I recently finished the first draft of the last of my books based on my 2010 blog posts (and also my Google buzzes), entitled “Compactness and contradiction“. The PDF of this draft is available here. This is a somewhat assorted (and lightly edited) collection of posts (and buzzes), though concentrating in the areas of analysis (both standard and nonstandard), logic, and group theory. As always, comments and corrections are welcome.

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## 10 comments

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1 July, 2011 at 2:27 pm

Bo JacobyFirst of all: Thank you for giving me this book. I look forward to studying it.

I looked at the Contents, and I would like to tell you about (my) Ordinal Fractions. They are effective for numbering chapters and paragraphs and anything else. They are too elementary to catch the attention of mathematicians.

The digit zero has two meanings. In ‘1770’ zero means ‘nothing’, and in ‘the 1770s’ zero means ‘everything’.

In order to avoid misunderstandings, zero should mean only one thing, and in an ordinal fraction, zero means ‘everything’.

There are only 9 one digit numbers left when zero doesn’t count. So this is not decimal, it is most nonal.

There is no dot between the chapter number and the section number. When there are more than 9 sections in a chapter the sections are numbered with two (nonzero) digits.

An ordinal fraction is something like ‘the third fourth’, because ‘the third’ is an ordinal number and ‘a fourth’ is a fraction. Like this:

00 whole

01 odd fourths

02 even fourths

10 first half

11 first fourth

12 second fourth

20 second half

21 first fourth

22 second fourth

The five ordinal fraction relations are these:

1 The first half is equal to the first half: 10=10

2 The first half is part of the whole: 1011

4 The first half is parallel with the second half: 10><20

5 The first half intersects the odd fourths: 1001

Ordinal fractions are and’ed like this:

10+10=10

10+00=10

10+11=11

10+20=Ø

10+01=11

10+Ø=Ø

Ø is the empty set or the impossible condition.

Ø is the improper ordinal fraction.

So your table of contents may look like below,

Note that 2122<2100, meaning that the section on circular arguments are part of chapter on logic and foundations.

Yours truly, Bo.

0000 Compactness and contradiction

1100 Preface

1200 A remark on notation

1300 Acknowledgments

2100 Logic and foundations

2111 Material implication

2112 Errors in mathematical proofs

2113 Mathematical strength

2114 Stable implications

2115 Notational conventions

2121 Abstraction

2122 Circular arguments

2123 The classical number systems

2124 Round numbers

2125 The \no self-defeating object" argument, revisited

2131 The \no self-defeating object" argument, and the vagueness paradox

2132 A computational perspective on set theory

2200 Group theory

2211 Torsors

2212 Active and passive transformations

2213 Cayley graphs and the geometry of groups

2214 Group extensions

2215 A proof of Gromov's theorem

2300 Analysis

2311 Orders of magnitude, and tropical geometry

2312 Descriptive set theory vs. Lebesgue set theory

2313 Complex analysis vs. real analysis

2314 Sharp inequalities

2315 Implied constants and asymptotic notation

2321 Brownian snowflakes

2322 The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

2323 Finitary consequences of the invariant subspace problem

2324 The Guth-Katz result on the Erd}os distance problem

2325 The Bourgain-Guth method for proving restriction theorems

2400 Nonstandard analysis

2411 Real numbers, nonstandard real numbers, and nite precision arithmetic

2412 Nonstandard analysis as algebraic analysis

2413 Compactness and contradiction: the correspondence principle in ergodic theory

2414 Nonstandard analysis as a completion of standard analysis

2425 Concentration compactness via nonstandard analysis

2500 Partial dierential equations

2511 Quasilinear well-posedness

2512 A type diagram for function spaces

2513 Amplitude-frequency dynamics for semilinear dispersive equations

2514 The Euler-Arnold equation

2600 Miscellaneous

2611 Multiplicity of perspective

2612 Memorisation vs. derivation

2613 Coordinates

2614 Spatial scales

2615 Averaging

2621 What colour is the sun?

2622 Zeno's paradoxes and induction

2623 Jevons' paradox

2624 Bayesian probability

2625 Best, worst, and average-case analysis

2631 Duality

2632 Open and closed conditions

3100 Bibliography

3200 Index

1 July, 2011 at 8:58 pm

Matthew N. PetersenPg. 18, you have “Gdel’s incompleteness theorem”, rather than “Gödel’s…”.

