Compactness and contradiction.

Terence Tao

American Mathematical Society

Publication Year: 2013

ISBN-10: 0-8218-9492-7

ISBN-13: 978-0-8218-9492-7

Last updated: Dec 10, 2016

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, and “Higher order Fourier analysis“.

A draft version of the MS can be found here.

Pre-errata (corrected in the published version):

- Page 78: In the first paragraph, “at least the” should be “at least one of the”. In the second paragraph, “generated by ” and “generated by ” should be “generated by ” and “generated by ” respectively. In the third paragraph, after the first sentence, add “We may take to be a normal subgroup of “. In the last paragraph, replace “cannot grow polynomially” by “cannot grow exponentially (as otherwise the number of subsums of for and would grow exponentially in , contradicting the polynomial growth hypothesis)”
- Page 79, footnote 12: replace the first sentence by “Proof: the algebraic integers for natural number have bounded degree and all Galois conjugates bounded, so the minimal polynomials have bounded integer coefficients and must thus repeat themselves after finitely many .”

Errata:

- Page ???: In Section 1.3, “Wittingstein” should be “Wittgenstein”.
- Page 92: In the third display, should be .
- Page 94: “quarternionic” should be “quaternionic” (two occurrences).
- Page ???: In the paragraph before (3.17), (3.13) should be (3.12).
- Page 110: “considing” should be “considering”.
- Page ???: After equation (3.24), should be . Also, all occurrences of in this section should be for consistency.
- Page 131: in Section 3.10.2, at the end of the treatment of the non-transverse case, “the exponent of here is positive” should be “the exponent of here is negative”.
- Page 220?: In Section 5.4.1, all occurrences of should be .

Thanks to Clément Caubel, Benjamin Sprung, Felix Voigtlaender, Bill Zajc, and an anonymous contributor for corrections.

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1 July, 2011 at 10:49 am

Compactness and contradiction « What’s new[…] 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 […]

14 July, 2011 at 10:01 am

TomSmall error (of a non-mathematical nature) on page 153

“each of the constants, operations, and relations as

anelements, functions, and relations”[Thanks, this will be corrected in the next version of the ms – T.]18 August, 2016 at 9:35 am

BriefcaseJoeThis comment concerns the first subsection “1.1 Material implication”:

Interpreting “If A, then B” as saying “B is at least as true as A” sheds bright light on how one can prove the equivalence of two or more statements, that is, that they have the same truth value: Given “If A, then B” and “If B, then A”, we can infer that A is both at least and at most as true as B, allowing us to conclude that A and B are equivalent. In general, if we have proven a chain of implication of the form

then we can interpret this as saying that is at least as true as , that is at least as true as etc. and that * is at least as true as *. Thus, the statements , , …, are equivalent.

Indeed, until learning this way of looking at implications I didn’t fully understand proof by contrapositive (which is also evident when interpreting “If A, then B” as “B is at least as true as A” and “If not B, then not A” as “A is at least as false as B”) and the method of proving the equivalence of several statements discussed above. Does this mean that I am stupid? Or are there other ways of understanding these proof methods?

I am also wondering why this way of looking at implications isn’t mentioned in other text books or on the internet. Can you guys find an explanation of that?

10 December, 2016 at 9:06 am

AnonymousFound a typo on page 4: “The early twentieth century philosopher Ludwig Wittingstein”. The austrian philosopher was called “Witt*gen*stein” instead.

At least this typo appears in the online version which can be found at

[Typo added to errata, thanks – T.]4 April, 2017 at 10:35 pm

AnonymousP2, LINE 5, Is should be “If”

p13, correct comma “,”

p17, remove “it” this type of …..

p 18, correct Gdel to “Godel”

p24, proof of 1.10.11, we define G “as” or “to be”

[These are corrected in the printed version of the book – T.]10 January, 2018 at 7:35 am

Thomas ArnoldHey ho, after reading your thoughts on your no-self defeating object argument, I was just wondering whether you have looked into Graham Priest’s account of paraconsistent logics and what your thoughts on it are. Self-defeating objects appear at what he calls “the limits of thought” (or conception or expression, depending on the context) and are quite a common theme within philosophy (e.g. there is a debate about whether the world is such an object and should therefore be considered not to exist).

Now, Priest argues that these objects do in fact exist (and can be constructed and explained through what he calls “inclosure schema”, cf. Priest, Beyond the Limits of Thought, 2nd ed., 135) and simply show that we should reject classical logic as there can be ‘true contradictions’ (he and other proponents of dialethism have argued for this at length in several books, but cf https://plato.stanford.edu/entries/dialetheism/).

Most mathematicians I know simply reject this without argument, so I’m very interested in your thoughts on this.

24 December, 2020 at 9:31 am

AnonymousSection 1.1 of the book says that “Material implication is not causal.” On the other hand, “if , then Riemann hypothesis is true” would not be considered as a proof.

Should one say that material implication in a “proof” of a mathematical statement is supposed to be “causal”?

24 December, 2020 at 12:36 pm

Terence TaoThe material implication “if , then the Riemann hypothesis is true” is a true statement, but does not prove the Riemann hypothesis, since the antecedent is false. It is an example of a vacuously true statement – one which is technically true but unlikely to be of substantive value (though sometimes vacuous truths are still useful in mathematical arguments, e.g., as a convenient base case for an induction).

It is certainly good practice to structure one’s proofs so that the implications one uses in the arguments do have a causal connection from their hypotheses to their conclusions, though it is not strictly necessary; for instance a formal proof assistant will happily accept any valid material implication in a proof even if the conclusion has no obvious connection to the hypothesis. But even when a proof is aimed at human readers rather than computers, it can sometimes be useful to use acausal implications. For instance, if the proof needs to split into two cases X and Y, and there is a statement Z that holds in both cases, it may still be preferable to organize the argument to explicitly state the material implications “X implies Z” and “Y implies Z” to manage the two separate cases (e.g., if one wishes to add Z to a long list of conculsions in a proposition exploring the consequences of X, and also add Z to a long list of conclusions in a separate proposition exploring the consequences of Y), even though there is no causal relationship between Z and either X or Y. (In these sorts of cases though one often adds a remark after the implication to the effect that the conclusion Z holds under more general assumptions than just X or Y though, to avoid the reader drawing an incorrect conclusion regarding the causal relationship. One could also consider structuring the argument so that Z is established before one splits into the cases X and Y, though sometimes for notational or organisational reasons this is not always feasible or desirable.)