If a supercomputer with depth ℵ₀ exists, shouldn't a chess move just return 6 type of evaluation?

(Note: There may be some ambiguity from the title alone, due to the 100 character limit.)

Since chess is finite and deterministic, it is theoretically possible to objectively label any position as winning, losing, or drawn.

Therefore, the current system of "score" that engines use to analyze a position is actually nonsense from an objective point of view. With enough depth, the "score" would always be +ℵ₀ (or whatever the engine uses for forced checkmate for white), 0 (drawn), or -ℵ₀ (forced checkmate for black). Are these not the only possible evaluations of a position?

If so, it follows that moves can be characterized in exactly 6 ways. If W, L, D represent won, lost, and drawn positions, the six possibilities can simply be written as ordered pairs of game states before and after the move (W, W), (W, D), (W, L), (D, D), (D, L), (L, L).

For example, a move that makes a won position into a drawn won could be (W, D).

Our intuition of a range of severity for mistakes is something that a theoretical supercomputer would not share. Is it then meaningless to speak of how bad a mistake is? It seems to me that the only relevant point is whether it changes if the game is objectively won/lost/drawn.

Fortunately for us humans, any advantage can be squandered and any deficit overcome.

Therefore, it is useful to receive such calculations after the game so that we know where to point when asking others to admire our brilliance.

Oops, sorry. For this case, I talked about a supercomputer playing another supercomputer, not about a normal chess game played by humans.

Well then how come Alpha Zero and Leela Chess zero are coming up with novelties and ingenuities sacrifices, even in the opening that computers like Deep Blue didn't know? What if the computer is wrong and it seems something but it is actually wrong? There is examples of this all the time where the engine bar goes haywire. May not be as technical as what you are saying but it's possible that these finite possibilities have not been fully calculated.

If chess was fully solved, we could predict the exact outcome (Win, Draw or Loss) of perfect play in any position. With perfect play, there could not be any transition: a move would be (W, W), (D, D) or (L, L), using your system; only a mistake could worsen the situation. Since chess is not fully solved, the outcome of a position can be only estimated. That's what we and engine developers and neural networks do, using our experience. The estimations of computers are very good on average, otherwise they would not be so strong.

You're describing tablebases, which already exist for positions with up to 7 pieces. They only have 3 types of evaluations (W, D, L), plus the exact number of moves needed for any "win".

Let me reiterate the title. "A supercomputer with depth ℵ₀"

Makes sense. But, why can't perfect play worsen the situation? It could, just be, that Black is always in zugzwang by perfect play.

Yes, chess is technically "finite", but there are around 10^120 games of chess. Good luck having a computer perfectly analyze the position such that it always evaluates the position as a draw (or a forced mate).

Such a supercomputer will not exist for the forseeable future. There are just far too many possibilities (although as @Rocky64 mentioned, tablebases only evaluate the position as a win, draw, or a loss). But tablebases are only for the endgame (with 7 or less pieces). When you put in all 32 pieces, there are just too many possible variations, so much so that a computer will not be able to evaluate the position perfectly unless one is willing to wait for over a googol years.

Even if such a supercomputer existed and chess is "solved", there will still be no way to memorize all the possible branches of moves, even for a computer.

But yes, you are correct in saying that such a computer should only display a drawn position or a forced mate

Aaaand there's your problem. Currently, it's thought that it may be impossible for us to ever solve chess. Personally I'm not convinced of that, betting that at some point (if civilization even survives) we'll come up with much better computing methods. But the point is, with any currently conceivable computer or algorithm, it seems hopeless. So, no, current engines are not "nonsense." They're the best we have. Your point is what is called "academic."

I dont know what that means, i smash things with hammers and make fires with torches.  please explain what that symbol means? is some type of statistical symbol?  your curt reply doesnt amuse me.  you think a computer can solve this game? it cant maybe you should focus on each and every move you make.  

ℵ₀ is another name for infinity

You need no depth ℵ₀: thanks to the 50 moves rule every chess game ends in 5898.5 moves at most.

Indeed there are only 3 game states: D, W, L.
Hence there are indeed 6 classes of moves: (W, W), (W, D), (W, L), (D, D), (D, L), (L, L).

Approximately:
-0.5 to +0.5 = D
> +0.5 = W
< -0.5 = L

The game starts with D.
After some good moves (D, D)
some side makes a mistake (?): (D, L)
Now the winning side has 3 possibilities:
He plays a good move (W, W), the opponent can only respond with (L, L)
He plays a mistake (?) = missed win (W, D)
He plays a blunder (??) (W, L)

#12
"it's thought that it may be impossible for us to ever solve chess"
GM Sveshnikov predicted chess solvable in 5 years.
With quantum computers and 8 men table bases it may be even faster.

#10
"there are around 10^120 games of chess"
Yes, but that does not matter.
There are about 10^38 legal and sensible chess positions.
Of these about the square root i.e. 10^19 are relevant to solving chess.
For comparison: there are 10^68 ways to shuffle a 52 card deck.

Well, true enough... now, assume there are no 50 move rule. Does every game ends in a draw?

technically, it's only the smallest infinity

"Only the smallest". True, but you're underestimating infinity.

#15
With the 75 move rule every game ends in 8848.5 moves at most.
With neither 50 nor 75 moves rule every game ends in a finite but huge number of moves thanks to the 3 fold repetition rule.
With neither 50, nor 75 moves rule, nor 3 fold repetition rule a chess game could go on infinitely and in that case its outcome is undecided.

If a chess game did, go on infinitely, and assume both sides maintain perfect play, can, and will, the outcome be decided?

#19
No, only finite games are decided.

Why so? Can't the outcome be decided by the quality of the moves played?