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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.
18
votes
When is 2 qualitatively different from 3?
Every group in which every non-identity element has order 2 is abelian.
6
votes
Papers that debunk common myths in the history of mathematics
Theodor Nenu and I have a paper addressing the question of whether Alan Turing proved the undecidability of the halting problem in his seminal 1936 paper on computable numbers, in which he introduces …
12
votes
Uniqueness results that follow from CH
Under CH, we have saturated models of size continuum of any consistent first-order theory in a countable language, and for a complete theory these are unique by the back-and-forth method.
(In my paper …
21
votes
What are some nice uses of ultraproducts/ultrapowers?
Here are a few common uses that come to mind:
Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by thei …
7
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
Here is a further example, which I find to be one of the rather more philosophically profound results in computability theory. Namely, Rice's theorem.
The theorem states, at bottom, that no nontrivial …
- 236k
12
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
Here is another of my favorite uses of the Kleene recursion theorem.
It arises from Turing's remarkable 1936 paper, "On computable numbers...", in which he defines Turing machines, provides a universa …
- 236k
20
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
My favorite use of the Kleene recursion theorem is the universal algorithm.
In the baby form, consider the program $e$ that (on any input) undertakes the following process: it looks for a proof from P …
- 236k
23
votes
Ur-elemental surprises
In his dissertation work, Bokai Yao has investigated the nature of urelement set theory, particularly in the context of a proper class of urelements. See a preprint at:
Bokai Yao, Forcing with urelem …
- 236k
5
votes
Decision problems for which it is unknown whether they are decidable
It remains open whether the won-position problem of infinite chess is decidable, the problem of determining whether a given finite position in infinite chess is winning for white or not. See Richard S …
21
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …
26
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …
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 …
35
votes
What programming language should a professional mathematician know?
My answer is: TikZ
This is a programming language, often used in combination with
LaTeX, for producing high-quality graphics.
I view this language as important for mathematicians, not because
mathem …
48
votes
Contemporary philosophy of mathematics
Let me mention a few current issues on which I have been involved in the philosophy of
mathematics. Of course there are also many other issues on which people are working.
Debate on pluralism. First, …
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 …