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15

With a strict enough definition of "non-foundational mathematics" I think the answer is probably "no" (although I would be very interested in seeing potential examples.) However, this shouldn't make mathematicians working on such mathematics feel safe about using unrestricted comprehension. The reason is that it's not always clear a priori what mathematics ...


13

It is impossible to find a rational number whose square is 2.


13

The proof that demonstrates the impossibility of trisecting an angle uses Galois theory. Galois theory can also be used to show that certain polygons cannot be constructed with compass and straightedge, and was originally used to show that, in general, polynomials of degree $\geq 5$ are not solvable. To be specific, an $n$-gon is constructible via ...


11

Initial comments: This is an excellent question in my opinion and is just what the proof-writing tag is for. Unfortunately, there are often many problems plaguing beginners when it comes to induction proofs: Why induction is a valid proof technique should be understood at the outset, and this is rarely the case. Less relevant in high school or undergrad, ...


8

This answer takes a bit of background, but I think it's worth it. Please bear with me! If you already have the necessary background, you can skip to the last section for the punch line. The surface of a sweet bun and the surface of a doughnut are both examples of two-dimensional manifolds: geometric spaces that look like planes from close up. An ant ...


7

The whole idea of using set theory as a foundational theory is that you want a theory that if you believe is consistent, the rest of mathematics is consistent. Naive set theory is inconsistent. So you can't really continue, you cannot trust it to give you the rest of mathematics. And it is not important that you "don't seem to appeal to the paradoxes". ...


7

I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level. Ph. Davis and R. Hersh tell the long and painful story of Fourier series. I quote from their book: "Fourier didn't know Euler had already done this, so he did it over. And Fourier, like Bernoulli and Euler before him, ...


7

First Isomorphism Theorem examples First Isomorphism Theorem (FIT) applies in different contexts: Groups, Rings, Vector Spaces, Lie Algebras and other structures. As follows, examples for the first three. Three Group Isomorphisms Let us consider $GL_2(\Bbb F_3)$: the group of $2\times 2$ invertible matrices with coefficients in $\Bbb F_3$ which is the ...


6

That there exist transcendental numbers. This was fist shown by Liouville, who proved that Liouville's number: $$\sum_{i=0}^\infty10^{-i!}$$ is transcendental. The "modern" proof would be due to Cantor: There are countably many algebraic numbers and uncountably many reals. Therefore there exists a transcendental number. Proving that Liouville's number ...


6

I think: This answer on MathOverflow is relevant. This paper is pointing out the counterexample. This paper is mending the original statement. The theorem is in algebra, and is about whether the $\lim^1$ functor vanishes on Mittag-Leffler sequences in abelian categories satisfying certain axioms. In order to correct the mistake, the author had to add a ...


5

Constructing an algorithm to solve any Diophantine equation has been proven to be equivalent to solving the halting problem, as is computing the Kolmogorov complexity (optimal compression size) of any given input, for any given universal description language. In general I think what you're looking for is either problems that are proven to be equivalent to ...


5

One problem which I like is the Bridge of Konigsberg problem. The challenge is to find a path that crosses each bridge exactly once. It was eventually proved to be impossible by Euler and is considered one of the early applications of graph theory. http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg


5

See the papers of: Dusart 2010 Axler 2013 Axler 2014 Büthe 2014 Kotnick 2008 Schoenfeld 1976 For the nth prime upper bound, Axler 2013 viii Korollar G is best when $n \ge 8009824$. Dusart 2010 page 2 is best from $688383$ to $8009823$. Below that, either Dusart 2010 page 7, Dusart 1999 page 14, Robin 1983 or tweak the values since this is a limited ...


5

A famous example is the law of quadratic reciprocity. Wikipedia says that several hundred proofs of the law of quadratic reciprocity have been found. Many details concerning proofs by different methods can be found at the question at MO.


5

This sum-of-squares theorem of Fermat may qualify as an example: An odd prime $p$ is expressible as the sum of squares $x^2+y^2$ if and only if $p\equiv 1 \text{ mod } 4$. You can read this Wikipedia article (as of the most recent update to this answer) to see the difference in mental effort in the original proof by Euler, as opposed to a modern ...


