Hi everyone!
As it is widely known, Zermelo–Fraenkel set theory with the axiom of choice, also widely known as ZFC, has several fatal flaws:
- Nobody remembers the axioms accurately;
- in ZFC, it is always valid to ask of a set ‘what are the elements of its elements?’, and in ordinary mathematical practice, it is not.
From now on, please use the Lawvere's Elementary Theory of the Category of Sets instead of ZFC:
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Only by switching to a superior set of set theory axioms we can save mathematics. Thank you for your attention.
P.S. On a more serious note, I think that their approach is quite interesting and it is useful to revisit the fundamentals once in a while. Overall, as highlighted in An Infinitely Large Napkin, we understand things much better when we think about them as "sets and structure-preserving maps between them" rather than just "sets and their elements", as suggested by ZFC.
ok but who asked?
people here don't even know that $$$(a+b)(aa+bb) = (aaaa-bbbb)/(a-b)$$$ and you are trying to talk about something complicated
Wth is this!
https://codeforces.com/problemset/problem/1188/B
all you need for that is (a-b)(a+b) = (a^2 — b^2) however
For this problem, knowing (a-b)(a+b) = (a^2 — b^2) is not enough.
care to elaborate? i solved it with that.
(x+y)(x^2+y^2)(x-y) = (x^2 — y^2)(x^2 + y^2) = x^4 — y^4
now you just separate variables and count frequency in map, did i miss something?
you miss the drama that started when i pointed to the author that this problem is not a cp problem but rather something between a lottery and a guessing game
he replied lol everyone knows the formula of square's difference therefore everyone should be able to solve it
the trick is actually quite common for me, its been repeatedly used in JEE, just yesterday there was a problem in a JEE mock about sum(i = 1 to 9999)(1/(i^1/4 + (i + 1)^1/4) (i^1/2 + (i + 1)^1/2))
so personally i dont see it as a guessing game.
I know we use ZFC in most places and the set theory is kinda related to programming languages, but how is it related to CP?
Set theory is literally the foundations of mathematics.
CP is just applied set theory.
I will never use any set theory foundation other than ZFC. Many mathematical statements are built on ZFC and require the axiom of choice. Moreover, they are impossible without the axiom of choice. For example, the measure theory and the theory of Lebesgue integral are built on the set theory based on ZFC.
By cancelling ZFC, we will cancel the traditional mathematics. The mathematics will be very different without ZFC... It will be different mathematics.
As mentioned in the paper, the suggested axioms are equivalent to "Zermelo with bounded comprehension and choice". In particular, the tenth axiom is essentially same as the axiom of choice. You could as well get a full ZFC-equivalent system by adding the eleventh axiom:
Though the author argues that for most applications it is not needed, unless you really need generic infinite disjoint unions and/or cartesian products.
P.S. But even if it was not the case, I would argue that having a different mathematics is not that bad in itself. Physics foundations are revisited every once in a while, and although you may say it's a different physics every time, it doesn't automatically mean throwing away older results, but rather rethinking them in new terms.
Applied mathematics is based on traditional measure theory, theory of the Lebesgue integral and functional analysis. In fact, on grounds that require the axiom of choice.
For example, existence of sets that are not Lebesgue-measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice. Many theorems of functional analysis (e. g., the Hahn–Banach theorem) is proved in general case with the Zorn lemma.
A different mathematics is not bad, but why? What problem does it solve? In its turn, the revision of the foundations of physics was caused by the impossibility of explaining certain phenomena in the 'old' physics.
In fact, this is not the first article, which I have read, where it is written about the need to cancel ZFC. But the authors of such articles usually do not write good reasons why we should cancel ZFC.
I'm not sure existence of immeasurable sets, or any other implication of the axiom of choice has any significance in applications? In pure mathematics sure, but hardly applied math.
But it doesn't really matter, because the tenth of the suggested axioms already has roughly the same meaning as the axiom of choice, it doesn't cancel it.
The reason here is roughly same as to why many modern geometrical subjects are studied coordinate-free. It allows for greater mathematical elegance, and is more consistent with how further algebraic subjects are studied and comprehended.
