No, these are not all equivalent. GCH implies AC, and T is equivalent to AC, but AC does not imply GCH. I believe this is provable in all standard formulations of set theory (i.e. just using naive notions).
Query: Is the relation < in T supposed to be the same as the relation < in WO, or is it just that if T holds, then there is also a WO, but perhaps with a different < ?
If T holds then for each pair of infinite cardinals (X, Y), XY. But this doesn't tell us anything about ordering sets that do not contain only infinite cardinals (which is what WO talks about). So, it must be that the < relation in T is a different thing than the < relation in WO.
Ok, confident now that they generally need to be a different relation. We could use the ordinary < relation on R to satisfy T, but if we expected to keep < constant, then by WO (0,1] would have a min, which is absurd. Thanks.
In fact, that has to be the thing about WO that keeps *it* from meaning what it seems to. We have sets, like R, that we understand only in terms of a useful ordering (the ordinary sense of R), but which are not WO. AC implies to us that these sets *are* susceptible to WO, but it's via a < so random in relation to the < by which we understand the given set as to be, in a sense, useless in handling it? It doesn't reduce sets to constructible sets, because the instructions for how to do so are not themselves readable constructively.
I posted the PSA because I often find that I am people's first intro to AC and GCH (either in class or casually), and I always forget to include this interesting detail in the discussion.
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If T holds then for each pair of infinite cardinals (X, Y), XY. But this doesn't tell us anything about ordering sets that do not contain only infinite cardinals (which is what WO talks about). So, it must be that the < relation in T is a different thing than the < relation in WO.
That's all I got ...
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In fact, that has to be the thing about WO that keeps *it* from meaning what it seems to. We have sets, like R, that we understand only in terms of a useful ordering (the ordinary sense of R), but which are not WO. AC implies to us that these sets *are* susceptible to WO, but it's via a < so random in relation to the < by which we understand the given set as to be, in a sense, useless in handling it? It doesn't reduce sets to constructible sets, because the instructions for how to do so are not themselves readable constructively.
What was the original motivation for your PSA?
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I posted the PSA because I often find that I am people's first intro to AC and GCH (either in class or casually), and I always forget to include this interesting detail in the discussion.
Reply
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