bio | website | people.su.se/~cesp |
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location | Stockholm, Sweden | |
age | ||
visits | member for | 3 years, 10 months |
seen | Dec 19 at 10:19 | |
stats | profile views | 1,248 |
I'm a graduate student deeply interested in foundations (logic, set theory, model theory, metamathematics) and category theory.
Dec 7 |
comment |
What are the current views on consistency of Reinhardt cardinals without AC?
...But it is conceivable that a large cardinal hierarchy could be mirrored in an intuitionistic setting where more notions of large cardinals can be defined without inconsistency (they will have to be non-classical though). |
Dec 7 |
comment |
What are the current views on consistency of Reinhardt cardinals without AC?
This is just speculative, but even if some large cardinals are found to be inconsistent with ZF, there could be weaker set theories where appropriate versions are consistent. I'm thinking more precisely of Aczel's constructive set theory CZF, based on intuitionistic logic, over which appropriate notions of regular, inaccessible and even Mahlo cardinals are defined (see "Inaccessibility in constructive set theories and type theories" by Rathjen-Griffor-Palmgren). In this case, the addition of AC turns the theory in just ZFC plus the usual notions...(cont.) |
Dec 6 |
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$\zeta(0)$ and the cotangent function
Given the fact that what you end up explaining why the series, considered as a Laurent series, predicts also the values of $\zeta$ at the negative even numbers, I think this proof is as simple as it reasonably can be expected to be. This because you are forced to consider the analytic continuation of $\zeta$ to the left half plane, not just to the critical strip, so it's reasonable that the functional equation is invoked. |
Nov 5 |
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Mysterious quotes (at least for me)
There is a very particular way in which those quotes make sense to me, but I'm afraid is not a mathematical way so I can't really answer :) |
Oct 16 |
revised |
Is it possible to formulate the axiom of choice as the existence of a survival strategy?
added 1981 characters in body |
Oct 15 |
answered | Is it possible to formulate the axiom of choice as the existence of a survival strategy? |
Sep 25 |
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An interpretation of not-Con(PA)
Careful! All that the the 2nd incompleteness theorem tells you is that $Con(PA)$ is unprovable, not that it is independent, and this provided you already accept the consistency of PA. If you apply that theorem to your theory $PA^+$ instead, you can only conclude that $Con(PA^+)$ is unprovable in $PA^+$, because in fact $\neg Con(PA^+)$ is actually provable there. |
Sep 24 |
awarded | Autobiographer |
Sep 22 |
awarded | Curious |
Sep 21 |
comment |
On a weak tree property for inaccessible cardinals
Excellent, thanks! That was exactly what I was looking for. |
Sep 21 |
accepted | On a weak tree property for inaccessible cardinals |
Sep 21 |
asked | On a weak tree property for inaccessible cardinals |
Aug 29 |
answered | construction of nonmeasurable sets |
Aug 17 |
comment |
How short can we state the Axiom of Choice?
It seems to me that François' solution is near optimal, considering that all usual equivalents of AC (which are gather in Rubin & Rubin's book rather than in Jech's), use notions which, when unwound in the language, are considerably larger. |
Aug 13 |
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Relationship between fragments of the axiom of choice and the dependent choice principles
Oh, good, then I won't have to go to the construction myself :) I just had read that it was Solovay who proved result 3 in this model. So now the real challenge is result 4, since the book only mentions permutation models for that. |
Aug 13 |
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Relationship between fragments of the axiom of choice and the dependent choice principles
Sure, the book refers to page 166 of Felgner, U. "Models of ZF set theory", Springer-Verlag, Berlin. I run into this model from the "Consequences..." web page, where I just searched for something validating form 40 ($\forall \kappa AC_{\kappa}$) but not form 44 ($DC_{\omega_1}$) and that was not a permutation model. |
Aug 13 |
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Relationship between fragments of the axiom of choice and the dependent choice principles
Oh, I'm referring to the model $\mathcal{M}13$, called in the book "Feferman/Solovay", it's just one model which extends $\mathcal{M}2$, called "Feferman" model there. Are we using different terminology or you meant to say something else? |
Aug 13 |
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Relationship between fragments of the axiom of choice and the dependent choice principles
I've just checked Howard-Rubin and it seems that at least for result 3 it can be done as Asaf says; Feferman/Solovay model does the trick. You have to add $\omega_1$ generic reals to the base model without collecting them in a set. For result 4, however, it seems the only available model is Jech's permutation model... |
Aug 13 |
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Relationship between fragments of the axiom of choice and the dependent choice principles
The references seem to be interchanged: 3 corresponds to Theorem 8.9 and 4 to Theorem 8.12 |
Aug 11 |
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Can one live without actual infinity?
I think the distinction lies on the cognitive aspect rather than on the mathematical one. I understand the quote as expressing that mathematical concepts are necessarily finitistic and that looking for examples in the physical world of an actual infinity is doomed to fail; saying space or time are infinite is not the same as saying they are unlimited. The second is a finitistic notion, while the first simply does not make sense. |