did his best to look like an ordinary tormented hero until she was gone. Everybody wants to punish my mother, for some reason.
  Some reason. This is lots of reasons. How dare she say—
  "I don't want to hear it again," Harry whispered. "Please, Regulus, don't make me live through it. She's been punished. It's enough. Everybody else has agreed to leave it alone." Well, except for Scrimgeour. And Lucius. And Narcissa. And Hawthorn. And Adalrico. But everybody else has.
• Oversaturated World: From the Pest Control description, "Ordinary", except for the ways that she isn't:
  Applejack was just an ordinary farmer.
  
  Well, true, she was best friends with an arguable deity, and yeah her little sister controlled part of fate subconsciously, and her social circle involved a crazy crew of characters that regularly got involved in wacky shenanigans, and she herself had helped to save the world a few times, and there was the possibility her principal was using the school to scout for super heroes, and technically she had acquired a skin condition that she had yet to see replicated by anybody else...
  
  ...but the point was, by most standards, she was an ordinary farmer. Not the kind of girl that went on adventures.
  
  And then she developed a unique form of magic, just in time for the giant bug invasion.
  
  Welp.
• Harry Potter And The Summer Of Change:
  Tonks: You know me too well, Harry. I can't get enough of you and that, good sir, is my only weakness.
  Harry: Except for stairs. Runners, curbs, cloaks some of the time, the covers after a really good...
• It All Started With A Visit To Gringotts:
  Lily's portrait: Thank you for apologising Harry and you are forgiven. It is kind of sweet you want to protect me. I am proud of you, you are a fine man. However you and I will have a talk later about stealing cars, breaking into teacher's offices, sneaking out of school and attacking teachers.
  Harry: The only teacher I attacked was Severus and he has forgiven me...
  Snape: And Quirrell...
  Harry: Yes well he was trying to kill me.
  Snape: And Lockhart...
  Harry: He was a git.
  Snape: And Lupin...
  Harry: He was in werewolf mode.
  Snape: And Umbridge...
  Harry: She was evil.
  Snape: And both the Carrows...
  Harry: ... Okay Severus stop helping, please.
• Hot Shower:
  Ginny: Sex is probably the only thing Fleur or her baby sister have any talent for. It's not like she had any other talents.
  Blaise: Aside from be a Tri-wizard contestant. Working for Gringotts, who are picker about their employees than Draco is about hair care products. And she's a pretty decent fighter.
• To Shape and Change:
  [Snape] shook his head, berating himself for being so selfish. It wasn't like he didn't have the funds himself to get whatever he wished, ingredients wise ... well, save for basilisk parts, unicorn blood, dragon eyes, vampire fangs, and banshee hair, but besides those things he could get whatever he wished.
• The Shock of it All:
  Hermione huffed, she still didn't believe in Divination, nothing controlled her life, except perhaps her parents, and the authorities in the school, and the government; oh bother.
• Trickshot:
  Harry: Rules are made to be broken Tom.
  Tom: Nothing's made to be broken.
  Lisa: Piñatas.
  Harry: Pistachios, well the shell at least.
  Lisa: Unhealthy relationships.
  Harry: Water balloons.
  Lisa: Bad habits.
  Harry: World records.
• Equilibrium:
  Harry still couldn't quite get his mind around that one. . . all his detentions weren't even detentions! Snape was just helping him study, because...well, he really didn't know why. Maybe Snape just liked making him work. Maybe he liked sneering at him and making out like Harry was dim or something. Except, Snape didn't make as much fun of him as he used to. . . he wasn't even very mean and spiteful, most of the time. . . .
  Except in class. . . or in the corridors. . . or the Great Hall, or on the Quidditch pitch, or anywhere else I run into him!
• In A Different Professor Fred states Harry could probably beat the entire Durmstrang dueling team while blindfolded with one hand tied behind his back. Harry replies that as a fourth-year, he couldn't beat them even if they were blindfolded and one-handed.
  Fred: Sure, you could, Harry, you face off against Professor Snape all the time, no one is scarier than he is.
  Neville: Dragons.
  George: Basilisks.
  Draco: Self-named Dark Lords.
  Harry: Moony during a full moon.
  Draco: My mother.
• In Lilies are Lovelier Than Narcissus Draco tries to court Theo by presenting him with a carved flower representing Draco's mother's name. Theo refuses him.
  "What's wrong with my flower?"
  "You didn't carve it—"
  "Besides that! You wouldn't have rejected me if it was just that."
  Theo wanted to say that he would have, but it was true that he might at least have considered Draco's suit if he had followed all but one bit of proper protocol. And if he hadn't spoken insultingly. And if he hadn't got there too late. And if he hadn't been Draco.
• In To the Letter Harry decides to self-study Potions during fourth year instead of attending classes.
  Harry: [Imitating Snape] Instructions are in the book. The only real difference is that the book has a lot more charisma. And less grease. And that I can read the whole thing and see if there are any safety instructions to keep myself from blowing up when the Slytherins throw something into my cauldron.
• Sympathetic Properties:
  Harry waited with bated breath as his best mate seemed to give it his full attention and he swore on his potentially-Slytherin grandparents if everything somehow worked out with his two best friends he'd never say anything bad about Slytherins again. Except for Malfoy, Crabbe and Goyle, he added to himself. And all those other idiots like them. And Snape; definitely Snape. The potions master had earned his spot as pride of place on his list of detestable Slytherins.
• In My Absolute Pleasure Harry complains about a bad blind date.
  Harry: And then he asked me if I 'get much tail, being a hero and all', and that was the point that my insides died and I gave up hope of ever finding someone to love me. I'm going to be one of those old cat ladies.
  Kingsley: Except for the fact that you're not yet even remotely old, you're not a lady, old or otherwise, and you don't like cats.
• Harry Potter and the Muggle's Daughter mentions that Lily refers to Hermione as Harry's partner in crime.
  Harry thought that a bit unfair, they didn't really cause that much trouble. Well, aside from the incident with the chemistry set and picnic table... and the anti-belching potion they'd reversed and tricked Sirius into drinking... and the time they'd washed his swim trunks in itching potion... and when they had tried to use a vanishing charm to clean the chicken coop and ended up plucking the chickens bare instead... The resulting feather tornado had been really cool though. But other than that there wasn't much; except maybe the diet coke and mentos incident with the six girls who had ganged up on Hermione as she'd been on her way to meet him and pushed her in the river. But they had all deserved it, and even though he'd been made to apologize to each of the girls for making them sick, Harry hadn't been repentant at all. Harry couldn't help smiling at the memory. Alright, he conceded, maybe he and Hermione did tend to cause a bit of mayhem.
• Harry Potter and the Chamber of Secrets:
  Down and down he went. The pipe was wet and nasty with slime and as he whizzed past he noticed other pipes branching off. But none were as big as this one. The drop steepened and Harry picked up even more speed. Really, if it weren't for the circumstances it would have been fun. It wasn't any different from the slides at the water park in Disney World after all.
  
