This is bizarre. You've just been corrected, you get shown a Wikipedia page that is about the way in which you are wrong, then you repeat the error and quote someone else who explains that you are wrong (from the MIT paper I linked). Your own figging quote says:
"PROOF of Zermelo's THEOREM"
not, say, "statement of Zermelo's axiom".
A proof by MATHEMATICAL INDUCTION is a DEDUCTIVE proof.
You are not the only one who has been confused by the two incompatible uses of the word "induction". I have explained this point earlier in this discussion.
Here is the last paragraph of the introduction to the wiki article on the topic. Sorry if it is difficult to understand, but this has been known since the 3rd century BC. The point is highly relevant to this discussion.
<<Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy, in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by a finite chain of deductive reasoning involving the variable 𝑛, which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability.>>
Anyway Elroch, you've been taken apart, comprehensively. As usual, when you've lost you make personal attacks, Tells me and others exactly what you are.
No, I doubt your blundering even fools those who have no understanding of the subject. Certainly no-one who has decent mathematical qualifications would side with you. They would understand all of the sources as well as me.
I do recall clearly that I learned how to do proofs by induction long before I went to University. An old memory, but quite clear.
[And if you are wondering if I understood it then I recall that in those days I got 100% in my maths exams 3 years in a row.]
"Looked at it. It's an hypothesis. It may be considered by Zermelo to be an axiom and there's no syllogistic proof to support it."
If it wasnt proved it wouldnt be called "zermelo's theorem"
"It was an inductive proof, you fool. Can't you even read? It wasn't deductive. Means it's an assumption."
in mathematics inductive proofs are literally logically equivalent to deductive proofs. "Induction" is just referring to the techniques used.
for example, one of the most basic inductive proofs is to prove that the sum of the first N integers is equal to N(N+1)/2.
let f(N) = N(N+1)/2. Basic arithmetic shows that f(N+1) - f(N) = N+1. therefore, if f(K) = the sum of the first K integers, then f(K+1) = sum of first K+1 integers (where K is a known constant).
then, we start by verifying that f(1)=1.
finally, mathematical induction refers to the step where N can be extended from 1 to all natural numbers. this too is mathematically rigorous, for any M that we claim is the lowest integer for which a statement is false, since M-1 must be true, M must also be true.
All in all optimissed i think your struggles come from imprinting different definitions to mathematical terminology and methods.
"The fact remains that Zermelo has not proven that chess can be represented mathematically.
but he literally did lmfao, chess falls under the class of games that zermelo addresses.
optimissed it would be prudent to admit your mistake on the rigor of mathematical induction
As you can see, Dio still inhabits a fairyland in which the meanings of words are forever obscured from him. Some say he's thick but it's where he lives. Really!!!
Fairies and fairylands are more of a British affectation than American. I would not insult those childhood wonderings/wanderings by lumping your default delusional state with them, however.
well he outsmarts u all day. and thats a fact !
esp when u bumble out s/t 1900's dum like this:
certainly not within our lifetimes with any foreseeable technology we are working on
its ok to be behind...we u/s lol !
It's cute that you think that, but we're in the real world here. *head pat*
Anyway, I asked Elroch to prove that it's correct, according to a theorem of combinatorial games theory, that chess can be solved mathematically.
He just told us that it wasn't an axiom, by the way, MEGA. I thought we were making some progress. However, you were more correct than Elroch, seemingly, because if he can't prove that it can be solved mathematically (and that proof has to be a syllogistic proof, which is what he always demands of others) then it's a axiom.
Having an axiom that chess can be solved mathematically is exactly equivalent to an axiom that states that mankind will reach other galaxies in their space exploration. Now, I'm not saying it's impossible but I strongly doubt it. The axiom is based on "mankind can travel, therefore mankind can travel to other galaxies" and is exactly equivalent to "simple combinatorial games can be solved mathematically: therefore chess can." I would require a proof, please, or you've lost the argument.
You would lose the argument since you would not have responded in kind. You required deductive proofs from tygxc but you cannot give them to defend your own far less reasonable claims.
Except it's not "solving mathematically" in the sense you misunderstand it to mean. You need a much longer talk with your data scientist son...and then you need to listen, not just cherry pick what you want to bring back here for your own "arguments".
