489 reputation
311
bio website colorado.edu/math
location Boulder, CO
age 31
visits member for 3 years, 11 months
seen 6 hours ago
I am a math instructor at the University of Colorado at Boulder. My interests have evolved from proof theory and philosophical concerns about knowledge to the upper reaches of set theory and the large cardinal hierarchy. I also occasionally dabble in combinatorics of the finite sort.

Feb
25
accepted Elementary Embeddings and Relative Constructibility
Feb
16
comment Antichains and the Knaster Property
@Paul McKenney: Thanks for your comment. I posed the question because I had a deep misunderstanding of the concept and you alerted me to this fact. Thanks!
Feb
16
revised Antichains and the Knaster Property
The background of the question was false and made for a confusing question.
Feb
13
revised Ultrapowers of ultrapowers
added set theory tag
Feb
13
suggested suggested edit on Ultrapowers of ultrapowers
Feb
13
accepted Antichains and the Knaster Property
Feb
13
comment Antichains and the Knaster Property
@MohammadGolshani: Yes, thank you!. I've edited my question.
Feb
13
revised Antichains and the Knaster Property
switched the ordering of my question to make sense
Feb
13
asked Antichains and the Knaster Property
Feb
2
comment What is the definition of a large cardinal axiom?
@Joel and Asaf: given your comments, how would you alter any of the proposed categories? I'm certainly willing to give up the full strength of ZFC, but I'm really not familiar enough with many of its fragments and their respective strength. Is ZFC or its strengthenings simply too strong a base to use when attempting a definition of "large cardinal"?
Feb
2
answered What is the definition of a large cardinal axiom?
Jan
24
comment What is the definition of a large cardinal axiom?
@StevenLandsburg: Couldn't it move toward a consensus? Sites like this provide a deeper and at the same time broader understanding of topics informed by experts working in the field, even if that understanding borders on what might be considered more philosophical concerns. If the only "definitions" of large cardinal are implicitly understood, so be it. But certainly we could benefit from having a single source collecting together some of these implicit "definitions", right?
Jan
24
comment What is the definition of a large cardinal axiom?
I think this is actually an important, if soft, question worthy of discussion by set-theorists with some interest in the subject. There is at least one formal definition that I know of offered by Woodin in part II of his article on CH in the AMS. I don't know if he himself considers the definition there satisfactory, but there doesn't seem to be any consensus on the topic (as far as I can tell).
Jan
23
awarded  Teacher
Jan
23
answered Very Large Cardinal Axioms and Continuum Hypothesis
Jan
23
comment Very Large Cardinal Axioms and Continuum Hypothesis
I think it's worth mentioning that Laver proved the converse direction of Levy-Solovay for rank-into-rank cardinals (you mentioned the other direction as folk-lore) in the same paper that introduces the definability of ground models result you and Johnstone extended recently.
Jan
23
comment Are superstrong stronger than strongly compact cardinals? (or vice versa)
@AsafKaragila: I find the "identity crises" phenomenon a little unnerving at times, and certainly an eyesore and unwanted complication when thinking about large cardinals. Would you (or really anyone here on MO) mind explaining why they find "identity crises" among large cardinal concepts "awesome"? My naive thinking on this suggests that one of two things follows from this phenomenon: Either certain large cardinal concepts are too robust to be very meaningful or concrete, or the structure of the universe is not only highly complex, but also a little inelegant. Thoughts?
Nov
16
comment Is the tree of large cardinals linear?
I want to point out that there are two reasonable large cardinals proposed very near the top. Both extend $I_1$ and involve incorporating more AC to the target model, specifically a non-trivial instance of uniformization holds in those models. Both are proposed by Laver and at least one is sensitive to small forcing, so it violates the Levy-Solovay theorem in an "unusual way". I stated the forcing-fragile axiom in a question here on MO. The other can be found in Woodin's Suitable Extender Models II, def. 1. It's not clear how these axioms relate to $I_0$ or Woodin's $E_\alpha$-sequence.
Nov
10
revised Infinite Partitions of the Primes and Sums of Reciprocals (Revised)
Revised to give two possible examples of a "nice" partition of the primes. No longer a question; mostly kept so as to share and have access to other users' comments and answers.
Nov
5
comment Infinite Partitions of the Primes and Sums of Reciprocals (Revised)
This is another (classic) construction that I hadn't thought about. Thank you!