  Well, except for the slime. And the stench. And the deadly snake that could kill you with just its eyes that was probably waiting for them at the bottom. And the Heir of Slytherin... But other than that, yeah, it was great fun.
• Harry Potter and the Lady of the Lake:
  "Be quiet." Harry tuned back in to Daphne hissing angrily at a group of third-years who hadn't turned their attention to the Deputy Headmistress quickly enough for her tastes. He hid a smirk as the group fell silent under her glare. Daphne could channel McGonagall's imperious demeanor quite well when she chose to. He was immune to it, thankfully. But then he was pretty much immune to everyone but his mum, Andi and Mali anyway. To a lessor extent Mr Granger could quell him, but that was about it. Well, maybe Ginny and Hermione too... and Luna, definitely her. Ok, maybe he wasn't as immune as he wanted to believe.
• In Harry Potter and the Interlude Luna mentions that she and her father went to Croatia and found a Wioprine, which is bright blue and has five horns.
  Harry had never heard of blue animals before, well except for some fish and whales, and some birds, and lizards, and frogs, and wow, there were a lot more blue animals than he had thought.
• Presence:
  Days went by, and Harry found himself back on the roof. But this time something was off. He could feel something different. He frowned, wondering what was wrong. He trusted his instincts. They hadn't been wrong yet.
  
  Okay, well, there was that one time, when he thought Snape was the one after the Stone... and then when he'd thought Malfoy had opened the Chamber... and when he'd thought that Sirius was really a murderer... and when he- Okay, so he's made some mistakes! But the point was, he was sure this time.

Films — Animation

• Madagascar: Escape 2 Africa: While courting Gloria, Moto Moto tries to express his devotion to her by saying "I promise, the answer will always be 'Yes'. Unless 'No' is required".

