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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2 Comments | 580: Topics in set theory | Tagged: Andras Hajnal, end-homogeneous set, Erdos-Rado theorem, James Baumgartner, Jerome Keisler, Michael Morley, Stevo Todorcevic | Permalink
Posted by andrescaicedo

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
Posted by andrescaicedo

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.

Leave a Comment » | 580: Topics in set theory | Permalink
Posted by andrescaicedo

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.

Leave a Comment » | 580: Topics in set theory | Permalink
Posted by andrescaicedo

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Mathoverflow

Answer by Andrés E. Caicedo for product of power sets July 13, 2017

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Choosing *two* representatives July 19, 2017

I assume by $\aleph$ you mean $\mathfrak c$, the cardinality of the continuum. You can build $D$ by transfinite recursion: Well-order the continuum in type $\mathfrak c$. At stage $\alpha$ you add a point of $A_\alpha$ to your set, and one to its complement. You can always do this because at each stage fewer than $\mathfrak c$ many points have been selected. […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Axiom of Choice: Ultrafilter vs. Vitali set July 6, 2011

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is negative. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${\mathfrak c}$ (This doesn't matter, all we need is that it is strictly larger. T […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for "Weakly" Woodin cardinals May 28, 2017

(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples: See here for $\Sigma^1_2$-Woodin cardinals and, more generally, the noti […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Completion of ZFC November 22, 2010

I am not sure which statement you heard as the "Ultimate $L$ axiom," but I will assume it is the following version: There is a proper class of Woodin cardinals, and for all sentences $\varphi$ that hold in $V$, there is a universally Baire set $A\subseteq{\mathbb R}$ such that, letting $\theta=\Theta^{L(A,{\mathbb R})}$, we have that $HOD^{L(A,{\ma […]

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Answer by Andrés E. Caicedo for Is there a permutation on $\mathbb{N}$ that, if repeated often enough, eventually shuffles the whole set? July 23, 2017

A simple example is the permutation $\pi$ given by $\pi(n)=n+2$ if $n$ is even, $\pi(1)=0$, and otherwise $\pi(n)=n−2$. It should be clear that $\pi$ is computable and has the desired property. By the way, regarding the footnote: if a bijection is computable, so is its inverse, so $\pi^{-1}$ is computable as well. In general, given a computable bijection $\s […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Clarifying a question on polynomials July 12, 2017

The question is asking to find all polynomials $f$ for which you can find $a,b\in\mathbb R$ with $a\ne b$ such that the displayed identity holds. The concrete numbers $a,b$ may very well depend on $f$. A priori, it may be that for some $f$ there is only one pair for which the identity holds, it may be that for some $f$ there are many such pairs, and it may a […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Minimum number of axioms for ZFC set theory July 1, 2017

The reflection principle is a theorem schema in ZFC, meaning that for each formula $\phi(\vec x)$ we can prove in ZFC a version of the principle for $\phi$. In particular, it gives us that if $\phi$ holds (in the universe of sets) then there is some ordinal $\alpha$ such that $V_\alpha\models \phi$. It follows from this that (assuming its consistency) $\math […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Direct bijection between $C[0,1]$ and $[0,1]$ June 23, 2017

All proofs of the Bernstein-Cantor-Schroeder theorem that I know either directly or with very little work produce an explicit bijection from any given pair of injections. There is an obvious injection from $[0,1]$ to $C[0,1]$ mapping each $t$ to the function constantly equal to $t$, so the question reduces to finding an explicit injection from $C[0,1]$ to $[ […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Is it possible to formalize the idea that large cardinal axioms don't limit us, while their negations do? November 25, 2013

One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]

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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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