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16h
revised What do named “tricks” share?
Improved exposition
17h
comment What do named “tricks” share?
In particular, I don't agree with the "used only once" joke idea for distinguishing tricks and methods. We have already many counterexamples to that view mentioned on this thread of posts. Some of these tricks are robust mathematical methods; but they are still tricks, because they involve artifice.
18h
revised What do named “tricks” share?
added 101 characters in body
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revised What do named “tricks” share?
added 200 characters in body
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answered What do named “tricks” share?
2d
comment Direct axiomatization of ordinal and cardinal numbers
Yes, sorry to have mislead. I was thinking of the presentation that Peter had made on this paper at our seminar, in which he had emphasized $L$, but for a different reason.
2d
comment Direct axiomatization of ordinal and cardinal numbers
Yes, @EmilJeřábek, that is right. From any model of their theory SO you get a model of ZFC, and every model of ZFC arises.
Feb
9
comment About the limitation by size
You seem to think that you have to go to Kelley-Morse set theory to refer to the class of all groups, but of course one can refer to that class even in ZFC, since it is after all a definable class. In particular, one can handle it satisfactorily even in ZFC, or in Goedel-Bernays set theory, which handles proper classes as objects, without going all the way to Kelley-Morse set theory.
Feb
7
comment Direct axiomatization of ordinal and cardinal numbers
Mirco, it's always nice to chat with you. The reason I view this work as one-step removed from your question is that it isn't just a theory of the ordinals, but a theory of sets of ordinals, which makes it already a little set-theoretic. But only one-step removed, and not two, since the theory does not have sets-of-sets of ordinals.
Feb
7
answered Direct axiomatization of ordinal and cardinal numbers
Feb
7
comment Least ordinal not embedded in a total order
The union of a chain of elementary extensions is an elementary extension, and so all the models have the same theory.
Feb
7
comment Least ordinal not embedded in a total order
Yes, those are different, since in the case of my $\kappa$-like model, the supremum will be $\kappa$. But the construction shows that this, too, can be singular.
Feb
7
revised Least ordinal not embedded in a total order
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Feb
7
revised Least ordinal not embedded in a total order
Added "not" in title
Feb
7
answered Least ordinal not embedded in a total order
Feb
4
comment Order-Perserving Bijection $f:A\to A^*$?
Oh, I'm sorry. Let me suggest that you simply update your math.stackexchange question to point to our discussion here, and ask your further questions about it over there. I'm sure that people will be able to explain those matters at the right level.
Feb
4
comment Order-Perserving Bijection $f:A\to A^*$?
Well, the answer to your question is no, there isn't necessarily such an order-preserving bijection, because I gave a counterexample in my answer. Meanwhile, in the comments, I described how you can have some affirmative instances, such as the case $A=\omega$, where there is such an order-preserving bijection.