Chess will never be solved, here's why

Regarding setss - all of the discussions about 'sets containing themselves' or not would not appear to work that well - without concrete examples.

It's a matter of definition. What is a set is determined by the axioms. And the versions of set theory that are used (because they are consistent) have NO sets that contain themselves.I guess I could get on chatgpt and Copilot and ask them to give concrete examples of sets that contain themselves versus those that don't - but Elroch and Martin seem quite well informed though.
Russell and other greats don't seem to give examples in their more famous pronouncements. Why not?

You can easily define such an example in Naive Set Theory (the formal system implicitly used before it was found to be inconsistent). Since in this system it is legitimate to define a set (of sets) by a property, you can define the set of all sets (the property being used is the one that is true for every set). This set can be deduced to be a member of itself.

Unfortunately, it is just as easy in this system to define the set of all sets that do not contain themselves as an element (that is a valid property), and we know that this leads to a paradox, proving Naive Set Theory inconsistent.

Well - back in those days who was their principle audience?
In other words the principle audience of the great mathematicians of the past?
The answer would appear to be: each other.
The more you go back - the more you see that the pioneers of maths and philosophy and science mainly talked to each other.

They still do: they are the people who can understand the details!Various causes of that. For one - very few could read or write.
For two - the pioneers were mostly nobles or bigshots or landowners. Whatever.
For three education was primitive or non-existent.
The fourth one - no internet - is more double-edged.
The internet is an instrument of information and education but unfortunately its also an instrument of disinformation and misguidance and 'reverse education' and negative indoctrination.
--------------------
'Sets that contain themselves'
That ice cube shouldn't be that hard to break.
Lets see first though - if an AI sledgehammer is needed to break it.
Shouldn't be.
If Daneel Olivaw was here maybe he might say 'Come on - use internet search to get the examples. Don't use AI.'

Correct. Or you could read the above and see that:

1. What is a set depends on the formal system you choose to use
2. The consistent set theories that are used don't have any sets that contain themselves as an element
3. Inconsistent set theories can include sets that contain themselves, but such systems are worthless. EVERY statement and its negation is "true" in such systems, so truth is meaningless.

e.g. - the below # is where my (12) leading zero sha-256 hash is stored in my virtual memory. sadly i cant convert that to physmem cuzza obvious restrictions. so im trying to make my own page table...unbeknownst to all but itty-bitty me lol !

11110111001110100100110000111001111110000 (virtual)

its here i start-stored my (12) leading zeros hash. they say that (8) were needed for a 50-bitcoin reward (tho the genesis block ended up w/ [10] for reasons not sure (01.03.2009). tale of my life. i can get there - only just as e/o's leaving :::/ (im wearing contacts AND readers right now)

try it !...take 4811533856641 (decimal) and use a sha256 converter to see. dum but kinda fun .

@Elroch
As usual we're having some disagreement as to how key points should or could be communicated to whoever.
Yes - science and math continue to have their 'elite' at the top but the communication of that science and math have changed over the centuries.
For example - in education.
Without even one single example of a set that 'contains itself' -
well I'll put it another way ...
you're suggesting there's no such examples because there's no such thing in modern set theory.

I gave an example in the (inconsistent) naive set theory, the "set of all sets". I also explained that the non-existence of sets which contain themself as a member was a theorem in all the formal systems I know that are used for set theory these days (specifically Zermelo-Fraenkel and Constructive Set Theory). The latter is a much more reasonable intuition as to what a set "really is", as it is at least consistent! 

While the modern axioms were chosen with a view to achieving consistency (an absolutely essential requirement of a branch of formal system used to formalise a branch of mathematics), a side effect is that it makes it impossible for a set to contain itself as an element.

There is an intuitive reason why this is a very good thing to happen - if such sets existed induction based on membership would not work - you could have an infinite sequence of steps, each leading from a set to a member of itself that happened to be the same set. This form of induction is (in my limited experience) used to prove most results in set theory, so stopping it working would be very bad!

With 'Naive' set theory being the predecessor and if I've got it right Bertrand Russell exposed the 'bug' in that by stating his famous paradox.

Yes. To be very precise, I think it was Frege's formalisation of naive set theory that Russell proved inconsistent. There was no consistent formal system for set theory until later.

Leading to 'unlimited composition' being set aside so that set theory would work better.

It's important to emphasise just how much better. In earlier formalisations like Frege's, Russell showed that ALL propositions can be deduced to be false and ALL propositions can be deduced to be true (this always happens when a formal system is inconsistent). This includes even simple things like comparing the sizes of finite sets. It is therefore entirely worthless as a formalisation of set theory, and it was merely a fortuitous accident that some correct results were deduced using it before Russell proved it inconsistent!