Tablebases already prove we could solve chess if we had the resources.
It's cute that you think that, but we're in the real world here. *head pat*
Are you?
I'm in a world where paranormal powers don't exist, and people cannot make sound arguments by unilateral declarations of uninformed opinions. So yes.
Dio, you made a decent argument a few posts back. I wasn't going to respond to you but I read it and I liked your thinking. I didn't think it "won" any points however but it's nice to see you can raise your game (said he, patronisingly). But I mean it. Thankyou for agreeing that my son is correct. The trouble is that you seem to have interpreted the mathematical thing as NOT depicting chess exactly by means of sets of equations; but as representing it by means of mathematical heuristics, which is EXACTLY what Elroch is criticising in others. I actually think he's changed his pitch on that and tacitly accepted an argument I made some time back. Anyway thankyou for that. I'll let you call me dumb if you like.
I never call you dumb. Your interpretation of what I said is not accurate, and your arguments are incorrect, which happens with alarming frequency.
optimissed it would be prudent to admit your mistake on the rigor of mathematical induction
Unless it can be proven that chess can be so represented, and Zermelo's Theorem is not a proof but a claim, then there's no need to believe that it can.
Just to bring you bang up to date, that time you were wondering about when that could happen was about 100 years ago. I have already explained this and underlined it several times using several independent sources.
I didn't bother to contest the calim [sic] that the Theorem itself can be proven because that's not the issue.
Correct. It's not the issue because it WAS proven before my father was born.
It needs to be proven that it can be used to prove that chess can be represented mathematically.
Zermelo explicitly dealt with the example game of chess. He was more interested in this than in general combinatorial games (the term did not even exist at the time. Indeed game theory did not yet exist and Zermelo wrote the first paper on the subject. About chess as a mathematically represented game.
Anyway, it's getting late, I won the argument [snip]
optimissed it would be prudent to admit your mistake on the rigor of mathematical induction
Unless it can be proven that chess can be so represented, and Zermelo's Theorem is not a proof but a claim, then there's no need to believe that it can.
Just to bring you bang up to date, that time you were wondering about when that could happen was about 100 years ago. I have already explained this and underlined it several times using several independent sources.
I didn't bother to contest the calim [sic] that the Theorem itself can be proven because that's not the issue.
Correct. It's not the issue because it WAS proven before my father was born.
It needs to be proven that it can be used to prove that chess can be represented mathematically.
Zermelo explicitly dealt with the example game of chess. He was more interested in this than in general combinatorial games (the term did not even exist at the time. Indeed game theory did not yet exist and Zermelo wrote the first paper on the subject. About chess as a mathematically represented game.
Anyway, it's getting late, I won the argument [snip]
Then I'm afraid Zermelo was wrong.
I was wondering if you would get a 3 digit score on an IQ test these days, based on your posts. I am not sure. I bet you don't want to find out!
A proof by MATHEMATICAL INDUCTION is a DEDUCTIVE proof.
You are not the only one who has been confused by the two incompatible uses of the word "induction". I have explained this point earlier in this discussion.
Here is the last paragraph of the introduction to the wiki article on the topic. Sorry if it is difficult to understand, but this has been known since the 3rd century BC. The point is highly relevant to this discussion.
<<Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy, in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by a finite chain of deductive reasoning involving the variable 𝑛, which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability.>>
You shouldn't use personal attacks, Elroch. And you are objecting to an unimportant aside. The fact remains that Zermelo has not proven that chess can be represented mathematically. The comparison is exactly as I stated. A claim that if a simple game can be so represented then chess can is exactly equivalent to a claim that if I can walk to the fruit shop on the corner then mankind can reach other galaxies.
Zermelo has made a claim, disguising it as a proof. It happens a lot doesn't it. I mean ... Cantor and his transfinite numbers?
I don't want to talk with you any more because try as you might, you can't keep a civil tongue in your head. You're a very bad loser and I often win our little differences although you have never admitted it once. And you call me a narcissist.
Your delusional worldview is showing through...
