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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.
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
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 …
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
4
votes
Viewing parts of $\mathbb{V}$ 'from the top down' or 'from the bottom up'
This is a central idea in many large cardinal axioms, which postulate the existence of a nontrivial elementary embedding of the set-theoretic universe $V$ into a transitive class $M$. $$j:V\to M$$
Thi …
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
5
votes
Critical points in $ZF$ without Choice
In ZF, many of the usual arguments about critical points still go through.
For example, every critical point $\kappa$ of an elementary embedding $j:V\to M$ is regular, since if $\kappa$ is the suprem …
- 236k
10
votes
Accepted
Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hier...
This kind of analysis is very well understood in ultrapowers, and one often sees this kind of thinking with ultrapowers, where one performs calculations with the representing function for an object. W …
- 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
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
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 …
6
votes
Proof by `universal receiver'
It is very common in set theory to prove that a particular model or structure is well-founded by mapping it into a fixed well-founded structure. The point is that if $j:\langle M,{\in^M}\rangle\to \la …
9
votes
Accepted
Is there a well defined subset of the integers that cannot be defined as a property of a rec...
Over at my answer to I. J. Kennedy's question about degrees of irrationality, I described several hierarchies of definable complexity that transcend computability. I have copied my answer below. Alrea …
- 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 …
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
7
votes
What are interesting families of subsets of a given set?
Another ultrafilter cousin is the concept of a majority
space. This is a family $M$ of nonempty subsets of $X$,
called the majorities, such that any superset of a
majority is a majority, every subset …
- 236k