Set Theory and Foundations of Mathematics

by bschne • 7 years ago

264 77 7 years ago

LeanderK• 7 years ago

I had a seminar on set theory and the foundations of mathmatics. It was very fun! But, coming from a computer science background, i found the foundations via set-theory unintuitive and a bit ackward to work with.

I know there are some type-system based approaches, how do they turn out in practice? How nice are they to work with "on a low level"?

gylterud5• 7 years ago

Type theories, especially dependent type theories a la Martin-Löf, are quite natural to work with. They are easier to implement on a computer, and there are active communities around several such implementations (Agda, Coq, Lean,⋯).

Most computer verified proofs are implemented in systems based on type theory, not set theory. Most of the time you do not need sets of sets of sets… You want types whose elements are the mathematical objects you care about.

In fact, if you want to do set theory, you can do that inside type theory as well. You can define the type of iterative sets and define a membership relation on this. The resulting stucture will satisfy set theoretical axioms. This was first done by Aczel in the 70s.

0815test4• 7 years ago

"Set theory" itself is a bit of an overloaded term; most practical uses of naïve set theory in math do not make use of properties like extensionality or the global membership relation, that characterize axiomatic set theory over TT. Thus, one could also describe the sort of "more natural" systems you reference in your comment as 'structural set theory' https://ncatlab.org/nlab/show/structural+set+theory

It's true however that one can do material-set theory inside structural-set theory or type theory, by defining a type of material sets (not necessarily "iterative" however, that's a matter of whether you want to model the axiom of foundation!) and a membership relation on them.

enriquto4• 7 years ago

Most of the time you do not need sets of sets of sets…

How do you construct the natural numbers, then ?

dwheeler3• 7 years ago

> i found the foundations via set-theory unintuitive and a bit ackward to work with.
Type theories, especially dependent type theories a la Martin-Löf, are quite natural to work with (and) are easier to implement on a computer...

Whether or not type theories are "natural" is probably a matter of taste.

But it's hard to argue that they are necessarily "easier to implement on a computer". Metamath is a general-purpose verifier, and its primary database (the Metamath Proof Explorer at http://us.metamath.org/mpeuni/mmset.html ) is based on classical logic and ZFC set theory. A Metamath verifier in Mathematica is only 74 lines, while another in Python is only 350 lines. The Metamath verifier written in Rust is longer (it's written for speed), but it manages to verify about 32,000 proofs in about 0.9 seconds. It's so easy to write one that there's a long list of verifiers (see http://us.metamath.org/other.html#verifiers ). Metamath can handle HOL, too. I don't know of any shorter verifier that can handle the full range of mathematics. So by both code size and speed it's hard to argue that type theory is necessarily "easier" than traditional set theory on a computer. They both simply process symbols given a small set of rules.

There are many possible foundations of mathematics beyond ZFC and type theory, e.g., Quine's "New Foundations" (NF) (see http://us.metamath.org/nfeuni/mmnf.html for a Metamath encoding of that), HOL, etc. And that's not even counting logic systems: Classical logic is the most common in mathematics (by far), but intuitionistic logic is in use (and tools like Coq are built on it), and there are many other logic systems as well (paraconsistent, minimal, etc.).

That said, I believe ZFC (combined with classical logic) is by far the most commonly foundation referenced in mathematics today. Wikipedia claims: "Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics." https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

That doesn't mean that other systems are "not as good" or "not mathematical" - but when people ask about mathematical foundations, ZFC is typically what people refer to first. It's probably good to at least know a little about ZFC when discussing foundations, even if you decide to do something different.

fnrslvr2• 7 years ago

Set theory as a toolkit is mostly geared towards questions of consistency/existence, size or "order" of objects, and raw mathematical explanatory power of theories or claims. It traditionally arose to explore ideas of infinity precisely, then grew to meet the needs of a mathematical community grappling with antinomies (e.g. Russell's paradox) and heated foundational divides (classicism vs intuitionism, etc). It still meets those needs well -- topics like large cardinals which chart the territory dangerously beyond the boundaries of (assumed) mathematical consistency are plied with set theory, and questions of consistency that arise in other fields tend to end up ported into set-theoretic terms.