The Zermelo–Fraenkel set theory with the axiom of choice allows for greater mathematical elegance. It allows us to construct very general facts. Usually, Zorn’s lemma appears in proofs of some non-constructive existence theorems. It allows us to prove some results in group theory, ring theory, linear algebra, and topology in very general cases.
For example, every vector space has a basis. Without the Zorn lemma, we cannot prove this statement, but we can prove a weaker version of this statement: "Every vector space generated by a countable number of elements has a basis". Moreover, if every vector space has a basis, then the axiom of choice holds.
Indeed, various authors propose their ideas of cancelling the axiom of choice. For example, one of them is the replacement of the axiom of choice with the axiom of dependent choice. But in this model (ZF + DC) every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property. But the built system of axioms builds a different mathematics...
P. S. In the first version of this comment, I used the wrong term.
The system proposed in the linked article does not cancel the axiom of choice...
Only if we consider 11 axioms, not 10, as in your blog, we will not cancel ZFC. Also in traditional mathematics, for many existence theorems, the Zorn lemma is necessary.
P. S. In the first version of the previous comment, I used the wrong term.
I agree with you that there is no reason to "cancel ZFC", but AC isn't necessarily required to develop analysis (or that much of math), and with enough care you can keep track of when AC is needed (or a weaker form like https://en.wikipedia.org/wiki/Axiom_of_dependent_choice) and I think you'd find that more than you expect doesn't require AC (obviously AC is powerful and gives you more still). In fact there have been interesting descriptive set theoretic results that explicitly use notAC, notably https://en.wikipedia.org/wiki/Solovay_model and research into https://en.wikipedia.org/wiki/Axiom_of_determinacy among others. Moreover the proposed system does have AC in axiom 10, so I don't understand why this conversation is happening around this point.
Replacement the axiom of choice with its weaker form usually leads to another analysis. For example, in analysis in the model ZF + DC we have that every set of real numbers is Lebesgue measurable, has the Baire property and has the perfect set property.
The Solovay model is actually the construction of another analysis, because in the Solovay model all sets of real numbers are Lebesgue measurable. The same can be said about the axiom of determinacy. The axiom of determinacy implies that all sets of real numbers are Lebesgue measurable.
This is not true. You cannot show that ZF+DC has all sets Lesbesgue measurable (in particular AC shows that this cannot be a theorem of ZF+DC), it's just that you cannot show there exists a set that is not Lebesgue measurable (in particular the Solovay model shows this cannot be a theorem of ZF+DC (contingent on consistency of certain large cardinal axioms)). This is a fundamental idea of model theory: if you can construct a model of a theory in which a statement notS holds, then S cannot be a theorem of that system, and doing so in both directions shows the statement is independent of the system. Same holds with Baire/perfect set property — the relative consistency of AC shows that "all sets have the property of Baire" is not a theorem of ZF+DC, and the result that shows existence of a model of ZF+DC in which all sets have the property of Baire shows that "not all sets have the property of Baire" is also not a theorem of ZF+DC.
Hmmm... But following your links in a previous comment:
1) "This follows because the Solovay model satisfies ZF + DC, and every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property."
2) "In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay (1970) in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable."
3) "Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable."
Yes, the Solovay model is a model of ZF+DC in which those statements hold, in particular showing that in some sense you "need" choice to definitely get nonmeasurable sets. Not every model of ZF+DC has those statements hold (ZFC for instance), so rather you get that the existence of a nonmeasurable set is independent of ZF+DC.
Axiom of determinacy is interesting for reasons other than just analysis which is why I mentioned it, and there isn't that strong of evidence to its consistency (yet).
EDIT: Maybe a simpler way of showing that we cannot prove all sets are measurable in ZF+DC is that we know that we can construct nonmeasurable sets in ZFC, so we cannot have that ZFC implies all sets measureable. Therefore, if we could show that ZF+DC implies all sets measureable, then since DC is just a weaker form of AC we have ZFC implies ZF+DC which in turn implies all sets measureable. But we know this cannot be true.
Never thought I'd see the day when adamant quotes Napkin.
