iiinova — where your digital self keeps evolving

Paul Fabel @paulfabel.bsky.social · 16/08/2026

Potential HW or test questions in math foundations. I give you SOME information about f:A-->B, e.g. A is infinite and f surjective. Then we can ask: Must B be infinite? Could B be infinite? Must be B finite? Could B be finite? Students can then justify with proofs and examples.

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Paul Fabel @paulfabel.bsky.social · 28/07/2026

Informative and fun to construct an abelian monoid of order 3 which is NOT monogenic (one generator). Since 3 is prime, this has no hope to factor as a product of monogenic monoids. It was NOT fun coaxing ChatGPT, free version, to find the example, a mixture of nonsense and turgid spoilers.

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Paul Fabel @paulfabel.bsky.social · 26/07/2026

Whether every quotient of every set admits a section is more or less equivalent to the axiom of choice, depending plausibly on your model of set theory. But a quotient of a 3 element abelian monoid might NOT embed in the starting monoid. {0,1,2,3...} mod {0,2,4...} is also a counterexample.

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Paul Fabel @paulfabel.bsky.social · 26/07/2026

The 3 element monoid with {1,f,ff} with ff not fff, and quotient {1,ff} , {f} shows 1) A quotient of a finite abelian monoid might NOT embed in the starting monoid. 2) The cancellative quotient of a finite abelian monoid might have cosets of different sizes. Useful to bring up in Foundations?

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OrcGirlBoss 🐇🏹 @orc.bsky.social · 02/07/2026

I would like a physical copy of my game please.

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Jennifer Ouellette @jenlucpiquant.bsky.social · 30/06/2026

Here’s Why Weather Forecasts Have Seemed So Inaccurate Lately. Meteorologists simply aren't able to gather as much data as they used to, and you're suffering for it. gizmodo.com/heres-why-we...

gizmodo.com

Here's Why Weather Forecasts Have Seemed So Inaccurate Lately

Meteorologists simply aren't able to gather as much data as they used to, and you're suffering for it.

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Paul Fabel @paulfabel.bsky.social · 07/06/2026

Given functions f,g :A-->A we can ask: 1) What is |<f>|? 2) Are f and g conjugate? 3) Are <f> and <g> isomorphic? Too easy? Warning: Questions about <f,g> are HARD, since `most' nonabelian groups have two generators.

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Paul Fabel @paulfabel.bsky.social · 07/06/2026

Teaching Foundations of Math in the Fall, but how to engage students in doable, compelling exercises, teachable for an exam, not totally trivial, and with long term healthy cognitive side effects? Monoids will work at lots of levels, starting with basic questions about f:A-->A with A finite.

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Maggie Wiggin @maggie162.bsky.social · 01/06/2026

My god, Octavia Butler, just miss ONCE

Page from a book saying:

From EARTHSEED: THE BOOKS OF THE LIVING
Choose your Leaders with wisdom and forethought.
To be led by a coward is to be controlled
by all that the coward fears.
To be led by a fool is to be led
by the opportunists who control the fool.
To be led by a thief is to offer up your most precious treasures to be stolen.
To be led by a liar
is to ask to be told lies.
To be led by a tyrant is to sell yourself and those you love into slavery.
t

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Paul Fabel @paulfabel.bsky.social · 29/03/2026

Can the right divisibility preorder on a noncommutative monoid be total? (a<=b if ac=b for some c). Yes, according to Eliahu Levy ( 1947-2023). arxiv.org/pdf/2006.00886

arxiv.org

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Fietser @americanfietser.bsky.social · 27/03/2026

We need a vehicle weight tax. If you can justify buying an $80k, 3,000 kg Wagoneer to haul one bag of groceries, taxpayers shouldn't be subsidizing your pavement wear and tear plus the added danger to our streets. If you can afford the payments, you can afford to stop asking society to subsidize it.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

If the abeliain monoid (M,*) determines a total order on (M,<), then the bounded rays comprise a totally ordered and left complete submonoid of 2^M. Now take the cancellative quotient, and we get the non-neg additive reals, if M is the non-neg dyadic rationals. All this is `easy'.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

f:M-->G is trivial if there is k in M so that km=mk=k for all m in M, if M is a monoid and G is a group, and f is a homomorphism.

