Chess will never be solved, here's why

I think you may be misrepresenting him in a subtle but enlightening way.

I don't believe I'm misrepresenting Russell, you can check with the horse's mouth here (archived copy of vol I of first edition of PM).

Note the highlighted sentence below, which is almost identical with what I said.

Note that  is Russell's notation for the class {z:φ(z)} (this is not the best editor for discussing the subject). Russell has no need of a distinction between classes and sets, the distinction has appeared subsequently owing to rival solutions to Russell's paradox e.g. NBG.

Before Russell discovered Naive Set Theory was inconsistent and inspired others (PM was his own attempt) to find a consistent formalisation of set theory, there was an axiom of Naive Set Theory that a property of a set (eg "S is not a member of S") always defined a set - the set of sets that satisfy that property. (I can't recall the name of the axiom, but it can be looked up, and that is its definition).

The axiom of universal (or unrestricted) comprehension, but I think the term appeared simultaneously with the realisation that some form of axiom of restricted comprehension was necessary rather than being part of a previously existing axiomatic system called "Naive Set Theory". (I always understood "Naive Set Theory" to mean just an informal development of set theory. Halmos published a book with the name "Naive Set Theory", which was, I think, just an informal development of selected set theoretic results from informally stated axioms of ZFC.)

So he could have observed that it could not be meaningful to define a set by the property mentioned in both our posts. But that does not mean the property itself is meaningless. For example, let's look at a typical set:

5 = {0, 1, 2, 3, 4}

Is 5 a member of 5? No. You can just check the elements one by one and observe they are not 5.

Here I think you are misrepresenting Russell. Since any class σ has a higher logical type than its elements, I believe he regarded the statements σ=x and ~(σ=x), where x∈σ, in the same way as φα in the above highlighted quote, as neither true nor false, but nonsense.

Indeed the property is meaningful for any set. "There exists an element E of S such that E = S" is a well-formed statement about a set S.

The relevant question, in relation to whether or not I was misrepresenting Russell is not whether you believe the property is meaningful or whether it's a wff of ZFC, but whether Russell believed it is meaningful. 

Anyhow, the bottom line is that in Zermelo-Fraenkel the above description is entirely valid. Agreed. I am just indicating it is intuitively sound (not a 100% reliable thing, to be fair!)

"S is a member of S" is a valid property of sets, and it is true for no sets at all in Zermelo-Fraenkel (because it can be proved to be false as I indicated in my previous post). Agreed again, but again not relevant to whether I misrepresent Russell.

It's worth remembering that the three volumes of Principia Mathematica were published between 1910 and 1913, and Zermelo-Fraenkel set theory was only arrived at in 1922.

Which may (or may not) indicate some progress in the interim.

Russell published a second edition of PM in 1927. I've only skimmed it but the main change appears to be a reduction of the number of primitive ideas and propositions. The book is not rewritten but just the changes necessary to accomodate the new primitive ideas and propositions discussed. I believe the theory of types is largely unchanged.

Wht? Russell's paradox or chess?

Hi Martin.
I asked AI about set theory. Some simple questions.
Like if the elements of a set A were just a single element and that element was also A - does that mean that A contains itself? I think the AI said no.

The AI was right.

But I would repeat - the best way to discuss these set things is to use very simple concrete examples rather than equations or to depend on terms with variable semantics.

That has its use - particularly for finite sets, but becomes less reliable with infinite sets, and good luck dealing with inaccessible cardinals or the like!

Set theory is almost 'out on its own limb' as it were. But its much more basic than many people think. Basic to life. Identification and so on.

It is a foundational topic.

@MARattigan, I accept that I was wrong to say that you misrepresented Russell, in that in his system the statement had no meaning. That doesn't mean he thought it could not have a meaning - and doubtless he accepted this was the case once ZF came along.

Some sad, lonely people on this website, aren't there?

I actually solved it the other day but forgot to hit save, fml. I'll try to recover the word file guys

Some sad, lonely people on this website, aren't there?

one major example is the person I'm quoting

Exactly. A person who is not in forums to discuss the stated subject of whatever forum.
But a solution to his problems does not lie in the forums and he doesn't grasp that.
Neither does the other one.
In other words the two people who have replaced the absent two.

i have returned good fellows

i would be truly beaming with happiness if one of you could help me catch up to the present