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bio website math.lsa.umich.edu/~ablass
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visits member for 5 years, 3 months
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1d
awarded  Necromancer
2d
comment When did people know that all real polynomials of degree greater than 2 are reducible?
If someone knew long ago that all polynomials over the reals can be factored into polynomials of degree $\leq2$, then all that would be needed to get the fundamental theorem of algebra would be that real quadratics have complex roots. That would be known pretty much immediately after complex numbers are invented.
2d
comment Terminology in combinatorics
The first property says that non-adjacency is a transitive relation (if we count each vertex as non-adjacent to itself). So if you need to invent a name, I'd suggest "co-transitive".
Jul
31
comment History of unstable formulas
The $n!$ count omits those $n$-types that say that two or more of the $n$ elements are equal. (Of course that doesn't affect the point you were making.)
Jul
30
comment Why only Normed Linear Spaces?
Norms on abelian groups have been used to characterize the free ones: "A Characterization of Free Abelian Groups" by Juris Steprāns, Proceedings of the American Mathematical Society Vol. 93, No. 2 (Feb., 1985), pp. 347-349
Jul
30
comment presentations of subalgebras
Just to make sure I'm understanding the question correctly: The presentation that you seek for the subalgebra $B$ need not use the known $y_1,\dots,y_m$ as its generators but rather can use completely different elements to generate $B$, right?
Jul
29
comment Recursion, Common Term, Combinatorics
It seems that the $n$-th term of your sequence is the largest $k$ such that $2^{k-1}$ divides $n$.
Jul
29
comment The status of 'the consistency of NF relative to ZF'
@AndresCaicedo Do you know whether the "everybody" in your comment includes Bob Solovay? The reasons I ask are that (1) I understand that he paid attention to some of Randall Holmes's earlier work on NF, so he would presumably be interested in the consistency result, and (2) his reputation for careful work leads me to think that, if he were to say that he's studied the proof and thinks it's correct, that would be a huge step toward general acceptance of the proof.
Jul
24
comment A question about sentences undecidable in Peano's Arithmetic
@DavidRoberts The facts that Con(SOA) is true, that all axioms of SOA are true, that logical inference preserves truth, and that therefore Con(SOA) cannot be refuted in SOA are all provable in the usual foundational system for mathematics, ZFC. (They're also provable in far weaker systems,but I don't think that's needed to answer "How does one get this?")
Jul
23
awarded  Nice Answer
Jul
23
answered Are there discontinuities in the large cardinal hierarchy?
Jul
23
comment A question about sentences undecidable in Peano's Arithmetic
Note that Gödel's paper on the incompleteness theorems is, as indicated in its title, about Principia Mathematica and related systems. Principia Mathematica is considerably stronger than SOA.
Jul
23
answered A question about sentences undecidable in Peano's Arithmetic
Jul
22
comment Are all polynomial inequalities deducible from the trivial inequality?
The algebraically prime model of the theory of real-closed fields is not the field of rationals (which isn't real-closed) but the intersection of the reals and the algebraic closure of the rationals.
Jul
20
comment What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?
@JoelDavidHamkins is right. You can't reasonably expect to get models of $T$ that are smaller than $|T|$.
Jul
18
comment A question on two modal formulas
One of your comments is sufficiently readable in my browser to address my concern; the $\lozenge$ at the beginning of your first comment was intended to be $\Box$, and then the semantics makes much better sense. The only remaining problem is that the formulas in the question are far too complicated for me to assimilate and say anything about. I hope some experts in modal logic will be able to grasp them and provide useful information.
Jul
18
comment A question on two modal formulas
I"m somewhat surprised by your comment saying that you use $\lozenge$ for what appears to be a necessity operator ("all worlds $w'$ which are better"), so that, for example, $\lozenge(p\land q)$ is equivalent to $(\lozenge p)\land(\lozenge q)$. Do you use the usual definition of $\Box$ as $\neg\lozenge\neg$, so that it would mean "holds at at least one world which is better"? I would expect hat this inversion of the usual roles of $\lozenge$ and $\Box$ would prevent any close connection between your modal logic and the traditional ones.
Jul
17
awarded  Good Answer
Jul
14
revised Propositional logic without negation
fixed a typo
Jul
14
comment Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?
@AsafKaragila Fortunately, it wasn't quite that bad. The combinatorial square came with a cardinal as a subscript, and The two squares were never in the same formula. (The paper is "Infinitary combinatorics and modal logic" JSL 55 (1990) 761-778.)