Current Reading:
Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander
Not too much of interest to report today. HOD is going well, everything is quite familiar and intuitive at the moment. The generation theorem in "Large Cardinals from Detemrinacy is going to be my weekend project. I finished the Glimm-Effros dichotomy chapter in IDST, and I am excited to move onto the general theory of countable Borel equivalence relations. The Introduction to Core Model Theory is, just like the HOD paper, quite familiar and intuitive at this point. It is nice to see the exposition still. Infinitesimal finally got a bit into the math, and now I'm hooked. Amir has elucidated some interesting differences between the Euclidean approach to geometry and the proto-calculus approach. It seems to be the start of rigor vs. inuition. I do, however, find it difficult to follow geometric proofs when they are presented in text. I feel if I am having to struggle with it, the average reader would probably just get lost. Maybe instead of completed geometric diagrams, it would be better to draw the diagram and add the labels in steps. The text brings up an interesting paradox: starting from axioms and chasing deductions is highly stable and very checkable, but the physical intuition seems capable of "justifying" the axioms and simplifying the work, whereas the one thing the axioms can never justify is themselves. Which approach provides deeper knowledge, or better, what's the proper combination of approaches?
Friday, July 25, 2014
Thursday, July 24, 2014
July 24 2014 Progress
Current Reading:
Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander
I am finally back to where I stopped in "HOD as a Core Model" before the slew of family drama started. I stopped at the wrong place when I went to deal with that. So I am almost done with HOD in L[x,G]. My advisor wants us to start reading the HOD in L(R) part starting next week, so the timing is almost perfect. I am at the main proof of "Large Cardinals from Determinacy," and its quite long. I am debating breaking it up into parts, as it is one proof. I think tomorrow I will just make a go of the whole thing. Tracks 3 and 4 are coming along nicely. I have dropped the bonus track for now, as I am not sure if I want to keep looking at Chang and Keisler. If I drop that I will go back to "Iterations of Rational Functions."
I decided to post this before I began reading Inifinitesimal today as I figured I would not be willing to make a post afterwards. Regularly updating this is harder than regularly doing research.
Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: An Introduction to Core Model Theory by John Steel
Pop Math: Infinitesimal by Amir Alexander
I am finally back to where I stopped in "HOD as a Core Model" before the slew of family drama started. I stopped at the wrong place when I went to deal with that. So I am almost done with HOD in L[x,G]. My advisor wants us to start reading the HOD in L(R) part starting next week, so the timing is almost perfect. I am at the main proof of "Large Cardinals from Determinacy," and its quite long. I am debating breaking it up into parts, as it is one proof. I think tomorrow I will just make a go of the whole thing. Tracks 3 and 4 are coming along nicely. I have dropped the bonus track for now, as I am not sure if I want to keep looking at Chang and Keisler. If I drop that I will go back to "Iterations of Rational Functions."
I decided to post this before I began reading Inifinitesimal today as I figured I would not be willing to make a post afterwards. Regularly updating this is harder than regularly doing research.
An Update on Updates
There have been a lack of updates lately. This started because there were a few days where I couldn't get work done because of family drama. Then I was getting work done but was too busy with friends in the evening to post on the blog.
In the meantime I have met with my advisor twice, and we finished reading HOD in L(R). We will be moving on to more current HOD analysis and then start on a research problem. I did ditch the mouse sets paper for now in favor of the introduction to core model theory. At this point its probably a little silly to read it, but I still think I'm better off with it under my belt before I read CMIP. I finished the section in Chang and Keisler on non-standard analysis. I would reccomend it, as the exposition of the material on superstructures there is better than I've seen it presented elsewhere. I have plans to read "Non-standard Anaysis for the Working Mathematician" in the fall. Anyways there should be more posts coming.
In the meantime I have met with my advisor twice, and we finished reading HOD in L(R). We will be moving on to more current HOD analysis and then start on a research problem. I did ditch the mouse sets paper for now in favor of the introduction to core model theory. At this point its probably a little silly to read it, but I still think I'm better off with it under my belt before I read CMIP. I finished the section in Chang and Keisler on non-standard analysis. I would reccomend it, as the exposition of the material on superstructures there is better than I've seen it presented elsewhere. I have plans to read "Non-standard Anaysis for the Working Mathematician" in the fall. Anyways there should be more posts coming.
