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 Strong partition property + DC + existence of non-principal ultrafilter on $\omega$ August 30, 2018

The key reference for this is MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985. There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (an […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets? August 12, 2018

This is consistent, at least under a rather tame large cardinal assumption. (One can also produce examples by manipulating Dedekind finite sets, but Asaf's answer addresses this. The answer here works even in the context of $\mathsf{DC}$.) For instance, see MR3612001. Conley, Clinton T.; Miller, Benjamin D. Measure reducibility of countable Borel equiva […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice? August 11, 2018

Let me add some comments to Asaf's answer. We are looking for situations with $|X|

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ May 10, 2018

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Does constructing non-measurable sets require the axiom of choice? October 14, 2010

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Andrés E. Caicedo

Math.StackExchange

Answer by Andrés E. Caicedo for Measure on ([0,1], P([0,1])) October 17, 2018

Surprisingly, the answer is no due to serious set-theoretic restrictions. If $X$ is an infinite set such that there is a ($\sigma$-additive) measure on $\mathcal P(X)$ that only takes the values 0 and 1, assigns 1 to $X$ and 0 to singletons, then the cardinality $\kappa=|X|$ of $X$ is much much larger than $\mathfrak c=|\mathbb R|$, the cardinality of the se […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Without the Axiom of Choice, must two nonempty sets have a surjection between them? October 9, 2018

No, you cannot show this. For instance, it is consistent to have infinite Dedekind-finite sets whose power set is still Dedekind-finite. Now, if there is a surjection from $A$ to $\omega$, then there is an injection from $\omega$ (indeed, from $\mathcal P(\omega)$) to $\mathcal P(A)$, so $\mathcal P(A)$ is Dedekind-infinite. Thus, if $\mathcal P(X)$ is infin […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Example of a concrete irrationality test October 7, 2018

First, there are some nice examples like $$ e=\sum_{n\ge0}\frac1{n!} $$ or Liouville-like numbers, mentioned in the answers by Wilem2, that can be easily proved to be irrational using the theorem, but for which typically there are simpler irrationality proofs: For $e$, we quickly get that $$0

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Set of infinite measure and subsets of finite measure. September 20, 2018

Note first that if $C$ is uncountable and for each $i\in C$ we have a real number $r_i>0$, then $\sum_i r_i=+\infty$. The point is that for some $n>0$, the family $\{i\in C: r_i>1/n\}$ is uncountable. Of course, in this case there is a countable subfamily whose sum is infinite as well. Now, let $A$ be measurable of infinite measure. If there is an u […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for What is the utility, exactly, of the Deduction Theorem? September 19, 2018

The theorem says that to prove an implication it is enough to assume the hypothesis and proceed to prove the conclusion. Proofs of that kind tend to be more natural than proofs that conclude the implication directly. Just as in regular mathematical practice: many theorems have the form "Assuming $A$, then we have $B$", and we usually prove them by […]

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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