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Paul Fabel @paulfabel.bsky.social · 22/03/2026

Ignoring order, the additive rationals are discrete, a colimit of cyclic groups. But if we restrict to the monoid of non-neg rationals, algebra gives us x<y iff x+z=y, we get order topology. Induced order on monoid of bounded rays, almost the non-neg reals. Mod out by [0,x)=[0,x] when x rational.

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Paul Fabel @paulfabel.bsky.social · 22/03/2026

Submonoid category. Exercises and examples. From S={a,b} we can create M(S), the monoid of functions f:M-->M under composition. From M(S) we can create the category C(S) of submonoids of M(S). |C(S)|=5 in this case. We can then ask, given x and y in C, are x and y isomorphic? |H(x,y)|=?

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Paul Fabel @paulfabel.bsky.social · 21/03/2026

Monoids are essential building blocks of groups. We sometimes get a useful partial order on a monoid declaring x<=y if zx=y has a solution. Sometimes this internal algebra creates a total order, e.g. on the natural numbers. But this NEVER happens in a group, since x <= y holds for all x and y.

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Leah McElrath @leahmcelrath.bsky.social · 06/03/2026

Malignant narcissists experience increased paranoia, denial, and rage as death nears. They often externalize these experiences through destruction and abuse of others in an effort to maintain a sense of control. I suspect all of these wars are in part related to Trump’s awareness of his mortality.

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A.R. Moxon @juliusgoat.bsky.social · 28/04/2024

Seven lessons learned from a quarter century in a war-oriented society. It's 2001—the year the movies promised we'd make contact with aliens—and the United States has rather recently been attacked by terrorists who flew passenger planes into buildings. www.the-reframe.com/war-or-nothi...

the-reframe.com

War or Nothing

A tale across the years; seven lessons learned from a quarter-century in a war-oriented society, where the greatest crime is any opposition to killing.

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A.R. Moxon @juliusgoat.bsky.social · 08/02/2026

Today I wrote about the racist accommodation to the racist response to Bad Bunny's Super Bowl show; the moral hazard of extending good faith to those who act in bad faith; and how alienating white bigots is actually part of the solution, not a part of the problem. www.the-reframe.com/hating-the-g...

the-reframe.com

Hating the Game

The cooperation game, the murder game, and acting in good faith with people you know are acting in bad faith.

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Paul Fabel @paulfabel.bsky.social · 04/02/2026

For cancellative abelian monoids, a nontrivial quotient map can have trivial kernel. Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving. Knowledge of the kernel of a quotient map is NOT one stop shopping.

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Paul Fabel @paulfabel.bsky.social · 31/01/2026

AI is getting good at math, but USING it still feels like looking up the answers in the back of the textbook. Useful as a sanity check but generally hopeless as one stop shopping for understanding. Exploring, developing tactics, finding examples and counterexamples, is not replaceable by AI.

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Paul Fabel @paulfabel.bsky.social · 31/01/2026

Unlike groups, point preimages need NOT all have the same size under monoid epimorphisms. There is an M_3-->M_2 counterexample with trivial kernel. Let AAA=A and BB=B, in the domain and codomain. Map A to B and id to id.

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Paul Fabel @paulfabel.bsky.social · 26/01/2026

The finite discrete space F is orderable but most automorphisms h:F-->F are NOT order preserving. Same holds if we replace F with Cantor space C. However each order preserving h:C-->C quotients to h^:[0,1]-->[0,1] preserving the `dyadic rationals' And vice versa, each h^ lifts to an h.

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cr0104.bsky.social @cr0104.bsky.social · 25/01/2026

Explore this gift article from The New York Times. You can read it for free without a subscription. www.nytimes.com/2026/01/24/o...

nytimes.com

Opinion | State Terror Has Arrived

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hannah gais @hannahgais.bsky.social · 22/01/2026

I wrote about how fucked up the past year has been as a person who's covered the far right for over a decade and who lives in DC. The guardrails are gone, and everyone knows it. Everything those of us on this beat warned about happened, and then some. postsfromunderground.ghost.io/the-not-so-g...

postsfromunderground.ghost.io

the not-so "golden age," one year on

Expo, a Swedish antiracist magazine that I contribute to, asked me to write a personal reflection how researching and reporting on the far right has felt during the first year of Donald Trump's return...

