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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 …
- 13.2k
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) …
- 2,467
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 …
- 13.4k
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 …
- 1,851
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 …
- 7,545
19
votes
2
answers
985
views
Which graphs are elementarily equivalent to their own disjoint sums?
In Stefan Geschke's recent
question,
one of the solutions observed that the graph consisting of
a single infinite beaded chain, a $\mathbb{Z}$-chain where
each integer is connected to its nearest neig …
- 4,713
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 …
- 4,713
8
votes
4
answers
938
views
Are there two computable binary trees such that each has a branch not computing any branch t...
It is a well-known elementary classical result in computability theory that there are computable infinite binary trees $T\subset 2^{<\omega}$ having no computable infinite branch. (One can build such …
CommunityBot
- 1
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 …
- 20.7k
76
votes
9
answers
6k
views
Can we unify addition and multiplication into one binary operation? To what extent can we fi...
The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary operati …
- 12.9k
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 …
- 35.5k
20
votes
7
answers
2k
views
Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?)
Let us say that a set B admits a rigid binary relation, if there is a binary relation R such that the structure (B,R) has no nontrivial automorphisms.
Under the Axiom of Choice, every set is well-or …
- 2,031
26
votes
3
answers
2k
views
Does ZF+AD settle the original Suslin hypothesis?
Everyone knows that the real line $\langle\mathbb{R},<\rangle$ is
the unique endless complete dense linear order with a countable
dense set. Suslin's
hypothesis is
the question whether we can replace …
- 237k
60
votes
8
answers
6k
views
Is the ultraproduct concept fundamentally category-theoretic?
Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.
My …
- 1,446
47
votes
4
answers
4k
views
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological sp …
- 11.4k