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Results tagged with big-picture
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user 1946
Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
136
votes
Accepted
Why do we have two theorems when one implies the other?
Some mathematicians seem to agree with you, and strive only to state and prove the most general versions of their theorems. I've had co-authors express that view. And I've sometimes had referee report …
95
votes
Examples of eventual counterexamples
The essence of the phenomenon of eventual counterexamples is that a certain pattern that holds among small numbers, turns out not to be universal. In the very best examples, such as the examples provi …
74
votes
Accepted
What's wrong with the surreals?
At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying ma …
- 236k
37
votes
Accepted
Is it necessary that model of theory is a set?
You seem to believe that it is somehow contradictory to have a set model of ZFC inside another model of ZFC. But this belief is mistaken.
As Gerald Edgar correctly points out, the Completeness Theor …
- 236k
33
votes
The concept of duality
There are various dualities arising in elementary logic:
the duality between $\forall$ and $\exists$, as
expressed by the validity $$\neg\forall x\ \neg\varphi(x)\iff
\exists x\ \varphi(x).$$
the d …
33
votes
Analogies between analogies
Let me give a very concrete analogy between analogies.
1:2 :: 2:4
is to
p:q :: kp:kq
as
x2+2x+1 : 0 :: x : -1
is to
ax2+bx+c : 0 :: x : (-b +- sqrt(b2-4ac))/2a.
And this analogy is called …
- 236k
32
votes
Equality vs. isomorphism vs. specific isomorphism
Automorphism groups are studied intensively in mathematics, and these groups explicitly track the difference between isomorphisms and equality. We aren't willing to say that every automorphism of a ma …
- 236k
24
votes
Logic in mathematics and philosophy
There is a general pattern of inquiry in mathematics and the sciences by which an investigation begins in philosophy, using philosophical ideas that may be initially quite vague, but which become incr …
- 236k
22
votes
When must it be sets rather than proper classes, or vice-versa, outside of foundational m...
Your question does not seemed aimed at set theorists, but let me
give a set theorist's answer.
I view the set/class distinction as analogous to and ultimately no
more problematic really than the othe …
20
votes
Axiom of Choice versus V=L in opposition to large cardinals
Consider the relativized constructibility hypothesis, which asserts that $V=L[A]$ for some set $A$.
This axiom is compatible with any locally verifiable large cardinal property, properties that can …
- 236k
19
votes
Accepted
What governs our "perception?" about the platonic realm of sets?
The recent developments on the consistency of NF bring welcome closure to the longstanding open question about whether NF was consistent. And this is naturally a very important matter for those who fi …
- 236k
16
votes
"$\kappa$ strongly inaccessible" = "every function $f:V_\kappa\to V_\kappa$ can be self-appl...
Unfortunately, your characterizations of the strongly inaccessible cardinals are not quite correct. The correct definition is that $\kappa$ is strongly inaccessible (also known as just plain inaccessi …
- 236k
14
votes
Is there a physically realizable inductive turing machine that can solve Hilbert's $10$th pr...
Let me try to answer the actual question that was asked. The Wikipedia
page
defines inductive Turing machines as follows:
An inductive Turing machine is a definite list of well-defined
instructi …
- 236k
11
votes
How are Modal Logic and Graph Theory related?
The ability of modal assertions to define natural and interesting
classes of frames (or digraphs) is indeed intensely studied and
constitutes one of the principal perpsectives of the subject,
pervasiv …
- 236k
11
votes
The unprecedented success of the “intersection” operator
There is a general sense in which any property that is closed under arbirtrary intersection is exactly a closure property.
To explain what I mean, suppose that $X$ has property $P$ and that the colle …
- 236k