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Results tagged with big-list
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user 1946
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
7
votes
Interesting applications of the pigeonhole principle
For most cardinals $\kappa \leq \lambda$, it must happen that the infinite symmetric group $S_\kappa$ satisfies exactly the same first order theory as $S_\lambda$. That is, the groups are elementarily …
9
votes
Interesting examples of generic behavior of mathematical objects being either unreasonably s...
Generic Turing machine programs are unreasonable: the computation head will fall off the beginning of the tape.
Basically, the situation is that on the usual one-way infinite tape model, a random pr …
4
votes
Properties of natural numbers such that there is a "very large largest" number with that pro...
The other examples are great! Meanwhile, there is a translation between this question and the eventual counterexamples question. Namely,
For any property $Q(m)$ with eventual counterexamples, the p …
8
votes
Examples of statements that provably can't be proved using a promising looking method
Gödel had conjectured that large cardinals might settle the continuum problem. He thought, for example, that perhaps the existence of a measurable cardinal would imply $\neg\text{CH}$. Such a perspect …
12
votes
Statements reliant on conjectures
Set theory is of course completely saturated with this feature, since the independence phenomenon means that a huge proportion of the most interesting natural set-theoretic questions turn out to be in …
13
votes
Arriving at the same result with the opposite hypotheses
Every proof by contradiction can be seen as following the template identified in the theorem.
Namely, when we've proved a statement $S$ by contradiction, then $S$ follows from $S$ and also from $\ne …
17
votes
Particular problem solved by solving a more general problem.
Cantor proved the existence of transcendental real numbers by proving that most numbers are transcendental. The set of algebraic real numbers is countable.
34
votes
A search for theorems which appear to have very few, if any hypotheses
Theorem. Every group has a terminating transfinite automorphism tower.
Start with any group $G$, compute $\text{Aut}(G)$ and $\text{Aut}(\text{Aut}(G))$ and so on, iterating transfinitely, mapping e …
30
votes
What is your favorite "strange" function?
The Ackermann function $A(n,m)$ is defined on the natural numbers by a very simple recursion, but the values grow enormously, almost beyond conception. This function completely transcends any simple-m …
32
votes
What are some examples of colorful language in serious mathematics papers?
In this MO answer, I mentioned Arnold Miller's lecture
notes, where he gives
an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the o …
31
votes
Interesting examples of vacuous / void entities
The usual axiomatizations of set theory (without
urelements) mean that every set in the entire set-theoretic
universe is ultimately built from copies of the emptyset,
in complex empty-box-in-a-box-in- …
19
votes
Interesting examples of vacuous / void entities
The Generalized Continuum
Hypothesis
is the assertion that $2^\kappa=\kappa^+$ for all infinite
cardinals $\kappa$, or in other words that the power set of
a set of size $\kappa$ has the next larger c …
104
votes
Theorems with unexpected conclusions
My favorite example of this phenomenon is Goodstein's Theorem.
Take any positive number $a_2$, such as the number $73$, and write it in complete base $2$, which means write it as a sum of powers of $2 …
30
votes
Nonequivalent definitions in Mathematics
A tree can be a very different thing in different parts of mathematics. It might be a certain kind of acyclic graph; or a partial order such that the predecessors of every node are linearly ordered; o …
6
votes
Suggestions for good notation
The three-dot notation $f\mathrel{\scriptsize\vdots}A\to B$ to indicate that $f$ is a partial function from $A$ to $B$, meaning that $\text{dom}(f)\subseteq A$ rather than $\text{dom}(f)=A$. Partial f …