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2d
comment Extension of a valuation on a function field
Doesn’t this follow from the easy standard fact that valuation rings are integrally closed?
2d
comment Primary structures in $\mathbb Q$
Q3 and Q4 are easy: Q3 follows from the fact that every positive rational is a ratio of products of primes, and Q4 from the fact that by D2, $S^*$ includes $\mathbb N$. Which actually implies a generalization of Q3: any primary structure is maximal (it is not properly included in another udp).
2d
comment Primary structures in $\mathbb Q$
OK, now it makes sense.
2d
comment Direct axiomatization of ordinal and cardinal numbers
OK, now I’ve seen the paper. Indeed, they actually show that ZFC is bi-interpretable with SO, as I would expect.
Feb
9
comment Direct axiomatization of ordinal and cardinal numbers
I wonder what you need L for. Sets of ordinals encode up to isomorphism arbitrary relations on (well-orderable) sets, and every set can be recovered from the $\in$ relation on its transitive closure (this is a standard argument), so the theory should be able to completely describe the ZFC universe. What am I missing?
Feb
5
comment Inconsistent theory with long contradiction
Hmm. $f_{9}^{\mathrm{Ack}}(f_{8}^{\mathrm{Ack}}(f_{8}^{\mathrm{Ack}}(254)))$ is so insanely huge number it squarely defeats human imagination, nevertheless it has a short and simple description, and it’s quite possible it could usefully appear in a short proof.
Feb
4
revised Examples of NIP fields of characteristic $p$
another typo
Feb
4
revised Examples of NIP fields of characteristic $p$
fix the definition
Feb
3
awarded  Yearling
Feb
3
comment Examples of NIP fields of characteristic $p$
Meanwhile, the definition is here: en.wikipedia.org/wiki/NIP_%28model_theory%29 .
Feb
1
comment A derivation in Tait calculus
Hmm. What I said applies when the $\forall$-rule is stated as $\Gamma,A(u)\mathrel/\Gamma,\forall x\,A(x)$, such as in Rathjen’s slides. I don’t think the weakening rule is admissible in full generality in the calculus where the eigenvariable $u$ is required to be $x$, as you stated it in the question.
Feb
1
comment A derivation in Tait calculus
@GabrielNivasch: The argument indeed needs to deal with variables in a more subtle way, but this is not too difficult. Basically, you first rename eigenvariables in the proof so that they do not conflict with $\Delta$, and then add $\Delta$ to each sequent.
Jan
31
comment A derivation in Tait calculus
No, you can't, in this calculus. It's a trivial induction on the length of derivation: all premises of all rules contain at least one formula, hence the only sequents you can derive from the empty sequent is itself, and sequents already derivable without the empty sequent, i.e., tautological.
Jan
31
comment A derivation in Tait calculus
It's not possible if $\Gamma$ is empty.
Jan
28
awarded  Nice Answer
Jan
28
comment Taller models of ZFC
In 1: “..without adding new axioms” – adding where? Can you clarify what does the sentence mean?
Jan
28
revised Are all the theorems true?
why doesn't this work?
Jan
28
revised Are all the theorems true?
fix MO 1 -> 2 formatting problem
Jan
28
comment Extracting a full rank matrix from a 0-1 matrix
Cross-posted from cstheory.stackexchange.com/questions/33672/…
Jan
27
comment How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?
... as the objects never live in the same model. With recursive ordinals as a common representation that is reasonably absolute, we can ask about the same ordinal in two different theories.