1. There were no personal attacks in that post, and certainly none that rise to your level of namecalling.
2. Zermelo's Theorem = proven
3. Cantor = well respected, not dubious as you would imply just becasue you skimmed something of his and failed to understand it.
I just took a quick refresher course on mathematical induction, since I learned it many decades ago. The course consisted of a really bad teacher, I think called Khan,
Do you mean Khan Academy, by chance?
explaining sums of consecutive numbers from 1 to n, as given by the formula n(n+1)/2. He managed to show how if it counts for a number k it counts for k+1, k+2, k+3 etc.
Firstly it's all just simple logic. Secondly, the average of a consecutive series of numbers to n is (n+1)/2 and so the sum is n multiplied by the average. OK so all very trivial.
The idea that you can use that kind of linearity to extend a mathematical depiction of a simple game like noughts and crosses into a proof that the same is available for chess is mistaken and Zermelo was wrong about it. Simple as that. Yes he's a famous mathematician. Yes, mathematicians always jealously protect their own. No, mathematical induction, which is a simple process of logic, does not and cannot be used to make a case that the impossible is possible. Noughts and crosses is not commensurable with chess.
I accept I'm making a claim. I already asked what my wife had to say about transfinite numbers. She's a psychologist. Mensa measured her IQ pretty high. 156 or 158. She's extremely bright. She thinks Cantor was a nutcase. Strangely enough, he was a nutcase. Then I asked my son when I was talking to him alone and he told me how important Cantor was for set theory. Nothing more or less. So he wasn't going to question him but perhaps nothing has caused him to question it so far. Maybe my question will have set him ticking. I would have liked to have been able to ask my father. Never mind. My wife's instinct was that Cantor was a nutcase. Using a bit more logic than that, I just thought he overstepped and he was describing an hypotheticality which he became caught up in and came to believe, which is what mentally ill people do. And my son supported Cantor because my son is a mathematician.
If I had a higher opinion of your ability to genuinely question, I would take you more seriously. I have never seen evidence that you are capable of it. Perhaps more tellingly, you never admit you lost an argument. That's the boy who cried Wolf! You aren't to be taken seriously in a situation where you are in danger of losing an argument.
You not only never admit you've lost an argument...you routinely claim to understand everything better than the most famous authorities on the subject. This should tell you and your crackpot fanbois something...
Einstein, Cantor, Zermelo, every authority ever in Thermodynamics...all hacks compared to you and your judgment rendered with 15 minutes of skimming over their ideas. Heck, you think the majority of philosophers (you own field of choice) are bunk and that you are inherently better.
I see your limitations then and must accept that it's beyond your control, because I know exactly what "informed" and "uninformed" mean to you. They have always stood as references as to whether an opinion is right or wrong, measured by whether it agrees with your own opinion. Nothing else.
That more properly applies to you...not an uncommon thing when you are arguing...well, anything.
I don't know why you choose to bring up the paranormal thing. It doesn't seem very relevant unless you think it will win you a couple of cheap votes.
We live in a World where many people disbelieve in the possibility of the paranormal or supernatural (they mean the same) and many people believe that it exists. I would think that the numbers believing it exists outweigh the numbers believing it doesn't and there are many undecided too. I'm an atheist for reasons that I would explain if it were allowed here but I do accept the reality of things like clairvoyance, some forms of telepathy, things that are variously called paranormal or miraculous etc. Again I could give a reasoned and detailed explanation of why but there's no need.
It's your attempt to win a point by means unrelated to this argument, since you probably suppose that only silly people believe that sort of stuff. You aren't doing very well but never mind, it's only to be expected.
Only silly people believe "that stuff", yes. And you believe in far more than clairvoyance...don't make me break out your crazy beliefs regarding your "abilities"...
