Rusell’s Paradox Reversed

Friday, January 16th, 2026

Bertrand Russell once wrote to the eminent mathematician and philosopher
Gottlob Frege to ask him how his system would handle the set of all sets that are not members of themselves.
Frege famously wrote this in the epilogue to the second volume of Basic Laws of Arithmetic:

A scientific writer can scarcely encounter anything more undesirable than, after completing a work, to have one of the foundations shaken. I became aware of this situation through a letter from Mr. Bertrand Russell as the printing of this volume neared completion.

In fact, today Frege is perhaps better remembered for this failure than for all his other work combined. At least this was an interesting mistake that taught mathematicians something. It’s much worse to publish a two-volume 500 page proof only to have an undergraduate notice an obvious mistake on page 2.

Russell’s paradox is a simple way of expressing the statement, “This statement is false” into the set theory Frege had laid out. It’s commonly responded to by removing the Principle of Unrestricted Comprehension, as well as some other changes eventually made in Zermelo-Fraenkel set theory instead of what mathematicians now call naive set theory. How successful this is I’m not sure since Godel’s Theorem still applies, even with ZF. Regardless, today I found myself thinking of an alternate problem with the Principle of Unrestricted Comprehension. What about the set of all sets that do contain themselves?
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On Fair Allocation of Scarce Resources

Tuesday, January 6th, 2026

There’s a common problem that arises when there are many more customers than available products. Examples include:

  • Event Tickets: Taylor Swift, Comic-con, the Superbowl, etc.
  • Government programs: low interest mortgages, below market rate housing, visas, etc.
  • Medical care: vaccine appointments
  • Limited goods: Furby, Tickle-me Elmo, the latest game console

In practice these goods are all very inefficiently allocated with badly designed systems that frustrate customers and waste people’s time with no benefit to anyone. People constantly refresh websites waiting to get into a queue. Football fans waited overnight in the cold when the Denver Broncos made Superbowl XII in 1977. And vaccines go first (or maybe second) to young, relatively well-to-do, tech-savvy people who have the skills and time to join Facebook groups and use special software to track down vaccine appointments.

For event tickets, there is an obvious solution in standard economics: raise the price to what the market will bear. Artists don’t always like this since they often prefer to have diehard fans in the crowd instead of one-percenter senior citizens who talk over the music instead of dancing, but it does work.

However, for more important goods like visas and medical care, it’s not OK to allocate first to the people most able to pay. (A few Chicago school economists and libertarians will claim that’s exactly what should happen, but they’re bad people and should feel bad.)

Sometimes it is reasonably possible to choose who’s most deserving or needy of the limited good and assign it to them. Artists sometimes allocate tickets to their fan club first. During Covid, vaccines went first to medical professionals, then to nursing home residents who were at much higher risk of catching the virus and dying from it, and then to people in different age and risk brackets.

However, that last step didn’t work. Once the vaccines started being administered to the general public, the system broke down. Shots went first to people who could navigate complicated online systems and had a PCP who could quickly write a letter attesting to a pre-existing condition that made them eligible. Or who were young enough, healthy enough, and uninfected enough to wait in line for hours to take the place of people with appointments who didn’t show up. Techies wrote Python scripts that pinged their phones when new appointments became available. Younger, healthier, and richer people dominated the vaccine clinics in the early weeks.

The same pattern repeated a couple of years later with monkeypox. Rich, mostly white, gay men got the vast majority of the early shots, while poorer gay men of color waited weeks for appointments. And that rollout worked as well as it did only because it was almost exclusively gay men who were seeking the vaccine. If the vaccine had been more popular with the general population, the allocation would have been far worse.

Another common solution is first come, first serve. Gen Xers still remember lining up outside Mushroom Records (or the local equivalent) for concert tickets. While most event tickets have migrated online these days, it’s still not uncommon to encounter a line of people camped out in front of a Gamestop or sneaker store to score the latest drop. Shakespeare in the Park still doles out tickets like this. Needless to say, this strategy highly prioritizes young, healthy people who can camp out for many hours or even overnight.

