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Results tagged with lo.logic
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user 6085
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
54
votes
Arguments against large cardinals
I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied.
[Reinhardt suggested an ultimate axiom of the …
- 32.5k
46
votes
Solutions to the Continuum Hypothesis
(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices pa …
- 32.5k
42
votes
Accepted
A set that can be covered by arbitrarily small intervals
The sets you are calling small are commonly referred to in the literature as "strong measure zero sets." The Borel conjecture is the assertion that any strong measure zero set is countable.
This is …
- 32.5k
34
votes
Accepted
Latest status of core model theory?
${}$Hi Ioanna,
I.
The answer probably depends on how we define core model. At the level of "there are no Woodin cardinals in any inner model", we can finally show that core models exist, provably in $ …
- 32.5k
32
votes
Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?
Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begi …
- 32.5k
32
votes
Is all ordinary mathematics contained in high school mathematics?
I believe this example may qualify. It is open whether Thompson's group $F$ is amenable.
We may present $F$ as $\langle A,B\mid [AB^{-1},A^{-1}BA]=[AB^{-1},A^{-2}BA^2]=\operatorname{id}\rangle$.
Amena …
- 32.5k
30
votes
Counterintuitive consequences of the Axiom of Determinacy?
Let's see: $\mathsf{AD}$ implies that all sets of reals are Lebesgue measurable, have the Baire property, and the perfect set property (so, a version of the continuum hypothesis holds). It is conjectu …
- 32.5k
29
votes
Accepted
Continuum Hypothesis
A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematic …
- 32.5k
25
votes
Accepted
Is the statement that every field has an algebraic closure known to be equivalent to the ult...
Qiaochu, using the link I provided in my answer to this question, you find that this question is still open (or was, as of the mid 2000s, and I haven't heard of any recent results in this direction). …
- 32.5k
24
votes
Inconsistent theory with long contradiction
This is quite possible, that a theory $T$ is inconsistent but any deduction takes so long that we do not know.
Hugh Woodin has a short, nice paper, that I recommend you take a look at, where he addr …
- 32.5k
23
votes
Can we have A={A} ?
1) No such sets exist under the usual axiomatization of set theory (ZF).
The axiom of foundation states that $\in$ is well-founded. The axiom states that if a set $A$ is non-empty, then it has at le …
- 32.5k
23
votes
Who introduced direct limits?
As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as th …
- 32.5k
22
votes
Accepted
Hahn-Banach without Choice
The ultrafilter theorem is the statement that any filter on a set can be extended to an ultrafilter. It is perhaps more common to see it sated as the (Boolean) Prime ideal theorem: Every Boolean algeb …
- 32.5k
22
votes
Accepted
(Non?)-linearity of the consistency strength ordering in ZF
Marios, this is indeed a fascinating topic.
The consistency strength hierarchy is not linearly ordered. One can produce counterexamples by variants of Gödel sentences or of Rosser sentences. It is a …
- 32.5k
22
votes
Accepted
A limit to Shoenfield Absoluteness
Noah: The sentence "there is a real not in $L$" is $\Sigma^1_3$: To say that $x\notin L$ means that for every $y$, if $y$ codes a model of the form $L_\alpha$, then $x$ is not in this model; but to sa …
- 32.5k