I'm not a type theorist, but I imagine the split between set theory and type theory is analogous to the way in which different models of computation appeal to researchers on different sides of the "Theory A/Theory B" split in computer science. Researchers in the "A" group who study the outer limits of what problems are/aren't tractable, how different kinds of computational resource compare in power, etc, tend to use Turing machines and RAM models and circuits, which allow computational resources to be easily measured, bounded, simulated, reduced into things, etc. Researchers in the "B" group who explore programming languages, type systems, program analysis, etc, tend to prefer models like lambda calculus which allow programs to be easily composed and annotated with types and whatnot. I don't think this is just about cultural differences -- if you try to prove time-space trade-offs with the lambda calculus or invent a new type parameterization system with Turing machines then you're going to have a bad time.

0815test4• 7 years ago

Ironically, large-cardinals and related devices give set theory a rather type-theoretic "flavor". One can definitely explore the very same questions of 'consistency/existence, size or "order" of objects, and raw mathematical explanatory power of theories' with type-theoretic tools, and arguably do it quite more elegantly, because the related formal devices (e.g. "universes") arise more naturally in a type-theoretic context.

The_rationalist4• 7 years ago

One should read about standard logic, propositional logic, term logic and nth order logics before, because set theory is founded by them.

enriquto4• 7 years ago

Not really. There are many reasonable places to start learning in the edifice of mathematics, without needing to be aware of the whole structure.

For example, you can do a lot of math without knowing the definition of real number.

rocqua3• 7 years ago

I'd say most intuition matches the reals, even if people's explanation of it would not match the definition.

You just need to walk someone through 'sqrt(2) is irrational ' and 'pi is transcendental' and you essentially end up with 'I want all converging sequences in Q to have a limit'. Which gives you the reals.

In some sense, the reals are the most naive number system. Just start with the natural numbers, and allow every sensible opperation.

The_rationalist4• 7 years ago

Yes indeed, one does not need to first learn zeroth order logic for doing maths. I was talking for one that want to learn from the root of mathematics, some people believe that set theory is the foundation while it is a composition of lower level primitives

fnrslvr2• 7 years ago

Set theory and logic are too interdependent to venture far in one whilst steering clear of the other. For instance, you'll only really skim the surface of model theory without a basic understanding of infinite cardinals.

The_rationalist4• 7 years ago

They can indeed be used together and are interdépendant in practice at "high level mathematics" But set theory is made of a lower level primitive (first order logic) itself being an extension of zeroth order logic. I was saying that in the context of learning the foundation of mathematics where set theory is not the first place to begin.

tempodox• 7 years ago

This contains some falsehoods, e.g.

  (A⇔B⇔C)

is not the same as

  ((A⇔B)∧(B⇔C))

and the former doesn't imply

  (A⇔C)

either, but the latter does.

drdeca• 7 years ago

What?

What do you mean by (A⇔B⇔C) then?

Because I’m pretty sure I’ve basically only seen it as shorthand for the conjunction of (A⇔B) and (B⇔C), when it is understood that ⇔ is transitive.

codethief• 7 years ago

How do you read

  (A⇔B⇔C)

?

The_rationalist4• 7 years ago

The error comme from the use of the equivalence operator instead of the material Implication operator =>

Buldak• 7 years ago

So what's the interpretation that will make one false and the other true?

rrobukef5• 7 years ago

My standard interpretation is ((A⇔B)⇔C). A=F,B=F,C=T differ in evaluation.

* edited T to F

gylterud5• 7 years ago

Interpreting A⇔B⇔C as (A⇔B)⇔C is utterly non-standard, but admittedly interesting.

A⇔B⇔C is usually not a valid expression in any mathematical formalism. But if used informally, A⇔B⇔C is usually meant to mean (A⇔B)∧(B⇔C), just as when you say A=B=C when you actually mean (A=B)∧(B=C).

rrobukef5• 7 years ago

https://en.wikipedia.org/wiki/Logical_biconditional

(A⇔B)⇔C has more in common with the other logical operators: due to it's associativity an interpretation that is, in my opinion, closer than A=B=C.

Edit, removed: Note that (A⇔B)∧(B⇔C)) is not one of the two ambiguous options: either (A⇔B)⇔C or (A∧B∧C)∨(¬A∧¬B∧¬C) this is wrong.

thomk• 7 years ago

Question: Would you use 'set theory' to find patterns of sets in a hierarchy?

markkm• 7 years ago

"The set theory formalism presented here differs from the traditional ZF system"

Do not, I repeat, DO NOT study nonstandard theories before familiarizing yourself with the standard theory. The standard set theory (ZFC) survived for over a century for very, very good reasons. Without an exposure to the accepted standards, you will not be able to tell what is reasonable and what is not in this set of notes.

