April | 2009 | A kind of library

580 -Partition calculus (6)

April 24, 2009

1. The ${\mbox{Erd\H os}}$ -Rado theorem

Large homogeneous sets (of size ${\omega_1}$ or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by ${\mbox{Erd\H os}}$ and Rado using tree arguments (with ${\kappa+1}$ lowered to ${\kappa}$ in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and ${\mbox{Todor\v cevi\'c},}$ which proves the stronger version. For ${\kappa}$ a cardinal let ${2^{<\kappa}=\sup_{\lambda<\kappa}2^\lambda.}$

Theorem 1 ( ${\mbox{Erd\H os}}$ -Rado) Let ${\kappa}$ be a regular cardinal and let ${\lambda=(2^{<\kappa})^+.}$ Then

$\displaystyle \lambda\rightarrow(\kappa+1)^2_\mu$

for all ${\mu<\kappa.}$

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305 -Homework set 8

April 24, 2009

This set is due May 1 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Solve exercises 11, 14, and 24 from Chapter 14 of the textbook.

2. Solve exercises 1, 9, and 17 from Chapter 15 of the textbook.

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305 -9. Groups

April 24, 2009

We want to define the notion of group that will be fundamental to determine which polynomials are solvable by radicals. This notion is very important and appears in every area of mathematics.

We motivate the definition through the example that most concerns us: automorphisms of fields. They are particular class of isomorphisms, so we begin with them.

Some of the arguments below have been discussed in previous lectures.

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Leave a Comment » | 305: Abstract Algebra I | Tagged: automorphism, isomorphism | Permalink
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580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that ${\omega\rightarrow(\omega)^n_m}$ (Ramsey) and ${\omega\rightarrow[\omega]^n_\omega}$ ( ${\mbox{Erd\H os}}$ -Rado) for any ${n,m<\omega.}$ On the other hand, we also have that ${2^\kappa\not\rightarrow(3)^2_\kappa}$ ( ${\mbox{Sierpi\'nski}}$ ) and ${2^\kappa\not\rightarrow(\kappa^+)^2}$ ( ${\mbox{Erd\H os}}$ -Kakutani) for any infinite ${\kappa.}$

Positive results can be obtained for larger cardinals than ${\omega}$ if we relax the requirements in some of the colors. A different extension, the ${\mbox{Erd\H os}}$ -Rado theorem, will be discussed later.

Theorem 1 ( ${\mbox{Erd\H os}}$ -Dushnik-Miller) For all infinite cardinals ${\lambda,}$ ${\lambda\rightarrow(\lambda,\omega)^2.}$

This was originally shown by Dushnik and Miller in 1941 for ${\lambda}$ regular, with ${\mbox{Erd\H os}}$ providing the singular case. For ${\lambda}$ regular one can in fact show something stronger:

Theorem 2 ( ${\mbox{Erd\H os}}$ -Rado) Suppose ${\kappa}$ is regular and uncountable. Then
$\displaystyle \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2,$ which means: If ${f:[\kappa]^2\rightarrow2}$ then either there is a stationary ${H\subseteq\kappa}$ that is ${0}$ -homogeneous for ${f}$ , or else there is a closed subset of ${\kappa}$ of order type ${\omega+1}$ that is ${1}$ -homogeneous for ${f}$ .

(Above, top stands for “topological.”)

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6 Comments | 580: Topics in set theory | Tagged: elementary embedding, Grigor Sargsyan, Jan Tryba, Jonsson cardinal, measurable cardinal, Ramsey cardinal, Reinhardt cardinal, Rowbottom cardinal, W. Hugh Woodin, weakly compact cardinal | Permalink
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305 -8. Irreducibility

April 20, 2009

In this lecture I want to present a couple of short results that are nevertheless very useful in practice when trying to show that a given polynomial in ${{\mathbb Q}[x]}$ is irreducible. Of course, we may assume that the polynomial actually has integer coefficients. In this case, it turns out that analyzing whether the polynomial factors over ${{\mathbb Z}[x]}$ suffices.