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Sarah Kendzior @sarahkendzior.bsky.social · 15/01/2026

New article! On life, death, and the rampage of ICE. sarahkendzior.substack.com/p/a-shining-...

sarahkendzior.substack.com

A Shining Mausoleum on a Hill

Life, death, and the rampage of ICE.

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Matt Wuerker @mwuerker.bsky.social · 15/01/2026

Testing our sanity

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emilyriehl.bsky.social @emilyriehl.bsky.social · 29/12/2025

In a talk called "A reintroduction to proofs" emilyriehl.github.io/files/reintr... I've speculated about teaching an undergraduate level introduction to proofs course but using dependent type theory as the implicit formal system in place of set theory and first order logic.

emilyriehl.github.io

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Paul Fabel @paulfabel.bsky.social · 02/12/2025

Strong Floor, No Ceiling, Can't Leak! Dibs.

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Paul Fabel @paulfabel.bsky.social · 10/11/2025

Suppose the free group G on countably many generators is a sequential Hausdorff topological group, refining the usual product topology so that sequentially closed sets are closed. What ensures some planar continuum X has pi_1(X) homeomorphic to G, with the natural quotient topology?

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Paul Fabel @paulfabel.bsky.social · 07/11/2025

Tread carefully in compactly generated free abelian topological groups. Closed subgroups need not be compactly generated. If A is the group of all bounded sequences of integers, so that uniformly bounded null sequences converge to 0, let B denote the subgroup with b in B iff n divides b_n.

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Paul Fabel @paulfabel.bsky.social · 01/11/2025

Hausdorff quotients of locally compact complete separable metric spaces need not be 1st countable. Start with X= [0,1] x {1,2,3,....} Now glue together (0,n) and (0,m). The quotient space is not 1st countable, despite being locally connected and uniquely arcwise connected.

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Nicholas Grossman @nicholasgrossman.bsky.social · 16/10/2025

In the near future economy, I bet the skill of “can type a prompt into ChatGPT and repeat whatever it says” will be less in demand than “understands things enough to catch when ChatGPT makes an error or hallucinates.”

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Robert Reich @rbreich.bsky.social · 13/10/2025

The 400 richest Americans are now worth a record $6.6 trillion. The entire bottom 50% of America is worth just $4.2 trillion. Read that back. When 400 people control more wealth than half a country’s population, we have a very serious problem.

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Paul Fabel @paulfabel.bsky.social · 08/10/2025

ONE of the tricks/ideas lying behind proofs of quadratic reciprocity, is the `half the units' trick. Suppose A is an abelian group. Suppose U is a ` not both u and -u subset', if u is in U then -u is NOT in U. Then vU is ALSO a ` not both u and -u' subset, for each automorphism v:A-->A.

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Paul Fabel @paulfabel.bsky.social · 22/09/2025

AI has a WAYS to go before it can do the basics reliably. 47^(73)= 932 mod 1009 , we can do this quickly with a cheap calculator, via repeated doubling, since 73=64+8+1. But BOTH chatgpt and google AI cannot stick the landing on the question: What is 47^(73) mod 1009?

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Carl T. Bergstrom @carlbergstrom.com · 20/09/2025

This is either engineered to create mass chaos in the cruelest possible fashion, or is a consequence of complete incompetence. I suspect both, though by different people in the chain of command leading to this executive order.

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Paul Fabel @paulfabel.bsky.social · 11/09/2025

Pass Line Craps EV should be 2(8/36)+ 2[ (10/36)(5/11) + (8/36)(4/10) +(6/36)(3/9) ) = 98.58585858....% Don't Pass: 2[3/36]+1[1/36]+ 2[ (10/36)(6/11) + (8/36)(6/10) +(6/36)(6/9) ) = 98.636363....%

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Paul Fabel @paulfabel.bsky.social · 08/09/2025

An approach to basic set theory and logic, Jerzy Dydak. arxiv.org/pdf/1305.3283

arxiv.org

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A.R. Moxon @juliusgoat.bsky.social · 07/09/2025

I wrote about the Third Way memo and its list of forbidden words. Comparing the alleged desire for simple persuasive talk and the way those who want it never seem to arrive at the simplest most persuasive "talk" there is: principled action. www.the-reframe.com/simple-talk-2/