It should also be noted that the author's field of concentration (geometric analysis) has nothing to do with set theory. A typical mathematician knows very little about axiomatic set theory and is prone to making imprecise statements about it.

gylterud5• 7 years ago

While I agree that one could take caution when studying proposed alternative foundations of mathematics (there are a lot of dysfunctional theories out there, and the link may be one of them for all I know), I think it is unfair to presume incompetence based on the fact that the author has another field as his specialisation. ZFC is also not the one true way, there are many sensible ways to formulate set theory, and even valid criticisms of ZFC.

jacobolus• 7 years ago

Set theory historically exists to serve analysis.

https://www.youtube.com/watch?v=hBcWRZMP6xs

korpiq• 7 years ago

To me this looks like an interesting attempt to holistically give some overview of mathematics to lay people and could be improved if eg. those with correction proposals here would have some way to contribute back.

I might immensely enjoy such a wikipedia of mathematics.

joe_the_user• 7 years ago

Honestly, I would suggest looking at Wikipedia proper. It's coverage of mathematical topics seems to be currently quite good. You can generally easily google the proofs and the details for topics presented without these. The web altogether is goldmine of mathematical knowledge today imo.

Most things aren't explained "down to the elementary level" but anyone who spends significant time on this stuff will reach a higher level and appreciate not having the same elementary explanations repeated over and over.

sridca• 7 years ago

Related: Category Theory for Programmers

https://github.com/hmemcpy/milewski-ctfp-pdf

bubble-074• 7 years ago

What is the first diagram on this page supposed to depict? Seems like a bunch of unrelated topics with random arrows drawn between them.

Also, the author seems more than a bit arrogant and deluded: http://settheory.net/life -- I would _not_ touch any of this stuff with a 40 foot pole.

travisjungroth• 7 years ago

That link looks like the writing of someone mentally ill.

crimsonalucard4• 7 years ago

A lot of geniuses have issues. I wouldn't use this to judge the rest of his work.

avinium• 7 years ago

I think "mentally ill" is pretty unfair. Passionate, depressed, maybe even self-righteous? Sure, all of these.

But his capacity for logic doesn't seem to be affected. English isn't his first language, but he still writes in full, grammatically correct sentences that are very clear. Wordy and emotional - yes. But not mentally ill.

I think we should be careful not to dismiss/overlook people because we don't share their emotional perspective on the world. In other words, just because he's a unique character doesn't mean his take on set theory should be ignored.

From a very brief click through (and from my own fairly superficial memory of the topic), it doesn't seem to be extremely unconventional. At the very least, it's not standing shoulder to shoulder with Timecube.

DyslexicAtheist2• 7 years ago

the whole thing you linked reads like he is going through some existential crisis triggered by spending too much time in his own head, or he is struggling to find himself. he calls others stupid but that might just be due to feeling misunderstood and a symptom of his alienation.

one thing I learned is that knowing more doesn't make us happy. knowledge can be the key that opens a door to insanity.

whether he is a lunatic or genius is hard to say without evaluating the quality of his work.

onemoresoop2• 7 years ago

But there are still many well-placed people who will never listen, cannot grasp this and that keep denying the right for young geniuses to decide for their own life.

This obsession with I am a genius is bordering on pathological narcissism, excluding oneself from the society because one's too smart to be understood by the world is a very dangerous slippery slope. On the other hand, while in a healthy way, I applaud when one sticks to their own ways. I just wish humility was mode widely practiced. We are not gods, we're just some creatures that have varying degrees of intelligence, but nothing special.

Good luck to you young thinker but remember we are to enjoy this life and the people around us. Don't be lonely and miserable trying to prove everybody wrong, it's pointless.

bubble-074• 7 years ago

The major red flag is in the "misunderstood genius" part -- 99% of the people I've ever seen with a claim like that haven't interacted with things enough to have been able to truly test their claims. "Genius" is only "misunderstood" when the person laying claim to it does not do enough to substantiate their argument.