Tuesday, July 15, 2014
July 15 2014 Progress
Current Reading:
Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
I finally bothered to look up the correct name for track 1, I should have done it earlier but I was lazy. It's more correct as technically the paper talks about L(R) and L[x,G]. Anyways, I covered pseudo-normal-iterates today in detail. My gut feeling is that this definition, like many others in this subfield is a highly refined, exactly what works definition. But the idea of describing the models that show up when following a strategy L[x,G] can't see in a way that L[x,G] can see is very much the same as the approach to HOD in L(R) and I find it intuitive.
Track 2 was not very exciting today, but I am building up important groundwork. It's just a generalization of the Turing cone measure. I say "just," but the generalization is intuitive and the results are strong. I now remember that I did start to cover strategic determinacy. I don't have a good feeling for this definition yet, I think it will require a good amount of meditation and use before it becomes clear, the excellent exposition in the paper not withstanding.
For track 3, I learned about the Effros dichotomy. I know this is an important result, and I am excited to see the applications of it to come. For whatever reason this material feels intuitive. Its probably because I have a good amount of DST experience.
Again I couldn't bring myself to look at track 4 today. I'm seriously considering leaving it for now and coming back to it when I can better understand its relevance. Its annoying to leave the paper so nearly finished, but I may have to follow my gut. If I do leave it I will start reading on the core model induction.
On the bonus track I waded halfway through the section on nonstandard universes. Chang has a comment about not liking the term nonstandard analysis, which he wants to replace with Robinsonian analysis. I get the historical context he's coming from, but the reality is that the method of analysis is nonstandard, and I don't see the name changing any time soon. So far the material has been review, but a second look never hurts.
"Infinitesimal" moved on to start telling the story of how math became more of a focus for the Jesuits. The story of the inception of the Gregorian calendar is fascinating, but I find this historical background to maybe be a bit too long. I hope the book starts getting to the controversy soon.
Track 1: HOD as a Core Model by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
I finally bothered to look up the correct name for track 1, I should have done it earlier but I was lazy. It's more correct as technically the paper talks about L(R) and L[x,G]. Anyways, I covered pseudo-normal-iterates today in detail. My gut feeling is that this definition, like many others in this subfield is a highly refined, exactly what works definition. But the idea of describing the models that show up when following a strategy L[x,G] can't see in a way that L[x,G] can see is very much the same as the approach to HOD in L(R) and I find it intuitive.
Track 2 was not very exciting today, but I am building up important groundwork. It's just a generalization of the Turing cone measure. I say "just," but the generalization is intuitive and the results are strong. I now remember that I did start to cover strategic determinacy. I don't have a good feeling for this definition yet, I think it will require a good amount of meditation and use before it becomes clear, the excellent exposition in the paper not withstanding.
For track 3, I learned about the Effros dichotomy. I know this is an important result, and I am excited to see the applications of it to come. For whatever reason this material feels intuitive. Its probably because I have a good amount of DST experience.
Again I couldn't bring myself to look at track 4 today. I'm seriously considering leaving it for now and coming back to it when I can better understand its relevance. Its annoying to leave the paper so nearly finished, but I may have to follow my gut. If I do leave it I will start reading on the core model induction.
On the bonus track I waded halfway through the section on nonstandard universes. Chang has a comment about not liking the term nonstandard analysis, which he wants to replace with Robinsonian analysis. I get the historical context he's coming from, but the reality is that the method of analysis is nonstandard, and I don't see the name changing any time soon. So far the material has been review, but a second look never hurts.
"Infinitesimal" moved on to start telling the story of how math became more of a focus for the Jesuits. The story of the inception of the Gregorian calendar is fascinating, but I find this historical background to maybe be a bit too long. I hope the book starts getting to the controversy soon.