Yes I suppose I'm a clever guy. Einstein was good and I only dislike him because of what he did to Mileva Maric and their son. There are lots of good and clever people but you managed to forget that I dislike Newton. Cantor and Zermelo aren't important. What they did was simple and if not they, then others would have easily achieved it, probably without all the deception. I think you're an extremely innocent person. There was a big amount of BS with many famous people. Ego and the lengths they would go to to perpetuate their fame .... you forget that I studied philosophy. It made me realise how tenuous is the fame of some past, famous people. You read one philosopher and he's saying the opposite of another. They're both mainly wrong. Maths is no different really. If it doesn't serve a purpose then it's useless but it can't be disproven. That's falsifiability. If something is not falsifiable, it isn't taken seriously. Cantor's transfinite series is useless. Zermelo made weak industive arguments amd presented them wrongly as mathematical, logical induction. You can't get to solving chess mathematically by logical induction and Elroch's problem is his ego. Not capable of self-questioning and neither are you. Like children.
Oh, I didn't forget about Newton, or Turing, etc. You routinely disparage anyone and everyone, alive or dead, in order to play king of the hill and protect your fragile psyche.
I'm waiting ...... or were you deliberately lying?
I'm cooking, you muppet. Try sitting on your hands, like when you have to pee. Your posturing is undignified.
A bit dishonest that I disparaged Turing but it does serve to show how completely dishonest you are. Let's do a test. How did I disparage Turing?
Lol. You already know that you did, ergo the "bit dishonest". You just said it in the past day or two, so I'm sure people will not have forgotten. You downplayed the impact of Turing's role at Bletchley. This might be possible with other various team endeavors in history, but not so much with Turing. There's a reason the machine is named after him.
I'm surprised given The Imitation Game that you did not come right out and say that the woman who helped him in the movie actually was key in the process, or behind the bulk of the work, as you implied with Einstein.
[Yes, I know that she was a fictionalized amalgam of all the women who worked on the project...let's see if Optimissed knows that.]
Anyway, I asked Elroch to prove that it's correct, according to a theorem of combinatorial games theory, that chess can be solved mathematically.
He just told us that it wasn't an axiom, by the way, MEGA. I thought we were making some progress. However, you were more correct than Elroch, seemingly, because if he can't prove that it can be solved mathematically (and that proof has to be a syllogistic proof, which is what he always demands of others) then it's a axiom.
Having an axiom that chess can be solved mathematically is exactly equivalent to an axiom that states that mankind will reach other galaxies in their space exploration. Now, I'm not saying it's impossible but I strongly doubt it. The axiom is based on "mankind can travel, therefore mankind can travel to other galaxies" and is exactly equivalent to "simple combinatorial games can be solved mathematically: therefore chess can." I would require a proof, please, or you've lost the argument.
You would lose the argument since you would not have responded in kind. You required deductive proofs from tygxc but you cannot give them to defend your own far less reasonable claims.
@Optimissed, this theorem even has its own wikipedia page.
If you want a formal proof, here is one (it's half a page long after the definitions have been made):
Zermelo's Theorem
The original paper was published in 1913 (in German), and apparently it was the first published paper on game theory. It needed an addition published in 1927 to be a truly rigorous proof for basic chess (where there are a finite number of possible positions, but games can be indefinitely long).
Interestingly, this is the same Zermelo who has his name attached to the Zermelo-Fraenkel axiomatisation of set theory that is perhaps the most popular foundation for mathematics.
[Remark: It strikes me that Zermelo's theorem can be easily generalised to a game where there is a general finite ordered set of outcomes].
Looked at it. It's an hypothesis. It may be considered by Zermelo to be an axiom and there's no syllogistic proof to support it.
Narcissism appears to continue to be a fatal obstruction to your understanding. It's a THEOREM that was PROVED in the early 20th century and which can be understood by an undergraduate. All you had to do to avoid that blunder was read what I said. Or the wikipedia article. Or the paper with the proof in it. Or any reputable source on the subject.
"Zermelo's work shows that in two-person zero-sum games with perfect information, if a player is in a winning position, then that player can always force a win no matter what strategy the other player may employ."
That is rather unfortunate, isn't it? Do you think it detracts from Zermelo's work? The commentator says that Zermelo shows that if a player is in a winning position then that player is in a winning position. There's no other interpretation.
No, it doesn't detract from the work: it is NOT the work. It is just worded in a redundant way. The key fact is the statement of the theorem itself in a general way for games that start at a general position:
For each legal chess position exactly one of the following is true:
We then define the three classes of position as winning positions for white, winning positions for black and drawing positions.
I hope my wording helps you.