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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.
135
votes
43
answers
38k
views
What are the most attractive Turing undecidable problems in mathematics?
What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on …
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
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 …
- 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 …
- 237k
54
votes
1
answer
3k
views
In the two-person Killing the Hydra game, what is the winning strategy?
My question is which player has a winning strategy in the
two-player version of the Killing the Hydra game?
In their amazing paper,
Kirby, Laurie; Paris, Jeff, Accessible independence results for P …
- 237k
49
votes
0
answers
3k
views
Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositi...
This question follows up on a comment I made on Joseph O'Rourke's
recent question, one of several questions here on mathoverflow
concerning surprising geometric partitions of space using the axiom
of …
- 237k
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 …
- 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
41
votes
3
answers
2k
views
What is the minimal size of a partial order that is universal for all partial orders of size n?
A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds
order-preservingly into $\mathbb{B}$.
For example, every partial order
$\langle\ma …
- 237k
41
votes
2
answers
2k
views
On the difference between two concepts of even cardinalities: Is there a model of ZF set the...
An interesting question has arisen over at this
math.stackexchange
question
about two concepts of even in the context of infinite
cardinalities, which are equivalent under the axiom of
choice, but whi …
- 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
36
votes
8
answers
2k
views
Does the truth of any statement of real matrix algebra stabilize in sufficiently high dimens...
This question is related to this recent but currently
unanswered MO
question
of Ricky Demer, where it arose as a comment.
Consider the structure $R^n$ consisting of $n\times n$
matrices over the real …
- 237k
34
votes
5
answers
1k
views
Does the exact pair phenomenon for partial orders occur in your area of mathematics?
Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if
Both $b$ …
- 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
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