Answer by Joel Adler for Can infinity shorten proofs a lot?
What about the puzzle named escape on Joel David Hamkins' homepage?http://jdh.hamkins.org/escape/That not all the squares occupied by the three stones in the initial configuration can be vacuated is...
View ArticleAnswer by Piotr Hajlasz for Can infinity shorten proofs a lot?
A good example is a solution to Hilbert's third problem: it is not possible to cut the unit cube into finitely many polyhedral pieces and reassemble then as the regular tetrahedron of unit volume. The...
View ArticleAnswer by Mohammad Golshani for Can infinity shorten proofs a lot?
Consider the following question of Erdos and Hajnal:Question (Erdos-Hajnal) Is there a finite $K_4$-free graph which, when the edges colored by $2$ colors, always contains a monocolored triangle.The...
View ArticleAnswer by Mikhail Katz for Can infinity shorten proofs a lot?
I once had a teaching assistant in calculus who admitted that he was unable to give an epsilon, delta proof that a the Heaviside stepfunction was not continuous. To take a slightly less trivial...
View ArticleAnswer by Pablo Zadunaisky for Can infinity shorten proofs a lot?
I am surprised no one has mentioned any examples from physics. Fluid mechanics for example obtains some beautiful results by assuming that a fluid is continuous, while at the molecular level it is a...
View ArticleAnswer by none for Can infinity shorten proofs a lot?
Maybe you want something more proof-theoretic, like Gödel's speed-up theorem. The example from the Wikipedia article says: the statement "This statement cannot be proved in Peano arithmetic in fewer...
View ArticleAnswer by Chris Pressey for Can infinity shorten proofs a lot?
Short version of "there exist functions not computable by Turing machine" using infinities:There are an uncountable number of predicates, but only a countable number of Turing machines, so there are...
View ArticleAnswer by Larry Rolen for Can infinity shorten proofs a lot?
Using Eilenberg-Maclane spaces (namely $K(\mathbb{Z},2)=\mathbb{P}^{\infty}(\mathbb{C})$) and cellular approximation,one can show that any 3-fold with a positive Betti number has a map to the sphere...
View ArticleAnswer by Johan Wästlund for Can infinity shorten proofs a lot?
An example that I came to think of even though it doesn't have to involve infinity is the existence of a finite field with a specified number $q=p^n$ of elements. One can show by an enumerative...
View ArticleAnswer by Andreas Blass for Can infinity shorten proofs a lot?
Hindman's theorem is an example where (slightly?) higher infinity simplifies a proof greatly. The theorem asserts that, if you partition the set of positive integers into finitely many subsets, then...
View ArticleAnswer by Gerald Edgar for Can infinity shorten proofs a lot?
Maybe "analytic number theory" belongs here as an answer. Certain number theory results are most simply proved using complex analysis.
View ArticleAnswer by Terry Tao for Can infinity shorten proofs a lot?
The Poincare conjecture might qualify as an example. One can phrase this conjecture in a purely combinatorial fashion, involving finite simplicial complexes, but the only known proof of this conjecture...
View ArticleAnswer by Igor Khavkine for Can infinity shorten proofs a lot?
A down-to-earth example: the Stirling approximation to the factorial.At least the derivation given on the Wikipedia page compares the explicit sum representation of $\ln(n!)$ with a Riemann sum...
View ArticleAnswer by Carl Mummert for Can infinity shorten proofs a lot?
The axiom system PRA of "primitive-recursive arithmetic" is finitistic, but it has been known for a few decades that it has the same set of $\Pi^0_1$ consequences as the infinitistic theory...
View ArticleAnswer by Michael Greinecker for Can infinity shorten proofs a lot?
Every finite partial order has a maximal element. One can prove that by induction and looking at cases, but I think the most natural way is to argue that if there is no maximal element, one must have a...
View ArticleAnswer by Jose Brox for Can infinity shorten proofs a lot?
While browsing Wikipedia, I came across this statement:"The introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the...
View ArticleAnswer by Gerald Edgar for Can infinity shorten proofs a lot?
Perhaps fitting in here is Lindenstrauss's proof of Lyapunov's convexity theorem.http://atlas-conferences.com/c/a/e/y/06.htm"the slickest proof to end all proofs" (Halmos)although the result is in...
View ArticleAnswer by Gil Kalai for Can infinity shorten proofs a lot?
There is a theorem that if a set of edge colored square tiles can be used to tile the positive orthant then it can also be used to tile the entire plane. (The edges are colored and in the edge-to edge...
View ArticleAnswer by Neel Krishnaswami for Can infinity shorten proofs a lot?
One way of generating very simple examples of this kind is to define a family of predicates $F(n)$ by recursion over the natural numbers. Then, there can usually be no proof of $\forall n. F(n)$ in...
View ArticleAnswer by David Corfield for Can infinity shorten proofs a lot?
Is there anything wrong with the simplest case - mathematical induction? If I want to show that the sum of the first million cubes is whatever it is, much easier to prove the general formula then...
View ArticleAnswer by Jonathan Wise for Can infinity shorten proofs a lot?
This is probably neither the sort of infinity nor the sort of simplification you're looking for, but Bézout's theorem is a challenge to prove, let alone state, without a line at infinity. I don't know...
View ArticleAnswer by Kevin H. Lin for Can infinity shorten proofs a lot?
I can't think of any good specific examples off the top of my head, but I'm sure that there must be lots of examples where you can much more easily prove something about an affine variety by adding...
View ArticleAnswer by Jose Brox for Can infinity shorten proofs a lot?
Maybe you could use some complex transformation related to Möebius functions (of the kind "circles and lines are the same") to prove several 2D-geometric statements at once.
View ArticleAnswer by Scott Carter for Can infinity shorten proofs a lot?
One of my colleagues (Silver) gave a general purpose talk last week. You can't untie a knot $A$ by tying another knot $B$ into the rope. Proof. First show that the connect-sum of long knots is abelian...
View ArticleAnswer by Dan Piponi for Can infinity shorten proofs a lot?
Nachum Dershowitz has lots of work on termination proofs and many of these make use of ordinal arithmetic. In particular, if you have a computer program that moves from one state to another, and you...
View ArticleAnswer by Kristal Cantwell for Can infinity shorten proofs a lot?
If you want a situation where there is only an infinite proof then there is the strengthened finite Ramsey theorem which is not provable in PA but can be deduced from the infinite Ramsey theorem....
View ArticleAnswer by Harrison Brown for Can infinity shorten proofs a lot?
This isn't particularly down-to-earth, but of course before February or March you had the situation where the density Hales-Jewett theorem (which can be phrased in completely finitary language) was...
View ArticleAnswer by Ori Gurel-Gurevich for Can infinity shorten proofs a lot?
I'm not sure whether this is more "down-to-earth" but take 0-1 laws of random graphs. The probability of a first order property to be satisfied by a random graph distributed $G(n,p)$ tends to 0 or...
View ArticleAnswer by Qiaochu Yuan for Can infinity shorten proofs a lot?
The precise asymptotics of the partition function $p(n)$ are, as far as I know, totally inaccessible by finitary tools. More generally the methods of complex analysis are a great way to obtain...
View ArticleAnswer by Mariano Suárez-Álvarez for Can infinity shorten proofs a lot?
A simple instance where infinity makes things simple is when proving that there exist transcendental numbers. Instead of having to come up with ways to tell a transcendental number from an algebraic...
View ArticleAnswer by Ryan Budney for Can infinity shorten proofs a lot?
The Mazur Swindle? See the Wikipedia page, for example.
View ArticleCan infinity shorten proofs a lot?
I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general...
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