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Leave a Comment » | 305: Abstract Algebra I | Tagged: Eisenstein criterion, Gauss lemma, irreducible polynomial | Permalink
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580 -Cardinal Arithmetic

April 20, 2009

Here is a pdf file with the contents of the lectures on cardinal arithmetic. As with the previous chapter, it follows closely the style of the notes. There are fewer typos than in the posts, and once again I made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

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580 -Some choiceless results

April 16, 2009

I reformatted the posts for the lectures on choiceless results. Here they are as a pdf file. It follows closely the style of the notes. I fixed a couple of typos, and made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

Update (Dec. 3, 2009). The pdf file still contains an error that was pointed out to me recently. In the third lecture on choiceless results, the correct argument is given.

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Mathoverflow

Answer by Andrés E. Caicedo for Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ January 16, 2019

Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Arguments against large cardinals October 29, 2010

I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Nonessential use of large cardinals December 10, 2010

There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a singular strong limit of uncountable cofinality. This is a consequence of a more general theorem that can be found in 4.6/6.7 of "Set Theory without choice: not everything on cofinality is possible", Ar […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for What definitions were crucial to further understanding? December 3, 2018

In set theory, definitely the notion of a Woodin cardinal. First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points of certain elementary embeddings. This is not the case here: Woodin cardinals need not be measurable. If $\kappa$ is Woodin, then $V_\kappa$ is a model of set the […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Examples of Mathematical Papers that Contain a Kind of Research Report November 21, 2018

The first example that came to mind was MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London. There, van der Waerden describes some of the history as well as his proof of his well-known theorem. Another example: MR224589 […]

Andrés E. Caicedo

Math.StackExchange

Answer by Andrés E. Caicedo for Does $2^{O(n)} $ mean $O(2^n)$? If not, what does $2^{O(n)} $ mean? April 24, 2019

A function $f:\mathbb N\to\mathbb R$ is $2^{O(n)}$ if and only if there is a constant $C$ such that for all $n$ large enough we have $f(n)\le 2^{Cn}$. We can think of the $O$ notation as decribing a family of functions. So, $2^{O(n)}$ would be the family of functions satisfying the requirements just indicated. In contrast, a function $f$ is $O(2^n)$ if and o […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Models of set theory where not every set can be linearly ordered April 15, 2019

An interesting example of a different kind is any model where all sets of reals have the Baire property. In any such set the quotient of $\mathbb R$ by the Vitali equivalence relation is not linearly orderable. See here for a sketch. Examples of such models are Solovay's model where all sets of reals are Lebesgue measurable, or natural models of the axi […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for What is the consistency strength of generalized failure of the continuum hypothesis. April 14, 2019

The precise consistency strength of the global failure of the generalized continuum hypothesis is somewhat technical to state. As far as I know, it has not been published, but I think we have a decent understanding of what the correct statement should be. The most relevant paper towards this result is MR2224051 (2007d:03082). Gitik, Moti Merimovich, Carmi. P […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Could P=NP be dependent with CH? April 13, 2019

P=NP is an arithmetic statement: we can code the relevant deterministic Turing machines by numbers in a fairly explicit recursive way (which also explicitly involves codes for polynomial upper bounds), and then the equality between both classes can be discussed by discussing numerical properties of the indices involved in the coding, and using a specific NP- […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Can Tarski's circle squaring problem be solved with measurable sets and/or without the Axiom of Choice? November 14, 2013

Update: The problem has been solved. See below for the original answer, with the state of the art in 2013. In 2017, Ł. Grabowski, A. Máthé and O. Pikhurko showed in Measurable circle squaring, Ann. of Math. (2) 185 (2017), no. 2, 671–710, MR3612006, that Tarski's problem can be solved using pieces that are both Lebesgue and Baire measurable. Their proof […]

Andrés E. Caicedo

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A kind of library

580 -Partition calculus (6)

305 -Homework set 8

305 -9. Groups

580 -Partition calculus (5)

305 -8. Irreducibility

580 -Cardinal Arithmetic

580 -Some choiceless results

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