4

There exists a set of Wang tiles for which it is undecidable whether they can tile the entire plane. Wang tiles are nice and geometric and easily described, but you can encode each Turing machine so that the machine halts if and only if the tiles can't tile the entire plane.


4

Well, I wouldn't trust a building whose structure was shown to be flawed. Note that it isn't a suspicion, it is a certainty. But I know what you mean by your question... consider the following "proof" (taken from another question at MSE): Theorem: Let $K$ be a field, then $K$ has an algebraic closure $\bar{K}$ (i.e an algebraic extension that is ...


3

It's impossible to define mathematics in a way that satisfies all mathematicians. This is likely not the answer you're after, but it stroke me when I first heard of it because definitions in mathematics are both exact and omnipresent, and I was not suspecting that providing a universal definition of mathematics itself could be such an intractable problem. ...


3

<< about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve >> This statement is false. Since it is now proved that "trisecting an angle using compass and straight edge is impossible", the problem is solved. So, the problem was not impossible ...


3

Here is my template for case #$1$: As an example, let's prove by induction that $\sum\limits_{k=0}^{n-1}2\cdot3^k=3^n-1$. First, show that this is true for $n=1$: $\sum\limits_{k=0}^{1-1}2\cdot3^k=3^1-1$ Second, assume that this is true for $n$: $\sum\limits_{k=0}^{n-1}2\cdot3^k=3^n-1$ Third, prove that this is true for $n+1$: ...


3

Take a look at this project http://www.formulae.org It is open source, it is about math and it is partially documented. There are several sources for documentation: the developer's guide (LaTeX, partially), the front-end user´s guide (LaTeX, starting), the API reference (JavaDoc, partially) and expression dictionary (online wiki, partially)


3

In the nineteenth century expressing the antiderivative of an elementary function as an elementary function was an open problem. Nowadays, Risch algorithm, which can be run on machines, decides whether such operation can be done and, if so, yields a version of the correct result. I cannot speak for past mathematicians, but I think this is a useful tool. ...


3

It seems like Stephen Abbott's Understanding Analysis is just what you are looking for. From page vii of the Preface: Each chapter begins with the discussion of some motivating examples and open questions. The tone in these discussions is intentionally informal, and full use is made of familiar functions and results from calculus. The idea is to freely ...


2

Great question! I would like to suggest you to have a look to some Python libraries specialized on Mathematics, like sympy or gmpy. I use them often to study and make my tests, and I miss always some extra explanations or samples in the online documentation. They are really great, and some extra samples and theory-related explanations would be imho a good ...


2

This is an answer covering a somewhat more exotic proof. The claim is one I used in my answer at prove or disprove $H$ is a subgroup but I will expand on the details here and write it as nicely as possible. For more about when induction can be done on these more exotic examples, see my blog post ...


2

Define $f : \mathbb{R} \to \mathcal{P}(\mathbb{Q})$ as $f(x) = (-\infty, x) \cap \mathbb{Q}$. It is injective, so $|\mathbb{R}| \leqslant |\mathcal{P}(\mathbb{Q})|.$


2

For each real number $x$ pick a strictly increasing sequence $x_n$ of rationals converging to $x$. Then $$f(x)=\{ x_n| n \}$$ is a one to one function from $\mathbb R$ to $P(Q)$. As $\mathbb R$ is uncountable....


2

$$\int_0^1\frac{x^4(1-x)^4}{1+x^2}=\frac{22}{7}-\pi$$ It's interesting how something so bizarre on the left hand side yields the tiniest of errors in one of the most famous approximations of $\pi$.


2

One place to look at is Topospaces (The Topology Wiki), which was suggested by a now-deleted user in a now-deleted answer. The description says This is a pre-alpha stage topology wiki primarily managed by Vipul Naik, a Ph.D. in Mathematics at the University of Chicago. We have over 400 articles including some material in basic point-set topology. As ...


1

As stated by JJacquelin, your example is wrong. However, the following theorem is impossible to solve with just the axioms of Zermelo–Fraenkel set theory: the Cartesian product of a collection of non-empty sets is non-empty. This led to the famous Axiom of choice being introduced (in another form) by Zermelo to address the issue. Previously, the axiom ...



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