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Questions tagged [logic]

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the following tags as well, if they fit the question: (model-theory), (set-theory), (computability), and (proof-theory). This tag is not for logical puzzles, use (puzzle).

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Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
213
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2answers
7k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\...
184
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4answers
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Why can a Venn diagram for 4+ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for 4 sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complete $4$-...
179
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4answers
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How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
158
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8answers
17k views

How do we prove that something is unprovable?

I have read somewhere there are some theorems that are shown to be "unprovable". It was a while ago and I don't remember the details, and I suspect that this question might be the result of a total ...
149
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11answers
18k views

Do we know if there exist true mathematical statements that can not be proven?

Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven ...
127
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6answers
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Using proof by contradiction vs proof of the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
118
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13answers
67k views

Can someone explain Gödel's incompleteness theorems in layman terms?

Can you explain it in a fathomable way at high school level?
117
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16answers
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Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
115
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9answers
26k views

Learning Lambda Calculus

What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus? Specifically, I am interested in the following areas: Untyped lambda calculus ...
107
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6answers
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What's the difference between predicate and propositional logic?

I'd heard of propositional logic for years, but until I came across this question, I'd never heard of predicate logic. Moreover, the fact that Introduction to Logic: Predicate Logic and Introduction ...
103
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8answers
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Are the “proofs by contradiction” weaker than other proofs?

I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the ...
97
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3answers
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True or false? $x^2\ne x\implies x\ne 1$

Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following: $$x^2\ne x\implies x\ne 1$$ I immediately ...
94
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21answers
14k views

In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \...
88
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2answers
4k views

Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on

Popular mathematics folklore provides some simple tools enabling us compactly to describe some truly enormous numbers. For example, the number $10^{100}$ is commonly known as a googol, and a googol ...
86
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10answers
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How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
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7answers
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In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I "...
79
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4answers
5k views

Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture

In the most recent numberphile video, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is ...
78
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13answers
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Why is “the set of all sets” a paradox, in layman's terms?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is "...
77
votes
7answers
24k views

“Which answer in this list is the correct answer to this question?”

I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct ...
76
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9answers
13k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
72
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11answers
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Infinite sets don't exist!?

Has anyone read this article? This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, ...
72
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6answers
6k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
67
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12answers
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What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language ...
67
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7answers
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Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...
65
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8answers
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Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign)...
62
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6answers
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Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. ...
61
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8answers
10k views

Is “The empty set is a subset of any set” a convention?

Recently I learned that for any set A, we have $\varnothing\subset A$. I found some explanation of why it holds. $\varnothing\subset A$ means "for every object $x$, if $x$ belongs to the empty set,...
61
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3answers
6k views

Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into ...
60
votes
8answers
14k views

Why do people lose in chess?

Zermelo's Theorem, when applied to chess, states: "either white can force a win, or black can force a win, or both sides can force at least a draw [1]" I do not get this. How can it be proven? ...
59
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5answers
25k views

Implies vs. Entails vs. Provable

Consider A $\Rightarrow$ B, A $\models$ B, and A $\vdash$ B. What are some examples contrasting their proper use? For example, give A and B such that A $\models$ B is true but A $\Rightarrow$ B is ...
57
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15answers
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Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
57
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14answers
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What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
56
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8answers
6k views

How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would ...
55
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5answers
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Doesn't the unprovability of the continuum hypothesis prove the continuum hypothesis? [duplicate]

The Continuum Hypothesis say that there is no set with cardinality between that of the reals and the natural numbers. Apparently, the Continuum Hypothesis can't be proved or disproved using the ...
55
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8answers
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Is it possible that “A counter-example exists but it cannot be found”

Then otherwise the sentence "It is not possible for someone to find a counter-example" would be a proof. I mean, are there some hypotheses that are false but the counter-example is somewhere we ...
54
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12answers
8k views

Are proofs by contradiction really logical?

Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. My question is this: How does one go from "so there's a contradiction if we don't have $A$" to ...
54
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2answers
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Is it possible to prove a mathematical statement by proving that a proof exists?

I'm sure there are easy ways of proving things using, well... any other method besides this! But still, I'm curious to know whether it would be acceptable/if it has been done before?
54
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2answers
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What does it take to divide by $2$?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That's because the axiom of choice allows for the definition of ...
51
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14answers
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An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
51
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3answers
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Is there any conjecture that we know is provable/disprovable but we haven't found a proof of yet?

I know that there are a lot of unsolved conjectures, but it could possible for them to be independent of ZFC (see Could it be that Goldbach conjecture is undecidable? for example). I was wondering if ...
51
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14answers
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In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?

Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&...
51
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5answers
12k views

Understanding Gödel's Incompleteness Theorem

I am trying very hard to understand Gödel's Incompleteness Theorem. I am really interested in what it says about axiomatic languages, but I have some questions: Gödel's theorem is proved based on ...
51
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3answers
22k views

First-Order Logic vs. Second-Order Logic

Wikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic ...
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5answers
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Say $a=b$. Is “Do the same thing to both sides of an equation, and it still holds” an axiom? [duplicate]

Recently I have started reviewing mathematical notions, that I have always just accepted. Today it is one of the fundamental ones used in equations: If we have an equation, then the equation holds ...
49
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4answers
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Why can we use induction when studying metamathematics?

In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction ...
48
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5answers
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What do logicians mean by “type”?

I enjoy reading about formal logic as an occasional hobby. However, one thing keeps tripping me up: I seem unable to understand what's being referred to when the word "type" (as in type theory) is ...
48
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12answers
29k views

Good books on mathematical logic?

I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
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6answers
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What would happen if we just made vacuous truths false instead?

It's well known that vacuous truths are a concept, i.e. an implication being true even if the premise is false. What would be the problem with simply redefining this to be evaluated to false? Would ...
42
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6answers
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“If everyone in front of you is bald, then you're bald.” Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...