Is there a fair solution that gives everyone equal access regardless of wealth, health, or youth? And doesn’t waste many people’s time on economically unproductive activities like sleeping on the sidewalk or constantly clicking the refresh button in Firefox like a mouse waiting for a food pellet to drop? Yes, and surprisingly it’s one that is sometimes used by government programs and almost no one else, though in many cases bureaucracies do it wrong.

That solution is a lottery.
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Why Yes, As a Matter of Fact You Can Prove the Parallel Postulate, and All the Others Too

Saturday, January 3rd, 2026

I can’t believe I got this far in mathematics without noticing that not only can the parallel postulate be proved, but that it’s been proved for hundreds of years. For the last hundred or so, it’s been completely provable, along with the rest of Euclidean geometry, in Zermelo-Frankel set theory. That is, since ZF was invented. You don’t even need the Axiom of Choice. It’s provable with naive set theory too, or just with the Peano postulates and basic algebra.

Are you surprised? I was. What about non-Euclidean geometry? It turns out we can prove that too, and it’s all consistent (assuming ZF is consistent). How about two thousand years of mathematicians trying (and failing) to prove the parallel postulate? Did they just miss it? In one sense, yes, but in one sense no. Let’s dig a little deeper.
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2025 The Year in Birds

Thursday, January 1st, 2026

Time for the annual wrap-up of what I saw where. Overall 2025 was a medium year. I spent about two months out of the country, though I didn’t take any major birding trips. I finished the year with 389 species total including 32 life birds including the very endangered and hard to find Ridgway’s Hawk:

  • Green-breasted Mango – Juan Jose Flores Park, Heredia Costa Rica
  • Stock Dove – St. Andrew Square Garden, Edinburgh Scotland
  • Willow Warbler – Yorkshire
  • Eurasian Skylark – The Coach House, Mesnes Lane, Leyburn Yorkshire
  • Red-legged Partridge – The Coach House, Mesnes Lane, Leyburn Yorkshire
  • Common Swift – Middleham | Yorkshire, England
  • Reed Bunting – Marfield Wetlands NR | Yorkshire, England
  • Rook – Marfield Wetlands NR | Yorkshire, England
  • Red Kite Road from Skipton to Masham | Yorkshire, England
  • Red Grouse – Road from Skipton to Masham | Yorkshire, England
  • Little Gull – Jamaica Bay Wildlife Refuge–West Pond | New York
  • Costa Rican Swift – Botanika Osa Peninsula | Costa Rica
  • Slate-headed Tody-Flycatcher – La perica | Costa Rica
  • Riverside Wren – Botanika Osa Peninsula | Costa Rica
  • Gray-cowled Wood-Rail – Botanika Osa Peninsula | Costa Rica
  • Ruddy-breasted Seedeater – finca san martin | Costa Rica
  • Gray-headed Tanager – Ecoturístico La Tarde | Costa Rica
  • Charming Hummingbird – Ecoturístico La Tarde | Costa Rica
  • White-crested Coquette – Ecoturístico La Tarde | Costa Rica
  • Great Curassow – Finca Miguel Sanchez, Rio Piro | Costa Rica
  • Laughing Falcon – Vía 245, Golfito | Costa Rica
  • Bare-throated Tiger-Heron – Vía 245, | Costa Rica
  • Golden-naped Woodpecker – Botanika Osa Peninsula
  • Greater Antillean Bullfinch – Westin Puntacana | Dominican Republic
  • Hispaniolan Lizard-Cuckoo – Westin Puntacana | Dominican Republic
  • Black-crowned Palm-Tanager – Punta Cana Ecological Reserve | Dominican Republic
  • Stolid Flycatcher – Punta Cana Ecological Reserve | Dominican Republic
  • Broad-billed Tody – Punta Cana Ecological Reserve | Dominican Republic
  • Vervain Hummingbird – Punta Cana Ecological Reserve | Dominican Republic
  • Hispaniolan Mango – Punta Cana Ecological Reserve | Dominican Republic
  • Palmchat – Punta Cana | Dominican Republic
  • Hispaniolan Woodpecker – Punta Cana | Dominican Republic
  • Ridgway’s Hawk – Punta Cana | Dominican Republic

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