1 July, 2011 at 9:07 pm

Matthew N. PetersenSame difficulty on p. 29

[Corrected, thanks – T.]1 July, 2011 at 11:30 pm

gowersA quick typo: in the statement of Theorem 4.3.11 you write “Szemerdi”. An easy exercise in reverse engineering suggests, in the light of the “Gdel” typo mentioned above, that something more general has gone wrong …

[Ah, some of the text I incorporated to the book contained accents which LaTeX then refused to recognise. I think the problem is fixed now and will be corrected in the next revision of the ms – T.]4 July, 2011 at 12:53 pm

GregCouple of typos right at the start. On page 2, in statement (8) you have “Is A, then B” which looks like it should be “If A, then B”. Lower down, penultimate paragraph (just before the footnote 1) your implication arrows seem to have gone wrong (-> rather than -> for A \implies B and B \implies A)

Otherwise, this looks good so far!

[Thanks, this will be corrected in the next version of the ms. -T.]4 July, 2011 at 1:12 pm

GregOne more before bed: on page 9, top paragraph, you have “the two are not symmetric, because in both cases is bounding an unknown quantity by a known quantity.” The “is” in the second clause is missing its subject — we’re not told _what_ is bounding the unknown quantity.

–g

[Thanks, this will be corrected in the next version of the ms. -T.]4 July, 2011 at 2:25 pm

JohnPage 80: “algberaic” -> “algebraic”

[Thanks, this will be corrected in the next version of the ms. -T.]5 July, 2011 at 12:11 pm

diego.maldona@gmail.comDear Terry, thank you for posting the book. It’s always insightful to read your unifying take on so many, seemingly disconnected, Math topics. Below are a few typos I found.

p.13, line 8, misplaced comma

p.20, line 16, “doesnt”

p.22, line -13, “and its proof is” should be “and its proof are”

p.22, last line, extra semicolon

p.23, “If B was an element” should read “If B were an element”

p.29. The sentence starting with “However…” goes on for ten and a half lines and there is a “then” that’s not quite working out.

p.29, first footnote, I believe “boolean” should be “Boolean” (as “Bayesian” reads “Bayesian”). “boolean” also appears in other parts of the manuscript. Same issue with “gaussian”; but, apparently, “abelian” is ok.

p.30, line 8, “that A was ﬁnite” should be “that A were ﬁnite” or “that A is ﬁnite”

p.48, line 8, “translation” should be “translations”

p.110, There are several “Erdos” and “Holder” (without the umlaut).

p.110, line 11, “considing”

p.192, lines -5, -4, the parameter “p” should be in math font

In Section 1.11, after discussing Richard’s paradox (p.32), you could mention that if a reader runs into a genie willing to grant three wishes, the instruction “I wish for 1000 wishes” won’t have any effect, since that’s not a wish, but a meta-wish. This situation could also be phrased as a paradox (writing, for instance, that the genie will grant exactly 3 wishes).

Regarding paradoxes, it will be illuminating to have your take on the “surprise test paradox”

http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

http://arxiv.org/abs/math/9903160

I loved the “digital-images analogy” when illustrating the concepts of extension and quotient.

[Thanks for the corrections! They will be incorporated in the next revision of the ms. The surprise test paradox is discussed on this blog at https://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/ -T.]9 July, 2011 at 11:51 pm

Hemant VermaHi All,

There are related mathematics and algorithms problem here, for those who love mathematics / algorithm problem solviing Mathalon , after all whats in mathematics without problems.

-Hemant

26 September, 2016 at 11:13 am

AnonymousDear Terry!

What’s the point of thinking of an implication A –> B as asserting that B is at least as true as A in order to understand the disjunction elimination? In know that “A or B” has the biggest truth value of the two truth values of A and B. Thus if C is at least as true as A and also at least as true as B, then “A or B” is at least as true as C (since “A or B” has the same truth value as A or as B). But why does this shed light on the disjunction elimination?