Films — Live-Action

• In Monty Python's Life of Brian: "All right, but apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, a fresh water system, and public health, what have the Romans ever done for us?"
• Used as a gag in Austin Powers in Goldmember, where Dr. Evil orders everyone to leave after a moment with his son Scott, then stops everyone except Mini-Me from actually leaving.

Literature

• Take A Girl Like You by Kingsley Amis, comparing sex to beer (also a Dissimile):
  Except that this is harder to get and the beer doesn't have to like you and you think about this more and you're proud of it afterwards and you're not supposed to have it, I suppose it is like a glass of beer. To some people. Except that this is much nicer.
• Horrible Histories used this to describe ancient Athenian democracy. It was the first democracy ever, and everyone had the vote. Except women, of course. And slaves. And poor people. And people under the age of thirty. Otherwise, everyone had the vote.
  "Groovy!"
• Discworld also parodies the Athenian example with the Fantasy Counterpart Culture of Greece, Ephebe, and again when democracy is introduced to the Kingdom of Lancre.
  • The philosophers in Ephebe scoff at the idea of gods as relics of an outmoded belief system. Then make exceptions for an increasing number of individual gods as they all send an Implied Death Threat based on their domain, such as thunder from a clear sky. After confirming that they are well away from the nearest snowy mountains, they take some time to mock the god of avalanches, until Brutha brings up that it seems unusually cold.
  • Nobby Nobs is insistent that he didn't sell all the army's weapons when he was quartermaster in the Pseudopolis army. The weapons were just sometimes taken away for cleaning and maintenance. When pressed, he estimates that they were in maintenance approximately 100% of the time.
• Winnie the Pooh: When Tigger is introduced, Pooh asks him whether Tiggers like honey and Tigger assures him that "Tiggers like everything". When he discovers he actually doesn't like honey, this becomes "everything except honey", so Pooh suggests he try haycorns, after which it becomes "everything except honey and haycorns", and eventually "everything except honey and haycorns and thistles". When the matter of finding something Tigger will like is brought to Christopher Robin, Tigger still insists that he likes "everything there is in the world", except everything he's actually tried.

Live-Action TV

• Blackadder:
  • Blackadder II
    • "Potato": When Captain Rum informs Blackadder that his ship doesn't have a crew, Blackadder replies that he was under the impression that it was common maritime practice for a ship to have a crew. Rum replies that opinions are divided on the subject: All the other captains say it is, he says it isn't.
    • Mrs. Ploppy the prison cook during Blackadder's stint as lord high executioner, who gave the prisoners anything they asked for their last meal provided they asked for sausages.
      Mrs. Ploppy: Otherwise they tend to get a tiny bit disappointed, sausages is all I've got.
  • Blackadder Goes Forth, "Captain Cook": General Melchett tells Blackadder to keep a mission top secret. Then Captain Darling provides the list of people with security clearance:
    Melchett: You and me, Darling, obviously. Field Marshal Haig, Field Marshal Haig's wife, all Field Marshal Haig's wife's friends, their families, their families' servants, their families' servants' tennis partners, and some chap I bumped into the mess the other day called Bernard.
    Blackadder: Quite so, sir, only myself and the rest of the English-speaking world is to know.
• Brooklyn Nine-Nine:
  Jake: Rules are made to be broken.
  Amy: They were made to be followed. Nothing is made to be broken.
  Jake: Uh, piñatas.
  Rosa: Glow sticks.
  Jake: Karate boards.
  Rosa: Spaghetti when you have a small pot.
• In Monty Python's Flying Circus, the Colonel says: "Now, nobody likes a good laugh more than I do, except perhaps my wife and some of her friends. Oh yes, and Captain Johnson. Come to think of it, most people like a good laugh more than I do, but that's beside the point."
• Inverted on Yes, Minister, when Hacker is arguing that high ranking civil servants don't need to receive special honours and knighthoods for doing a job they're already paid for. Humphrey states that civil servants work for a modest wage and deserve the added incentive, prompting Hacker to counter that Humphrey makes more than him. Humphrey then tries to amend it to "relatively modest wage."
  Jim Hacker: Relative to whom?
  Sir Humphrey: Well... Elizabeth Taylor, for example.
• In the second season of Galavant, it's revealed that King Richard's people have turned the country into a proto-democracy. But obviously they're not going to let women vote. Or the poor, or non-Christians, or the handicapped, or the illegitimate, or anyone with radical ideas... Galavant does take a moment to note that it's still pretty progressive for the Middle Ages.
  "The landed, and the wealthy, and the pious, and the healthy, and the straight ones, and the pale ones, and we only mean the male ones. If you're all of the above, then you're okay!"
• In Red Dwarf, when Lister loses Rimmer in a card game, the Cat assures Rimmer that the other three members of the crew are really upset about it "except for me and him and him."
• In Schmigadoon!, Melissa tries to reassure Josh, fearing for his life for breaking off a Shotgun Wedding, telling him that "nobody gets killed in a musical"…only to immediately defeat her own point by citing four well-known musicals in which someone does get killed (Oklahoma!, Carousel, South Pacific, and West Side Story). An even more alarmed Josh replies, "that sounds like all the musicals!"

