Book of Proof (2018) | Hacker News Reader

Book of Proof (2018)

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Book of Proof – An introduction to the methods of proving mathematical theorems - https://news.ycombinator.com/item?id=12101328 - July 2016 (27 comments)

Also this related ongoing thread:

Basics of Proofs (2017) [pdf] - https://news.ycombinator.com/item?id=36353322 - June 2023 (32 comments)

A claim that it is a book of proof makes it hard to get past the first sentence being a lie: "All of mathematics can be described with sets"... Set theory cannot be proven to describe all of math as set theory is a branch of math and has been objected to as its base since it was proposed https://en.wikipedia.org/wiki/Set_theory#Objections_to_set_t...

First, I don't think it's fair to characterize it as a lie.

Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything in math can be described by sets.

[1] https://math.stackexchange.com/questions/2674701/how-why-doe...

>Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything in math can be described by sets.

It doesn't. In fact it has no relation to what is and isn't a set.

The problem is that "can be described" is not a mathematical question* and neither is "set" a universally defined term.

You definitely can define a mathematical object which isn't a set in ZFC, e.g. {x: x is a set}, which is a universe ( https://en.m.wikipedia.org/wiki/Universe_(mathematics) ), but obviously it is not a set in ZFC.

Notoriously {x: x is a set} is a set in naive set theory (which consequently is an incoherent theory).

In a broad sense Category Theory considers objects which can not be sets under certain set-theories, but in a meta-mathematical way that is pretty much the point.

Of course the correct point the author is trying to make is that standard mathematics is founded on set theory and is essential to almost every area you study in mathematics. "2" is a set "f:R \to R, x -> x^2" is a set and so on.

*: Actually it is a mathematical question (see representation theory) but only makes sense if you ask whether something is representable as a specific set theory, which is not a precisely defined domain. N is a set in ZFC (or ZF-C) but definitely not in ZF-I, so "which set theory" is the real question.

Re: "not everything in math can be described by sets”

This is probably a better example: Russel's Paradox. Does there exist a set of all sets that do not contain themselves?

I think it's a clear example of something that can't be described by standard set theory. "Set Theory", without further qualification, I think that it's safe to assume that they are talking about ZFC (or maybe Naive) set theory. Also, this is a reply to a post, so I need to make some assumptions about the audience, while keeping the post brief and coherent. I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math.

>I think it's a clear example of something that can't be described by standard set theory.

It absolutely can be described by naive set theory. You literally just did it.

It also is not a mathematical object in ZFC, there is no way to construct it.

>I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math.

But that is the only way the statement could ever be false. In standard mathematics, founded on ZFC, every object is a set. So the only way the statement could ever be false is if you look at different foundations and broader theories.

It is an intentionally false statement stated without qualification that it should not be taken as true. It is not ignorance as the author knows what (logical) qualifications are as the statement itself starts with the qualification "All" and the statement alone is false with no preceding context. To not make it false, it could be further qualified with "Almost" at the start, or use "Most" instead of "All".

Set theory has traditionally been used as the foundation of maths from which you can model everything else. See the Bourbaki project.

(Nowadays it seems there are a lot of people interested in type theory taking that place.)

The claim here is not that "all of mathematics is set theory", just that sets are sufficient to describe everything else. I don't know if that's correct, but it's certainly a more defensible claim.

You need to read it as: "Set theory is a language through which all of math can be described."

Think of this example:

    Set A is the set of all maximas and minimas of the polynomial function f(x).

Now, to populate the set you'll need to do Calculus. But it is described in the language of set theory.

There are many more examples that can be given:

    Set X is a set of all non-prime integers between 2,000,000,000 and 3,000,000,000. (Number theory)
    Set Y is a set of all the eigenvectors of the transformation matrix A. (Linear Algebra)

And so on.

This is just one of many ways set theory describe math. In my understanding, it makes human-to-human communique on Math much more efficient, but it is not limited to that.

If there is a new field where you can show that you can satisfy tenets of set theory, then rules of set theory applies to the new field as well.

This pattern is most profound in Linear Algebra. You use it in Deep Learning, Geometry, Equations, Non-Linear Dynamics, Quantum Computation/Physics, etc.

That's indeed an exaggerated claim, but it's not far from the truth. I suspect the author didn't want to introduce the concept of the class right away, which is very similar to the concept of set.

Group theory can be described with sets. Analysis can be described with sets. Geometry can be described with sets.

Before calling something a lie, please inform yourself.

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