Comments on: Playing Games in the Transfinite: An Introduction to “Ordinal Chomp” http://xorshammer.com/2008/09/29/playing-games-in-the-transfinite-an-introduction-to-ordinal-chomp/ Some things in mathematical logic that I find interesting Mon, 06 Sep 2010 21:42:48 +0000 hourly 1 http://wordpress.com/ By: mkoconnor http://xorshammer.com/2008/09/29/playing-games-in-the-transfinite-an-introduction-to-ordinal-chomp/#comment-111 mkoconnor Thu, 02 Oct 2008 11:08:13 +0000 http://xorshammer.wordpress.com/?p=163#comment-111 Quite right. Thanks for the correction. Quite right. Thanks for the correction.

]]> By: Kenny Easwaran http://xorshammer.com/2008/09/29/playing-games-in-the-transfinite-an-introduction-to-ordinal-chomp/#comment-108 Kenny Easwaran Thu, 02 Oct 2008 04:23:12 +0000 http://xorshammer.wordpress.com/?p=163#comment-108 In your discussion of non-non-increasing sequences I think you mean "min" instead of "max". In your discussion of non-non-increasing sequences I think you mean “min” instead of “max”.

]]> By: mkoconnor http://xorshammer.com/2008/09/29/playing-games-in-the-transfinite-an-introduction-to-ordinal-chomp/#comment-107 mkoconnor Wed, 01 Oct 2008 11:54:13 +0000 http://xorshammer.wordpress.com/?p=163#comment-107 Actually, it's just that an $latex \alpha\times \beta$ board has a column for each ordinal $latex \alpha'$ where $latex 0\leq \alpha' < \alpha$ (and similarly for the rows with $latex \beta$). I believe that works for all cases, but I definitely should have made it more clear. Actually, it’s just that an \alpha\times \beta board has a column for each ordinal \alpha' where 0\leq \alpha' < \alpha (and similarly for the rows with \beta).

I believe that works for all cases, but I definitely should have made it more clear.

]]> By: GR http://xorshammer.com/2008/09/29/playing-games-in-the-transfinite-an-introduction-to-ordinal-chomp/#comment-106 GR Wed, 01 Oct 2008 11:42:26 +0000 http://xorshammer.wordpress.com/?p=163#comment-106 No doubt it is a silly point and perhaps I am just being thick here, but here goes anyway... Now, if we speak of an n x m board where n and m are finite, this uniquely picks out both 1) a board whose highest column/row are specified by the finite ordinals n and m, and 2) a board whose column/row count has cardinality the cardinality of the ordinals n and m. You get both, because these descriptions pick out the very same board. But this nomenclature breaks down when you turn to speak of an alpha x beta board where alpha and/or beta are limit ordinals. The board whose highest column has ordinal alpha is just one of the many board(s) whose column count has the cardinality of alpha. If when you speak of an alpha x beta board, you mean to pick out board(s) by cardinality, then you will have picked out not a single board but a rather large collection of boards, and in particular have not distinguished between the board with a column for every n < alpha and no more, and the board whose highest column has ordinal alpha. So, in that case, your nomenclature cannot distinguish between these two rather different sorts of board. Notice, it would then be a mistake to assert that there is no upper right square on an alpha x beta board, because one of the boards thus picked out does have a highest column/row, namely the one whose highest column has ordinal alpha, and whose highest row has ordinal beta. On the other hand, if your talk of an alpha x beta board is meant to speak of a board whose highest column/row have the ordinals alpha and beta respectively, then in this case it would again be a mistake to say that there would be no upper right square. It is the square at column alpha, row beta. Finally, maybe the nomenclature is supposed to work differently for limit and non-limit ordinals. One might say that an alpha x beta board means a board with a) columns for every n<=alpha for non-limit alpha, and b) columns for every n<alpha for limit alpha; & ditto for rows. But this would not be quite happy either, since we would be skipping a board at every limit ordinal, namely the one whose highest column has ordinal alpha for limit alpha (ditto for rows). Maybe I am just trying to think too early in the morning... No doubt it is a silly point and perhaps I am just being thick here, but here goes anyway… Now, if we speak of an n x m board where n and m are finite, this uniquely picks out both 1) a board whose highest column/row are specified by the finite ordinals n and m, and 2) a board whose column/row count has cardinality the cardinality of the ordinals n and m. You get both, because these descriptions pick out the very same board.

But this nomenclature breaks down when you turn to speak of an alpha x beta board where alpha and/or beta are limit ordinals. The board whose highest column has ordinal alpha is just one of the many board(s) whose column count has the cardinality of alpha.

If when you speak of an alpha x beta board, you mean to pick out board(s) by cardinality, then you will have picked out not a single board but a rather large collection of boards, and in particular have not distinguished between the board with a column for every n < alpha and no more, and the board whose highest column has ordinal alpha. So, in that case, your nomenclature cannot distinguish between these two rather different sorts of board.

Notice, it would then be a mistake to assert that there is no upper right square on an alpha x beta board, because one of the boards thus picked out does have a highest column/row, namely the one whose highest column has ordinal alpha, and whose highest row has ordinal beta.

On the other hand, if your talk of an alpha x beta board is meant to speak of a board whose highest column/row have the ordinals alpha and beta respectively, then in this case it would again be a mistake to say that there would be no upper right square. It is the square at column alpha, row beta.

Finally, maybe the nomenclature is supposed to work differently for limit and non-limit ordinals. One might say that an alpha x beta board means a board with a) columns for every n<=alpha for non-limit alpha, and b) columns for every n<alpha for limit alpha; & ditto for rows. But this would not be quite happy either, since we would be skipping a board at every limit ordinal, namely the one whose highest column has ordinal alpha for limit alpha (ditto for rows).

Maybe I am just trying to think too early in the morning…

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