Monday, July 14, 2014
July 14 2014 Progress
Current Reading:
Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
Today was good work day, not perfect, but good. "HOD in L(R)" continues to be interesting, but I find it difficult to keep the myriad versions of iterability straight in my head. I'm trying to form some intuition for the various forms, but I suspect that will come later. As for "Large Cardinals from Determinacy," I find the level of generality breathtaking, and I haven't even reached the most general point yet. What's nice is how this and "HOD in L(R)" are overlapping right now. My studies for a while now have been centered on L(R), its nice to be able to spend time in HOD. "Invariant Descriptive Set Theory" has finally started to get into gear, today I covered E_0 and there are interesting Dichotomies to come. Track 4 got no love today unfortunately.
I finally reached the section in "Model Theory" where they cover regular filters. As much as I love my complete filters, studying interesting filters that more naturally exist has its own appeal. Tomorrow I cover the section on nonstandard universes, I hope the more complete model theory background will help make these wonderful objects more accessible. Finally I got through the first chapter of "Infinitesimal." The book certainly starts off tediously enough, I felt as though I were watching a discovery channel documentary for a while there. Finally though he starts telling the story instead of trying to convince me the story is worth telling, and the first chapter is a fascinating, if brief, history of the fracturing of the Catholic church and consequent formation of the Jesuit order. No math has been discussed yet, but I do feel like I understand the sociological motivations of the anti-infinitesimal side a bit better. As far as learning what kind of writing allows a math book to be popular, I fear I am so far only able to reach the sad conclusion that excluding the math and emphasizing the people and the history is the answer.
Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
Today was good work day, not perfect, but good. "HOD in L(R)" continues to be interesting, but I find it difficult to keep the myriad versions of iterability straight in my head. I'm trying to form some intuition for the various forms, but I suspect that will come later. As for "Large Cardinals from Determinacy," I find the level of generality breathtaking, and I haven't even reached the most general point yet. What's nice is how this and "HOD in L(R)" are overlapping right now. My studies for a while now have been centered on L(R), its nice to be able to spend time in HOD. "Invariant Descriptive Set Theory" has finally started to get into gear, today I covered E_0 and there are interesting Dichotomies to come. Track 4 got no love today unfortunately.
I finally reached the section in "Model Theory" where they cover regular filters. As much as I love my complete filters, studying interesting filters that more naturally exist has its own appeal. Tomorrow I cover the section on nonstandard universes, I hope the more complete model theory background will help make these wonderful objects more accessible. Finally I got through the first chapter of "Infinitesimal." The book certainly starts off tediously enough, I felt as though I were watching a discovery channel documentary for a while there. Finally though he starts telling the story instead of trying to convince me the story is worth telling, and the first chapter is a fascinating, if brief, history of the fracturing of the Catholic church and consequent formation of the Jesuit order. No math has been discussed yet, but I do feel like I understand the sociological motivations of the anti-infinitesimal side a bit better. As far as learning what kind of writing allows a math book to be popular, I fear I am so far only able to reach the sad conclusion that excluding the math and emphasizing the people and the history is the answer.
Friday, July 11, 2014
First Post
Current Reading:
Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
I didn't get too much reading today since I took my girlfriend to have her wisdom teeth removed. I'm currently feeling a little frustrated as I am at the start of a new section in all of my readings. The theorem of Woodin on mouse sets paper should be finished next week, and then I will start reading on the core model induction. "Infinitesimal" has already surprised me: I had no idea that there was important infinitesimal work done before Newton and Liebniz.
The weekend is ahead, and I plan to get a lot of recreational reading done, and maybe make an opinion post. We'll see.
Track 1: HOD in L(R) by John Steel and Hugh Woodin
Track 2: Large Cardinals from Determinacy by Hugh Woodin
Track 3: Invariant Descriptive Set Theory by Su Gao
Track 4: A Theorem of Woodin on Mouse Sets by John Steel
Bonus Track: Model Theory by Chang and Kiesler
Pop Math: Infinitesimal by Amir Alexander
I didn't get too much reading today since I took my girlfriend to have her wisdom teeth removed. I'm currently feeling a little frustrated as I am at the start of a new section in all of my readings. The theorem of Woodin on mouse sets paper should be finished next week, and then I will start reading on the core model induction. "Infinitesimal" has already surprised me: I had no idea that there was important infinitesimal work done before Newton and Liebniz.
The weekend is ahead, and I plan to get a lot of recreational reading done, and maybe make an opinion post. We'll see.
Subscribe to:
Posts (Atom)