Here, I read some of what they had to say about "infoliberalism" -- the immediate issue that I have with their position is that they make some kind of claim that a few participants in it would naturally lead to a revolution. They then go on to claim that the system is completely decentralized, but also complain that not one other person is willing to join. If their system were truly decentralized, and also something that would provide significant benefits to participants over the existing social/economic/political order, why is the number of practitioners exactly one? All of this screams quackery.

ben5092• 7 years ago

He's definitely angry, but from skimming it, it sounds like he had some ideas, put them out there, and no one bit.

And he's uspet that it didn't pan out, and he's quite clear on a fairly reasonable, if hyperbolic, explanation of why the ideas didn't work.

I read it as him coming to grips with the common blind spot of "my ideology would work just fine except for all the people who don't hold my ideology". You hear it from proponents of all major ideologies; liberalism "they're on the wrong side of history", conservatism "what happened to common sense?!" and libertarianism "but in a libertarian society..."

His stuff is ranty and tone-deaf, and he probably shouldn't include it on a page on set theory, but he's not delusional.

bubble-074• 7 years ago

For all of the political ideologies you listed, we have concrete examples of societies which trend in those directions. Sure, most real societies do not trend toward radical versions of those ideologies, but at least we can each get an okay sense of the first derivative, so to speak, of "goodness" with respect to variation in political leanings.

Correct me if I'm wrong, but it doesn't seem like anything like "Infoliberalism" has ever been tested (and the author even _says_ as much). Before jumping on the boat of a radical political ideology, I'm sure that any reasonable person would at least like to have an _idea_ of what it leads to.

crimsonalucard4• 7 years ago

Eh. Donald Knuth is Lutherian. I'm atheist and I believe it is unscientific and deluded to believe in things from an ancient book of fairy tales.

This doesn't discount Knuths' major contributions to the field. No one is perfect. Even the most logical, intelligent person could have aspects of himself that are illogical.

If the OP's set theory stuff is good, I wouldn't let his emotional rants affect my judgement of this.

dkarl• 7 years ago

I don't see anything there that's incompatible with a high level of skill at mathematics. Honestly, I don't know of any opinion outside the realm of mathematics that could be deluded or disorganized enough to make me think that the person holding it couldn't be competent at math. That said, before I trusted this web site as a source I'd look for evidence that it has been reviewed and improved by other people, because even a competent person is bound to make mistakes, and if he's working entirely alone his mistakes are likely to go uncorrected.

fesoliveira• 7 years ago

I wish someone could give this a beauty treatment to make it look like Immersive Math, for instance (http://immersivemath.com/ila/index.html). This typography and formatting makes it very hard to read these notes, although the content seems to be very interesting.

onemoresoop2• 7 years ago

I love this. I wish I started math with these tools back in the day!!

meuk• 7 years ago

There are links to pdf files near the headers of the chapters, although the pdfs are called "obsolete", so I guess they are not the latest version or miss content.

formalsystem• 7 years ago

Seems like it's hit by the HN hug of death. Am wondering is there any way for me to help out random people who I think have good content by mirroring their content and increase scale on their behalf?

crimsonalucard4• 7 years ago

Category theory is an alternative view on the foundations of mathematics that literally gets rid of sets.

rocqua3• 7 years ago

Well, there is the rather prototypical category Set. So I would not say it gets rid of sets.

Koshkin2• 7 years ago

Also, there are many examples of concrete sets in there. For example, "the set of morphisms from A to B" (in a locally small category).

IngoBlechschmid• 7 years ago

It's more subtle than that.

True, given any reasonable well-behaved notion of "set", there is a category Set of all sets. (Its properties will depend on the specific notion of "set" chosen.)

But this observation isn't really relevant to issues of formalization. Mathematics can be formalized in lots of systems, set theory being one, type theory being another, and category theory (more specifically ETCS) providing yet another alternative.

All of these foundations are interesting, enjoy unique features, and are interrelated. In particular, ETCS and set theory are very similar from the point of view of foundations: Basically, ordinary set theory and ETCS are two different flavors of set theory: ordinary set theory rests upon a global membership predicate ∈, while ETCS rests upon the notion of a morphism.

Closest to mathematical practice is probably type theory. (But keep in mind that the points above still stand, and that most mathematicians work informally, not knowing any details about set theory, type theory or ETCS.)

crimsonalucard4• 7 years ago

Yeah, you are right. Maybe I should say gets rid of sets as a the foundation of mathematical theories?

Set Theory and Foundations of Mathematics

Related Stories