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Results tagged with lo.logic
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user 1946
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.
17
votes
2
answers
2k
views
Do the surreal numbers enjoy the transfer principle in ZFC?
The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ga …
- 237k
11
votes
2
answers
1k
views
An infinite hat puzzle variation—if we don't know our place, can we still be almost all corr...
An evil demon is holding uncountably many set theorists captive. He explains to us how he will presently arrange us into a well ordered sequence, with everyone facing the same direction upward in the …
- 237k
14
votes
2
answers
978
views
If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set …
- 237k
15
votes
2
answers
871
views
Which are the hereditarily computably enumerable sets?
My question is about sets that are computably enumerable with respect to their hereditary membership structure. Specifically, let me define that a hereditarily computably enumerable (h.c.e.) set is on …
- 237k
7
votes
1
answer
490
views
Normal form for terms in language with two ring structures
Suppose I have two different ring structures on the same domain $\langle R,+,\cdot,0,1\rangle$, $\langle R,\oplus,\otimes,\bar 0,\bar 1\rangle$ and I throw the structures together into a single common …
- 237k
34
votes
2
answers
2k
views
What is the logical status of the sentence combining the ideas of Löb and Rosser, "this sent...
Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic:
The Gödel sentence, "this sentence is not provable", which indeed is not provable in w …
- 237k
19
votes
2
answers
1k
views
Is the theory of a partial order bi-interpretable with the theory of a pre-order?
A partial order relation $\leq$ on a set $A$ is a binary relation that is reflexive, transitive, and antisymmetric.
A preorder relation $\unlhd$ (also sometimes known as a quasi order or pseudo order) …
- 237k
12
votes
3
answers
881
views
Is there a simple instance of intransitivity for implicit definability?
This question continues the theme of some recent questions on implicit definability.
A relation $R$ is implicitly definable in a first-order structure $M$ if there is a property $\varphi(\dot R)$, exp …
- 237k
44
votes
2
answers
4k
views
Is multiplication implicitly definable from successor?
A relation $R$ is implicitly definable in a structure $M$ if there is a formula $\varphi(\dot R)$ in the first-order language of $M$ expanded to include relation $R$, such that $M\models\varphi(\dot R …
- 237k
33
votes
2
answers
3k
views
Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
There are many interpretations of arithmetic in set theory. The
Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor:
$$0=\{\ \}$$
$$1=\{0\}$$ …
- 237k
39
votes
3
answers
3k
views
Can one show that the real field is not interpretable in the complex field without the axiom...
We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number p …
- 237k
11
votes
1
answer
703
views
Can we separate the almost-disjointness sunflower numbers?
This question concerns a new cardinal characteristic of the
continuum that arose out of issues in my answer to the question,
Sunflowers in maximal almost disjoint
families.
A family $\cal A$ of infini …
- 237k
12
votes
1
answer
771
views
Does every countable set of Turing degrees have an upper bound, without AC?
It is easy to see that every countable collection of sets $A_n\subseteq\mathbb{N}$ has an upper bound in the Turing degrees, since we can just take a copy of their disjoint sum $\oplus_n A_n=\{\langle …
- 237k
16
votes
2
answers
1k
views
Are the vertical sections of the Ackermann function primitive recursive?
The Ackermann function $A(m,n)$ is a binary function on the natural numbers defined by a certain double recursion, famous for exhibiting extremely fast-growing behavior.
One finds various slightly dif …
- 237k
80
votes
4
answers
9k
views
Who first characterized the real numbers as the unique complete ordered field?
Nearly every mathematician nowadays is familiar with the fact that
there is up to isomorphism only one complete ordered field, the
real numbers.
Theorem. Any two complete ordered fields are isomorphic …
- 237k