Music

• From "If Yr Famous and You Know It Sack Yr Band" by Custard:
  And the only time I cry now is when I'm sad...which is most of the time...

Theater

• In The Marriage of Figaro, during Figaro's rant near the ending:
  Figaro: They tell me a new system of freedom of press has been instated, and as long as one doesn't talk of <long list> one can say pretty much anything.

Video Games

• In City in Fear if you ask Clancy for further info on murder victims Carlos, Michael and Larry, he states his belief that they were aliens.
  Clancy: Yeah boss, hear me out. The plaza is a well known hotspot for paranormal investigations.
  Player character: Is it?
  Clancy: Yes, boss. It says so in my latest copy of the Alien Hunters Guide and that's never been wrong before - oh, except about the bit where it said George Bush was going to be abducted by aliens and replaced with a cabbage during his first year in office, oh and the time they said Bill Clinton was actually Richard Nixon in disguise, oh and maybe a couple hundred other things... Anyway, they said the plaza is a hotspot for aliens so I reckon these three guys were aliens and they came here to meet their alien cronies only this alien assassin - and that's some guy who assassinates aliens, boss, not some guy who's an alien who assassinates if you get my drift - I reckon this alien assassin came here an' assassinated 'em. So whatdo you think, boss?
• Oracle of Askigaga: The first Guard in the game, when talked to with "nothing", except "[this stuff]":
  Guard: Nothing to report, General!
  [Sweatdrop-type Pictorial Speech-Bubble]
  Well, except for the thief that tried to get by. And her accomplice that tried to distract us by being some merchant trying to sell... whatever.
  Oh yeah, there was that gal that was dressed as a maid, but there totally wasn't any maid-shifts scheduled for a while! Other than that, nothing to report!
  Oharu: You call that "nothing"?
  Hiroji: They are not matters that should concern a person of your position. I believe that's all she meant.
  Oharu: I suppose there is that point, but...
  Hiroji: Then don't concern yourself with it!
• The Simpsons Hit & Run:
  • If you engage Moe the bartender one of his responses is "Oh, my life is in a crapper. But other than that, ah, you know, I'm fine".
  • One of Lisa's Dummied Out lines is "I'll eat anything! As long as it's organic, vegan, and cruelty-free".
• Spider and Web: Checking your inventory after entering the building will tell you that you're carrying nothing important, except for all the spy equipment.
  You are carrying nothing worthy of attention, except a lockpick and a toolcase (which is open) (in which are a minilamp, a scan scrambler, an acid pack, a blast tab, a voice module, a radio module, a timer, a toggle switch, a pushbutton, a voice transmitter and a key transmitter).
• Team Sonic Racing had a promotional video featuring Arin, The Completionist, and Jun Senoue singing a song about Big the Cat, with a lyric saying that everyone loves him except for most people.

Web Animation

• Gooseworx, in her 2023 video "No April Fools Day Video This year...", claims that she has "never missed an April Fools day upload except for 2017 and 2015 and 2013 and 2011".
• Homestar Runner: In Strong Bad's Dangeresque movie series, the protagonist/title character has an example as one of his catchphrases.
  Dangeresque: I work alone! 'Cept when I work with Renaldo... which is all the time.

Webcomics

• Paranatural: Jean tries to decline the delinquent trio's offer to hang out by saying their relationship is "professional", but undercuts his claim by acknowledging the myriad of ways he's enabled their delinquency.
  "Look, guys. What he means is... I'm fine, y'know, not ratting you out for slackin' off, and letting you smoke, and smoking with you, and lying and saying it was bears when you break stuff and also shoplifting with you and giving you rides and stuff, but, like..." He pushed a stray hair back behind his ear. "What we have is a professional relationship."

Web Original

• In one episode of House to Astonish, they discuss how both DC Comics and Marvel Comics have a tendency to sign creators to "exclusive contracts" with a whole bunch of exceptions for creator-owned work, work for indie publishers etc., and if it would be easier to just say they can do what they like as long as they don't work for those guys.

