Edit: We can even deduce from the rule itself that the series has to be finite, because an infinite series of moves can't end in checkmate, because checkmate terminates the series.

Prelim note: I don't think that is a valid argument because between the finite and the infinite is the undecidable - not being able to determine whether or not the answer is finite or infinite by proper algorithm. There is no number of moves N such that you can be sure that - when no checkmate is found - you will not find a checkmate at a move number >N. So you must go on looking forever. Unless of course you introduce one of a million meta-observations to reflect your (non-axiomatic) understanding of the game.

I don't see how that would make the argument invalid. If a series of moves ends in checkmate, then it is always a finite series of moves. So you don't have to know if a series is finite or infinite. Only the series which turn out to be finite can potentially end in checkmate.

Edit: We can even deduce from the rule itself that the series has to be finite, because an infinite series of moves can't end in checkmate, because checkmate terminates the series.

Prelim note: I don't think that is a valid argument because between the finite and the infinite is the undecidable - not being able to determine whether or not the answer is finite or infinite by proper algorithm. There is no number of moves N such that you can be sure that - when no checkmate is found - you will not find a checkmate at a move number >N. So you must go on looking forever. Unless of course you introduce one of a million meta-observations to reflect your (non-axiomatic) understanding of the game.

Since it is relevant, I chose to interpret your question freely as raising the issue of the finity of the chess game.

I started a reply and it grew and it grew and it will end up quoting system theory A-Z. So I aborted and decided on a short synopsis: This is the decidability question which depends on the finiteness of positions, not of move series. They are indeed finite -

by our understandingof the game semantics - but not necessarily for different chess rules or different stipulations as in the examples of my forelast post. If you succeed in enteringour understandingof the finiteness of positions in the definition of the axioms (plus language, alphabet ...), then you can indeed theoretically determine the scoring status of every position. If you can not or do not, you can enter it in the form of a repetition axiom - which has already been done in standard chess. Algorithmically, the use of a repeat axiom is highly preferred, since it is funny to base the diagnosis that a position will end in a draw, on the complementary elimination of all positions which might end in checkmate.