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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.
3
votes
When is $\mathbb{L}$-rank definable in inner models of $\mathbb{V} = \mathbb{L}$?
[Edit: The question is more subtle than I originally understood. I am leaving this here so as to avoid it being repeated by others.]
You can define $<'$ internally only if $M$ is a model of $V=L$, th …
- 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
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
6
votes
The disjunction property in Peano Arithmetic?
One can in fact prove a bit more than Joel did. For example:
Let $X$ be any r.e. set such that for any $\phi\in X$, PA$+\phi$ is consistent. Let $\psi$ be any true $\Pi^0_1$ sentence. Then there …
- 32.5k
7
votes
Accepted
Effect of large cardinals on the value of $\omega_1^L$ in $L$
The answer is no. If there is a transitive set model $M$ of set theory (and this is all you need), then if $\alpha$ is its height (that is, if $\alpha=\mathsf{ORD}^M$), then $L_\alpha$ is a model of $ …
- 32.5k
1
vote
A question about ordinal definable real numbers
(I am replacing prior nonsense with a completely different suggestion. I am also turning this into CW so details can be added by somebody with time (which, sadly, most likely won't be me). Comments pr …
8
votes
A question about Transfinite Induction
@Dong:
As I pointed out in comments, and Pete mentioned in his answer, well-orderability is all you need, so yes, if $P$ is a property of cardinals as you describe, then it holds of all cardinals. Y …
- 32.5k
14
votes
Accepted
Use of indiscernibles in model theory
Eran,
As far as I know, indiscernibility is used in two ways in model theory. One, as you say, is to obtain many non-isomorphic models. This is for sure the classical use of indiscernibility.
Anothe …
- 32.5k
13
votes
Accepted
Is it consistent to have a function that is sensitive to subset relation from the power set ...
No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of …
- 32.5k
17
votes
Accepted
Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$
Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of mo …
- 32.5k
6
votes
Accepted
Ultrapowers, and Models of Set theory.
Michael,
the existence of a well-founded $M$ and an embedding $j:V\to M$ different from the identity
is equivalent to
measurability,
which is equivalent to
the existence of an $\omega_1 …
- 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
7
votes
Condensation for L[U]
Eran, there is some ambiguity in your question, as there are several possible interpretations of what is meant by "the $L[U]$ hierarchy". The fine details of the argument depend on the version you cho …
- 32.5k
3
votes
Basic results with three or more hypotheses
A nice example from recent work in set theory.
Theorem (Viale). Assume Martin's maximum, and that every limit cardinal is a strong limit. Suppose that $N$ is an inner model, that $N$ has the same …
10
votes
Cohen reals and strong measure zero sets
Andy:
A good reference for your questions is "Consequences of adding Cohen reals" by J. Steprans, in "Set Theory of the reals", Judah ed., Bar-Ilan University, 1993, pp. 583-617. Another reference i …
- 32.5k