Things I have learned include: (1) doing propositional logic via axioms and proofs, rather than truth tables (e.g. translating the propositional logic chapter of Principia Mathematica to a formal proof system which is metamath-like in the sense of not having it built in), (2) intuitionistic propositional logic (if there is a good textbook on this, I haven’t yet found it, but as long as I start from a correct set of axioms, I feel like I won’t go too far astray), (3) the difference between a class and a set (hinted at in my undergraduate days, but which actually has subtleties that come out, at least for me, much more in a formal proof system than in any amount of trying to explain it in terms of concepts), (4) predicate logic: I knew very little of this, beyond what the average non-logician uses in an intuitive way without trying to explain it in detail. Now I have proved enough predicate logic to use in, say, arithmetic or set theory, from two different axiomizations. (5) Formalizing geometry (in the point/line style of Euclid’s elements or Tarski’s geometry axioms). When I started I had no idea how the diagrams of high school geometry could be translated to a formal language with predicates like “x is between y and z” and “x y is congruent to z w”. (6) Logical concepts like the deduction theorem (although I haven’t tried to express these in a formal proof system, or even read the work of people who have, it is hard to work in a system like metamath without gaining some insight into what kinds of informal reasoning translate into what kinds of formal proofs).

I’m happy to use a relevant (non-computer-formalized) textbook, so I wouldn’t say my learning has been exclusively by means of formal proofs, but the bulk of it has been focused on writing formal proofs and many of the source materials I have been reading are formal proofs.

Also, you can list the post titles to every post by using a WordPress shortcode. The instructions are on this page:

http://en.support.wordpress.com/archives-shortcode/

You would put the command “[archives]” in a new page, then drag a text widget into the sidebar, and then put a link to the page in the text widget. That’s one way to do it.

Or, you would use a template that provides links to your pages (pages rather than posts).

It’s just a suggestion.

My quote above by Andrews is what makes me paranoid. Naive sets rule, and he indicates that people occasionally define something as a set and violate the Axiom Scheme of Specification. If I’m trying to duplicate math where that has happened, then maybe there’s an easy way to fix the problem with the right framework.

I have to check out of here permanently. I can only use what people like you produce. I forget that because I’ve been in a dormant state, having had to work.

I’ll be working soon to try and produce some social-moral-political commentary-music-entertainment. That doesn’t mix well with the professional scene.

However, out of this series of comments, I think I’ve gotten one answer and one future possibility.

It appears that NBG set theory is not a practical solution to the “more than a set” problem. If you get too far from imitating naive set theory, it’s probably a clumsy solution.

I put the idea about inaccessible cardinals in the back of my mind, as a future possibility.

Happy proving grounds.

Not wanting to talk unintelligently about what I don’t understand, I had set aside part of what I originally wanted to ask about as a possible way to get proper classes with ZF.

In the wiki article I linked to in my first comment, it says this:

Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as “classes”.

It seems to indicate that by simply adding an axiom, we get to mess around with classes like we get to mess around with sets, all in ZF or a model of ZF.

I thought, can it be that simple? But then I also asked, “What’s a model of ZF?”

Grothendieck unverses, models of ZF, that’s too many things to ask about, so I decided.

But, there is this link:

http://mathoverflow.net/questions/6423/set-theory-for-category-theory-beginners

And an arXiv article linked to in one of the answers, the pertinent sections being 6, 7, and 8:

http://arxiv.org/pdf/0810.1279v2.pdf

Those indicate that maybe I was on the right track. I think I’m going to write up a question for math.stackexchange.com to ask about these things. Maybe it’s so simple as adding an axiom assuming an inaccessible cardinal, whatever that means.

I don’t really like putting all these lengthy comments on your blog, but these type of activities can help me work through things sometimes. That regrettable thread I started on the mailing list helped me think some things through faster than I would have without it.

I’ve only barely scraped the surface of axiomatic set theory, but as I understand it, the axioms absolutely prevent you from building certain types of sets, otherwise, you would get a paradox, such as Russel’s paradox. I barely went through a certain exercise which showed that one of the axioms, call it Axiom X, prevents you from creating a set of sets which don’t contain themselves, or whatever it did, it prevented Russel’s Paradox. I’m too lazy to look that up again…

Actually, I’m not, and it’s the Axiom Schema of Specification, referred to as the Axiom of Comprehension by Andrews, who I quote in my first comment. Not allowing unrestricted comprehension prevents the paradoxes. Here’s the wiki link:

http://en.wikipedia.org/wiki/Axiom_schema_of_specification#Unrestricted_comprehension

I’m not worried about axiomatic set theory at this time, but I would like to make a particular request of someone familiar with Isabelle, where my request is in the context of the quotes above about proper.

The request would be this:

Can someone please give me an example of a theorem involving a proper class that’s implemented in Isabelle/ZF?

I don’t make that request, because I don’t want to be a pest anymore than what I am, and who says anyone will answer it? But if you or Larry Paulson created a little example involving a proper class, then that would make me feel better about the future. I assume everyone has higher priorities than making me feel better about the future.