Web Videos

• Binging with Babish: The description of the biscuits and gravy episode of "Basics with Babish" opens with "It is nearly impossible to turn down biscuits and gravy. Unless you are vegetarian, vegan, celiac, lactose intolerant, kosher, or have even a modicum of appreciation for or understanding of your basic human health."
• In the history of japan, when Kyoto became such a dreamworld the Imperial government stopped giving a shit about running the country, the only way for people to protect their shit from criminals was to hire a samurai!. Everybody was hiring samurai! Correction: Rich important people hired samurai. Poor people who could not afford to hire samurai did not hire samurai.
• Kitboga is a scambaiter popular on Twitch and YouTube. In "I Got Married... To A Scammer!," Kitboga's character promises the scammer to watch any television channel he wants once they're married, so long as it's HGTV, Discovery Channel or FOX News.
• The Scott The Woz video on Mega Man noted that the character is Capcom's golden child, right after Ryu, and Leon Kennedy, and Monster Hunter, [fastforwarding through every other major Capcom franchise], and Arthur, and the Snow Bros.
• Title Pending: Caleb says he'd fight a bear for Bayden, as long as it's neither black, brown nor white. Meaning any of them.

Western Animation

• On Daria after a confidence building assignment by Mr. O'Neill has backfired rather spectacularly, Daria uncharacteristically tries to cheer him up.
  Daria: We know you feel responsible because some of your students, or more specifically all of your students, got depressed over the assignment.
• Robot Chicken parodies FDR's "Only thing we have to fear is fear itself" speech in a skit (broken up into two half with other skit between) where he has to regularly amend actual phobias in at audience inquiry until he has a Long List ending with fear itself.
• Family Guy:
  • In an episode about the founding of Quahog, the settler (played by Peter) mentions his plans for the new land.
    Peter: We'll have a happy new life and we will have equal rights for all...except blacks, Asians, Hispanics, Jews, gays, women, Muslims, uhmm...everybody who's not a white man. And I mean white-white, so no Italians, no Polish, just people from Ireland, England, and Scotland. But only certain parts of Scotland and Ireland. Just full blooded whites. No, you know what? Not even whites. Nobody gets any rights. Ahhh...America!
  • In the episode "Short Cuts", Lois considers getting a short haircut, which Peter forbids. When Lois protests that he cannot forbid her, Peter retorts, “Yes I can. That's the one thing men are allowed to control about women. That and your uterus and how much money you make and also everything else.”
  • In “Wild Wild West,” Peter claims that nothing bad has ever happened because of men in power…
    Peter: And don't say the economy or Iraq or income inequality or racism or Brett Kavanaugh or air pollution or Vietnam or slavery or Watergate or capitalism or #MeToo or homelessness or police brutality or homophobia or Monica Lewinsky or school shootings or Native American genocide or FOX News or Tim Allen or climate change.
• Futurama: In "A Fishful of Dollars", Fry is shocked to learn that in the 31st century adverts are transmitted into people's dreams, which he considers unacceptably invasive, as in the 20th century ads were "only on TV and radio. And in magazines. And movies. And at ball games, and on buses and milk cartons and T-shirts and bananas and written on the sky. But not in dreams! No siree!"

Real Life

• Winston Churchill once described democracy as "the worst form of government except all those other forms that have been tried from time to time".
• Henry Ford reported in his autobiography that he told his management team in 1909 that "any customer can have a [Model T] painted any color that he wants so long as it is black." (In reality, early Model Ts could be painted grey, green, blue, or red; black was not an option until 1914.)
• The commonly cited spelling rule of "I before E except after C" has more exceptions in the English language than it has words that follow it. Even with the addition of "or sounding like AY as in neighbor or weigh," there are still so many exceptions that the "rule" may as well be forgotten.
• In mathematics, such a situation is called ω-inconsistency. A theory that is ω-inconsistent may prove that a natural number with certain properties exists – but simultaneously that it’s not 0, it’s not 1, it’s not 2, nor is it any particular number you can actually name. This doesn’t mean the theory is inconsistent – that is, that it allows you to derive an actual contradiction. It turns out, it’s not always possible to postulate that every natural number must be possible to name, which opens the door for the existence of ‘non-standard’ numbers that are larger than any single number that can be uniquely specified by a formula. This concept played an important role in the proof of the famous incompleteness theorems.

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