In a million years we will come here and someone will say "Hurrah, we have finally solved 9 dimensional tic-tac-toe!" .

And we will say "but what about chess?".

And they will say "what's 'chess'?".

@Elroch
I think we're making more progress now.
Example of a set that 'contains itself' ...
The 'set of all sets'.
So there is at least one.

"Is" is not as trivial a word as you might think. Here it means that "there are ways to define set theory in which 'the set of all sets" is a set, and this provides an example of a set that is a member of itself". Unfortunately, the example of a way to define set theory in which that is the case is inconsistent, so doesn't really define anything. If that is not obvious, note that it is also true in this system that there is no set which is a member of itself. The system is inconsistent, so nothing about it is of substance.

But another thing - how 'sets' work in the background.
People (and creatures) aware of sets in a primary way even over math and language.

Of course. But it seems obvious that as soon as you need to define infinite sets, abstraction is required. We don't deal directly with infinite sets of anything, whether it be apples or atoms.--------------------
When I was in school (before I went to university) I had some exposure to 'the math of sets'.
It was under the heading of 'additional mathematics' ..
and concerned things like Venn diagrams and 'intersection and union' of sets.
I would expect that most high school graduates around the world know what a Venn diagram is. (even in the US).
Although in some schooling systems they're way ahead - with even 8th graders having had exposure to basic calculus and the like. Derivative of x^2 = 2x for example and integral of 2x with respect to dx would be (x^2) + C.
Integration not quite the exact reverse of differentiation.
Or rather they were. Were way ahead. Now? I don't know.

Yes. In such useful elementary introductions, you generally have a problem-dependent concept called "the universal set" in which you work for a particular problem. For example, if you are drawing diagrams about people, your universal set might be "the set of all people".

You might even deal with specific infinite universal sets, such as drawing diagrams about numbers, where the universal set might be the natural numbers (or even the real numbers).

There is no problem making such things consistent. You avoid anything like Russell's paradox by never dealing with really big sets.

@Elroch
"You avoid anything like Russell's paradox by never dealing with really big sets."
Yesterday I was discussing that paradox with chatgpt and it put Russell's paradox in one quick line.
"Does the set of all sets that do not contain themselves, contain itself?"
There it is. 13 words.
It could be further condensed by getting rid of 'the' and contracting 'do not' to don't.
A cutdown from 13 to 11.
---------------------------------
(reminds me of the cutdown when considering all possible chess positions - each square can only have up to 13 states - but you can cut that down by factoring in there can only be two kings and they're oppositely coloured and must always be there.)
Then you get 11 states instead.
For every square.
The first Cutdown is from 13^64 down to 64 x 60 x (11^62).
Such cutdowns eventually lead to the Tromp number. Appr. 5 x (10^44).
A daunting number of possible legal chess positions that is prohibitive of chess ever being solved even in billions of years. Trillions of trillions of years? Maybe.
And noting that Lola 'having fun' talking about big numbers.
---------------------------
But: "Does the set of all sets that do not contain themselves, contain itself?"
Fun. 
Russell came out with that in 1901 but it wasn't until the next year that he hit Frege with it.

Yes. Of course now you understand that the English sentence hides some assumptions. These assumptions may or may not be valid in a given formalisation of set theory. In the consistent formalisations of set theory they are not all valid, which is how Russell's Paradox is avoided.Einstein came out with E = mc² three years after that. In 1905.

It was a good decade, the first of the 20th century.But it wasn't until Fifty Years Later that Russell and Einstein had their famous collaboration.
They did talk to each other.

Which is interesting, especially seeing as their fields were quite distant. But physics requires consistent mathematical theories in order to have something to build models with!Russell formed a letter signed by many great men including Einstein who signed it shortly before his death the same year. 1955. He was first to sign. They signed in this order:
Albert Einstein (Germany/USA)
Bertrand Russell (England)
Max Born (Germany)
Percy Bridgman (USA)
Leopold Infeld (Poland)
Frédéric Joliot-Curie (France)
Hermann Muller (USA)
Linus Pauling (USA)
Cecil Powell (England)
Hideki Yukawa (Japan)
Joseph Rotblat (England)
-----------------------------------------

You didn't say, but of course the letter was the famous one warning of the danger to mankind from nuclear weapons.

Adding a temporal coincidence, as well as that letter and Einstein's death, 1955 is the first year for which we have chess games by Fischer, at the age of 12 or 13.