For questions about proofs within a formal system, such as natural deduction or Hilbert system.

For questions about proofs within a formal system, such as natural deduction or a Hilbert system.

For questions about proofs within a formal system, such as natural deduction or Hilbert system.

For questions about proofs within a formal system, such as natural deduction or a Hilbert system.

765 questions

votes

I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this:
A mathematical proof is a finite sequence of mathematical assertions which forms a valid and convincing…

Stephen

- 3,522
- 1
- 14
- 30

votes

I'm near the end of Velleman's How to Prove It, self-studying and learning a lot about proofs. This book teaches you how to express ideas rigorously in logic notation, prove the theorem logically, and then "translate" it back to English for the…

Raiden Worley

- 479
- 3
- 6

votes

The question that I saw is as follows:
In the Parliament of Sikinia, each member has at most three enemies. Prove that the house can be separated into two houses, so that each member has at most one enemy in his own house.
I built a graph where…

user 6663629

- 144
- 8
- 17

votes

Goedel's incompleteness tells us that any system containing Robinson arithmetic is incomplete. OTOH, Presburger Arithmetic, which contains only the successor and addition, is complete. I'm pretty sure that I have read that it possible to define a…

Itai Seggev

- 253
- 1
- 6

votes

I've been learning natural deduction recently. I've seen many problems and am starting to be able to solve problems more easily.
For some reason I feel the need to ask what high school math students always ask about mathematics.
What is the point…

John Smith

- 1,345
- 8
- 25

votes

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents.
Definition. If $\Gamma$ is a set of formulas and $\phi$ a formula, then $\Gamma\vdash\phi$ is called a…

ooooooo

- 193
- 7

votes

I found the following proof arguing for the irrationality of $\sqrt{2}$.
Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q > 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::=…

Eulerian

- 473
- 3
- 9

votes

Classical theorems like the irrationality of $\sqrt{2}$ or the infinitude of the primes have lots of proofs. But one theorem in particular, which I studied years ago in an introductory course of Number Theory, called the Quadratic Reciprocity Law,…

Filburt

- 1,986
- 2
- 14
- 42

votes

I'm interested in formalising mathematics and logics in a proof assistant, both to get to know a proof assistant and to make an archive of proofs for myself (nothing too fancy, mainly first order logic, set-theory, group-theory and so on).
To be…

jules

- 221
- 1
- 3

votes

This is from a talk by Edsger W. Dijkstra, "How Computing Science created a new mathematical style", 4 March 1990:
Almost all formalisms used daily by the classical mathematician are at least ambiguous. But that does not hurt the classical…

JonathanDavidArndt

- 493
- 3
- 6
- 21

votes

How do we formally prove (e.g. in type theory) that
$$\sum _ix_i=\sum_ix_{f(i)}$$
For any bijection $f:I\to I$ for any finite set $I$?
I might be overcomplicating things, but I’m having trouble seeing how to best define $\sum$ formally, and do the…

user56834

- 11,887
- 6
- 32
- 91

votes

In everyday math, if we ever prove the existence of a unique object with a certain property, we tend to give it a name and refer to it as "the" such-and-such object moving forward. For example, in ZFC set theory, an important early theorem…

WillG

- 4,878
- 1
- 13
- 33

votes

Suppose I have a statement $X$, for which I do not know whether it is true or false. And suppose further that I want to prove a statement $Y$:
I first assume that $X$ is true, and I construct an involved argument that shows that $Y$ follows.
I then…

M. Winter

- 28,034
- 8
- 43
- 92

votes

While reading some old paper on the foundations of set theory, I came across a symbol $\mid$ that I eventually determined was the Sheffer stroke, which is a fancy word for NAND.
Wikipedia, and also the paper, has only this to say about working with…

Misha Lavrov

- 113,596
- 10
- 105
- 192

votes

Given a first-order logic theory $T$ and and a formula $F$, suppose I have semantically proved that $T\vdash F$. That is, I have proved that any model $M$ of $T$ satisfies $F$ and I conclude by Gödel's completeness theorem.
Do I have a general…

V. Semeria

- 961
- 4
- 8