If everything is required to be a set, then certain sets can’t exist, which puts limitations on what kind of objects you can have that contain sets. That sounds to me like a dead end. The other possibility is that it ends up being such a hassle to have these kind of objects, that it’s not worth using them in ZF set theory.

So I’ll quote you on “there is no dead end”, and hold you to it. But that’s a joke, because no one can know what the dead ends are or aren’t, unless they’ve formally proved what the dead ends are or aren’t for a particular theorem prover.

As far as being able to learn math from a proof, In my mind, I was relating that to the idea that in mathematical writing, people have an expectation that they’re going to be able to learn something that they don’t know from studying a proof. On mathoverflow.net, they emphasize that even when you ask a question, the way that you ask it should be informative:

So when you ask for the help of those who know more than you, please also extend a hand to those who know less than you by explaining why you are asking the question.

A person throwing out cryptic, mathematical statements that only serve his or her purpose is obnoxious to people.

A decently explained proof is the completion of something like a three or four step process: an informal discussion, an example, the formal statement of the theorem, and the proof. It’s the understanding of the proof that takes you to a different level of understanding. The scope of mathematical writing runs the gamut, but the more unfriendly an author gets, such as going to a pure theorem, proof, theorem, proof, theorem, proof format, the less their book is going to be read.

Additionally, if an author gets to a point where they’re not filling in huge logical steps in their proofs, it becomes totally unacceptable to the mathematical masses. The use of “by auto” and other automatic proof methods is the equivalent of a textbook author not providing a proof, or providing a cryptic proof. The level of mathematics in a book will determine what steps can be left out, but still, you don’t get to say, “Proof: It’s true. Figure it out yourself.”.

As it is, to learn an Isabelle statement applied to math, I have to understand the math to figure out the logical reason underlying why the person used that statement. If I already knew how to use Isabelle, I would just start proving the standard, textbook theorems I want to prove.

For me, learning Isabelle can’t be a three-month process. Part of my complaint is people’s use of commands like “by auto”. For me to be able to fill in the gaps of “by auto” and “simp”, I’ll have to know where the theorems are that those methods use. That will require me going through a certain amount of foundational code.

Additionally, I need to learn how to use natural deduction and the Isabelle methods that go along with that, which all starts down in Isabelle/FOL. The fact that Isabelle/ZF and Isabelle/HOL are based on natural deduction is part of what keeps me coming back.

The idea of matching conclusions of theorems with hypothesis is something I like. Backward proof is not considered good form when writing up a formal proof, but it’s an important part of proving math. You figure a proof out by going backwards, many times, and then you write it up forwards style. Not knowing a lot, it still appears to me that natural deduction tactics are simple, but yet potentially very powerful.

SECOND PART

I don’t really like all that social-networking style voting, especially because of what I see on mathoverflow.net. Math.stackexchange.com is more tolerant of questions, but even on that forum, I suspect you end up with math supercops wanting to police everyone. I haven’t been watching math.stackeschange to know how bad it is.

I can get pedantic with the best of the pedantics, but if someone gets snippy with me, then I want to go inflammatory on them, and that’s not good on a site like that. These days, I don’t like to see my name out on the web much, so you might see me ask questions under a pseudonym, not that JF is a pseudonym, though it’s not my full name. I don’t even like the idea of opening up an account on stackexchange.com, but a man has to do what he has to do to get ahead in life.

THIRD PART

You have to understand that I have no interest in making any contribution to IsarMathLib. I only care to learn from what you’ve written, both in how to use Isabelle, in reviewing math I’ll be familiar with in IsarMathLib, and learning math I don’t know as a result of trying to figure out a proof.

All I’m interested in right now is proving theorems in textbooks that I’ll be studying. Eventually that’s supposed to lead into something original, but that’s not of ultimate importance to me. If something can be extracted out of what I do, or if I want something I don’t know how to get, and it causes you to add definitions and theorems to IsarMathLib, then that’ll be good, but having that happen is not my primary goal.

I’ll be marking up what you’ve done to suit my purposes. Renaming definitions and theorems, maybe. Expanding steps in proofs to make the logic more clear, hopefully. Anything to help me out and put what I study into a more verbose format.

Creating a fork based on your IsarMathLib wouldn’t be desirable for a number of reasons, but forks are a fact of life, because people want what they want, the way they want it. ProofPeer is a fork off of both Isabelle and HOL-Light (I guess I’m using “fork” in the right way). I wouldn’t want to create a fork off of IsarMathLib, but I want what I want. I just tell you these things.

This is all lofty talk, but after I get laid off from my current job, I expect to make faster progress. I should start making some progress even now. This all assumes I won’t look again in the Isabelle/ZF files and again say, “Forget this. I’m not going this route.”

I hope to get out of “opinion mode” here and go purely technical. Expressing my opinions and talking about the future suck up my time in a bad way.