Elementary Epimorphisms Talk

Elementary epimorphisms between models of set theory
Norman Lewis Perlmutter, LaGuardia Community College
October 17, 2014
This talk concerns joint work with Robert S. Lubarsky.
Elementary epimorphisms were introduced by Philipp Rothmaler. A surjective homomorphism f: M –> N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters.
Philipp asked me whether nontrivial elementary epimorphisms between models of set theory exist. We answer this question in the negative for fully elementary epimorphisms between models of ZFC, but in the positive under weaker assumptions.
In particular, we show that every Pi_1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Sigma_1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^-.

LaGuardia Wins $2.9 Million “First in the World” grant

Excerpted from an email from President Mellow:

U.S. Secretary of Education Arne Duncan announced that LaGuardia was selected to receive a $2.9 million “First in the World” grant from the Obama Administration and the U.S. Department of Education (DOE) to fund our student success initiatives.

This project grant will allow us to:

●   Support thousands of high-risk students as they move from LaGuardia’s non-credit programs to academic enrollment.

●   Integrate new discipline-based curriculum with co-curricular innovation to launch more than 20,000 new students towards graduation as part of our new First Year Experience seminar.

●   Transform advisement for all LaGuardia students by training and activating College-wide faculty/staff/peer mentor teams.

For more information about our grant check out LaGuardia’s press release http://www.laguardia.cuny.edu/Home/News/LAGCC-Awarded-%E2%80%9CFirst-in-the-World%E2%80%9D-Grant-by-U-S–Department-of-Education/

Starting out at LaGuardia

My new job at LaGuardia has been exciting and a lot of work. There’s a lot of new information to take in and procedures to figure out, but I’m adapting to it all. LaGuardia hired 70 new instructional faculty this year, 12 of them in the math department (including computer science and engineering). The school is growing by leaps and bounds. It’s great to be part of this large group of incoming faculty; it feels like we’re undertaking a grand adventure together, and we are available to support each other through the challenges of the experience. Like the student body, the faculty exhibit great diversity in ethnicity and in life experience. Even by CUNY standards, LaGuardia is a very diverse place, and that’s really saying something.

I’ve had a few opportunities to hear President Mellow address the new hires, and she is a very charismatic leader. She really seems to have a strong vision for the direction of the school, and she makes us feel that our work is very important.

The facilities of the school are much better than I was expecting. From the outside it looks like a series of ugly brown bulidings . . . but it’s the inside that matters, as that’s where I spend more time. The bathrooms are reasonably clean, and the elevators operate reasonably quickly. I have a good-sized private office, painted in interesting colors, with almost-new furniture including plenty of bookshelves, a filing cabinet, another cabinet, and pegs to hang my coat, backpack, etc. No windows to the outside, though some small high windows let in light from the hallway. Offices with outside windows are assigned based on seniority.  I have a much larger desk than I had at FAU.  The hallways are decorated with an eclectic collection of artwork, much of it by LaGuardia students, and painted in bright colors. There is a beautiful courtyard with greenery and benches, and a hall of flags with a high ceiling from which hang flags from many nations of the world. Is it all the nations of the world or just the ones that LaGuardia students come from? I don’t think there would be a big difference actually.

Got a tenure-track position at LaGuardia Community College

On May 22, I accepted a tenure-track professor position at LaGuardia Community College. LaGuardia is part of the City University of New York (CUNY) the same university system where I went to graduate school. It is located in Long Island City with in the borough of Queens, in the far southwest corner of Queens, south of Astoria and west of Sunnyside. The college is exceptional among community colleges in that the majority of faculty have PhDs and faculty are expected to do some research.

Several people have asked me whether LaGuardia Community College is located near LaGuardia airport. They are about five miles apart. They share the same name because they’re both named after a former mayor of New York, Fiorello LaGuardia. LaGuardia was mayor during the 1930s. Although he was a Republican, he was also a progressive and an ally of FDR. He spoke several languages and stood up for immigrants. Therefore, it is particularly appropriate that LaGuardia Community College is named for him, as the college has an extremely ethnically diverse student body even by CUNY standards, which is really saying something. This is meaningful to me personally because my great-grandparents were immigrants, and I like the idea of helping immigrants and first-generation college students to improve their lives.

I am very happy to be moving back to New York City, as I really love the culture and lifestyle there.  It will also be nice to socialize and work with my many friends and colleagues in set theory at CUNY. CUNY has one of the most active set theory research groups in the world.

It’s also nice to have the security of a tenure-track position. Moving around every few years for postdocs would be exhausting, I think.The move down to Florida really took a toll on me, although in retrospect it was fun to live for a year in Florida, and I learned a lot about myself and my personal and professional life goals through the experience. There are a few people, places, and lifestyle aspects that I’ll really miss from Florida, but overall, I think I’ll be much happier back in New York. I’m planning to move around the end of July.


On extensions of supercompactness

Robert S. Lubarsky and Norman Lewis Perlmutter

To appear in Mathematical Logic Quarterly.

Read a preprint here.

hyp and enh scs June2014


We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study $C^{(n)}$ -supercompactness.

Elementary epimorphisms between models of set theory

Bob Lubarsky and I have been working on this during the current academic year, and today we submitted it for publication. (Update: the paper has been accepted to the Archive for Mathematical Logic.) You can read a preprint here: ElementaryEpimorphisms.




We show that every $\Pi_1$-elementary epimorphism between models of $ZF$ is an isomorphism. On the other hand, nonisomorphic $\Sigma_1$-elementary epimorphisms between models of $ZF$ can be constructed, as can fully elementary epimorphisms between models of $ZFC^-$. Elementary epimorphisms were introduced by Philipp Rothmaler.. A surjective homomorphism $f: M \to N$ between two model-theoretic structures is an elementary epimorphism if and only if every
formula with parameters satisfied by $N$ is satisfied in $M$ using a preimage of those parameters.

This was freaky

I’m currently teaching a unit on probability. One of the homework problems that I wrote was roughly follows. Consider the experiment of drawing four cards without replacement from a thoroughly shuffled deck of cards. What is the probability that the first card is a heart, the second is a diamond, the third is a heart, and the fourth is a spade?

A student, Michael, came into my office for help on the homework problems. While he was asking about various problems, I was shuffling a deck of cards. When we got to the problem given above, I first asked him for his intuition about the problem –was the probability likely or unlikely. He recognized that it was extremely unlikely. Then I used the deck of cards to illustrate the problem. So, I draw a card from the deck. I asked, “What’s the probability that it’s a heart?” “13/52” he answered. I drew a card from the top of the deck. It was a heart. “How about that, it was actually a heart,” I said. “What’s the probability that the next card is a diamond?” “13/51”. I drew the next card. It was a diamond. Wow. “Okay, what’s the probability that the next card is a heart?” “12/50”. I drew the next card. It was a heart! At this point we were pretty surprised. Michael made some comment about he really hoped the next card wouldn’t be a spade. I drew the next card . . . it was a spade! Weird.

The probability of this happening . . . about 0.4%.

Of course, this far from the first time that I had illustrated an experiment like this, so it’s not so surprising that eventually a coincidence would occur eventually. But still, it was pretty exciting.

Paper accepted

My paper, “The large cardinals between supercompact and almost-huge,” has been accepted into the Archive for Mathematical Logic, subject to minor revisions. The referee was very generous with his time, going over the paper carefully to make many corrections and improvements. This is my first solely-authored paper to be accepted into a peer-reviewed journal. You can read more details about it under the publications tab.