It's not really a quote, but the book clearly means it spiritually :)
In my opinion coming up with some new set of statements to "base math on" isn't that interesting if it just ends up being equivalent to some weakened form of ZFC. ZFC works, is pretty easily understandable, and within a couple steps you can get to the objects most mathematicians care about. At best, this is just some weird argument against the axiom of replacement which kills (among many other things) ordinals as Hartog's no longer holds without axiom 11 in the paper, so transfinite induction stops working (iirc transfinite induction is part of the reason why set theory was initially developed). This means that AC (equivalent to axiom 10) no longer implies Zorn's lemma, which is used by working mathematicians and has a lot of useful consequences in normal work. Adding back in axiom 11 just means this is just a funky way to write something equivalent to ZFC.
Otherwise if this is "we should teach these instead of ZFC", I don't understand why. In most teaching the legal "things" are made explicit and I doubt most undergrads need / care for exactly what the axioms of set theory are, with the exceptions of being in a course on set theory or to understand when AC / transfinite induction is needed / possible. Once a student is in a set theory course I don't think how the axioms are stated will matter more than a couple extra "o yeah this implies this" before showing the important results — in fact notably this system does not include (or even seem to provide a nice way of describing) the axiom of foundation.
BUT, that doesn't mean that if you're interested in this sort 'new axiom system' stuff that there isn't interesting work being done at the moment. Woodin (among other set theorists) have been working on building a (roughly) "canonical inner model for supercompactness", which roughly corresponds to "what additional axioms can and should we add to ZFC to get some particularly nice behavior with very large sets". A (supposedly) "accessible" overview can be found at https://arxiv.org/abs/1605.00613 but requires a decent amount of background in sets and models to understand.
Thanks! Speaking of the axiom of foundation, and set theory expansions, what do you think about set theory based on Aczel's anti-foundation axiom? It looked like an interesting concept to me.
I don't know, maybe someone has actually thought about this more, but my gut reaction (on top of not really seeing the point) is that you could have two sets each of which only contain themselves, and they could both be equal or not equal to one another, which seems annoying.
All my life I considered myself to be a mathematician, at least in cp community (in other words, I tend to prove my solutions). But if that's what mathematicians do...
We only use corollaries from ZFC in real life, so let's just put those corollaries as axioms. Proceeds to choose 10 random properties of sets and functions to be axioms.
Why these 10? It is still an arbitrary choice of axioms (pun intended). What is the difference? Real mathematics still will use all the properties of sets they need, not only these 10.
Why do they need to pretend that it matters for real mathematics? Set theory (and basically any mathematics) is purely abstract bullshit, a thing in itself, what's wrong with that? Yes, real mathematics has fundamental stuff like sets and functions underneath. How exactly is that underneath constructed is not that important, is it? Like, I learned Peano arithmetic, it was fun, but it didn't change how I do arithmetic operations on positive integers.
I'm not saying that axiomatic set theory is useless. I just don't think that "this set of axioms is closer to what we are going to use" is a valid argument in the question of choosing the axioms.
I think the paper is authored by a category theory fan, who thinks that making set theory axiomatics spiritually adherent to category theory principles and concepts is a valid reason in itself. And the chosen properties are not random for category theorists, of course.
Sorry, I don't know what is ZFC. But I have an impression that someone who talks about it are not too social.. And they usually talks about it when nobody cares..
The title seems to be a reference to a Dijkstra’s famous article about goto.
“Considered Harmful” Essays Considered Harmful.
In all seriousness, though, I agree with the linked article (and some of the scenarios mentioned in that article pop up here too).
I think the "“Considered Harmful” Essays Considered Harmful" essay has a several fatal flaws that are highlighted in the "“Considered Harmful” Essays Considered Harmful" essay. They should have also mentioned benefits of “Considered Harmful” Essays or provide a side-by-side “compare and contrast” essay that mentions the possible alternatives and their benefits over “Considered Harmful” essays.
Given that the essay itself is a "Considered Harmful" essay, it's not contradictory with its contents. One of the few examples where a diagonalization argument doesn't lead to a contradiction. (/s)
My reaction. I'm just too much of a "practice into theory" kind of guy, so unless I actively use things, I won't get them...
Thanks for an interesting blog about set theory I like very much. As I understood, the author of the article about categorical set theory said that the axiom of foundation is the source of the flaw in ZFC. In my opinion, this problem can be solved clarifying the definition of intersection of sets: if one intersects a set with a math object that is not a set, the empty set is the result. The axiom of foundation is still true and there are no problems like “what are the elements of 2?”. P.S. Sorry for my English