0
votes
4answers
70 views

If I want to avoid quantifiers?

In mathematics quantifiers always is used with restrictions $\forall x\in A$ etc and mathematicians often write: $$x\in A\implies p(x)\;\;\;\text{instead of}\;\;\;\forall x\in A:p(x).$$ Are there a ...
-1
votes
0answers
31 views

Properties that only hold for 'small' infinities

I don't know the right terminology, so I'll pose it this way: Let $\alpha_1=|\mathbb{N}|$ e.a. countable. Let $\alpha_2=|2^\mathbb{N} | = |\mathbb {R}|$ There are a lot of properties that hold only ...
1
vote
1answer
55 views

Are all transcendental numbers theoretically accessible?

I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do. I've been thinking ...
1
vote
2answers
105 views

Incompleteness and Counterexamples [on hold]

How can one know if an unprovable statement is true or false? What I want to know is if it is possible to construct a counterexample for a statement which cannot be proved within the theory.
-1
votes
0answers
247 views

Are paradoxes a threat against mathematics? [closed]

Or are they just mathematical tools? Given a binary predicate $R$, written in infix notation, there are unitary predicates $p$ such that: $\qquad \exists y \forall x: x R y \leftrightarrow p(x)$. ...
1
vote
2answers
67 views

Books/subjects for proof practice

So I want to practice writing proofs. I've studied general proof-writing but now I want to learn how to apply that to mathematics. From what I understand, the best and most accessible subjects for ...
4
votes
2answers
181 views

How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
1
vote
1answer
89 views

Quantifier problems of equations in physics [closed]

Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. To assert that static energy equals mass ...
2
votes
1answer
125 views

Are there any “obviously” true propositions in number theory?

After all efforts spent on wrong proofs of famous number theory conjectures and theorems like Goldbach's or Fermat's last theorem, could one find some simple statements (might be correct ones) whose ...
6
votes
1answer
179 views

How much mathematics should a student of mathematical logic know?

I would like to know what areas of mathematic are directly related to mathematical logic, besides the usual courses on model theory, proof theory and computability. If you suggest only one book on ...
2
votes
2answers
127 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
2
votes
0answers
48 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
32
votes
5answers
2k views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
2
votes
1answer
152 views

Why do some universities offer mathematical logic in different departments? [closed]

I'm thinking of pursuing mathematical logic after my undergraduate work and I have noticed that some universities offer mathematical logic in their philosophy department while others offer it in their ...
0
votes
3answers
167 views

How do we define equality in real numbers?

How do we define equality in real numbers? I know in logic we define equality by Leibniz's law. $$ \forall x \forall y[x=y \rightarrow \forall P(Px \leftrightarrow Py)] $$ But how do we define the ...
1
vote
2answers
100 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
7
votes
2answers
281 views

Graduate level elementary logic books

I've done two courses on Logic during my Bachelor course, but they were very basic. Now I'm going to start by PhD, and I'm interested in learning "real Logic". Could you please provide some references ...
1
vote
1answer
21 views

Notation for Model-Relation of formulae with free variables

Lets assume we have a formula $\mathsf{path}(x,y)$ with free variables $x,y$, and $\mathsf{acyclic}$ with no free variables on the signature $\tau = \{E\}$ (i.e. Graphs). Informally, what the formula ...
1
vote
2answers
55 views

Formalizing the Fallacy of Composition

Is there a well-known formalization of the fallacy of composition? More generally, where in mathematics is it true that if a property holds for all of some elements of a set it holds for the whole ...
9
votes
1answer
146 views

A graph of all of mathematics

In mathematics, one often makes (proves) statements on the basis of: Previously proven statements Axioms I like to think of these dependencies as a directed graph, with edges from the accepted ...
3
votes
0answers
46 views

Efficient software for producing (expression) trees “on the fly,” with good editing (e.g. cut/copy/paste) facilities?

A formula like $\forall x \exists y(x+y=0)$ can be represented as a tree. Something like so: ...
8
votes
3answers
1k views

What Maths are the most important for Artificial Intelligence?

I am just curious about this. Please don't include anything about programming.
0
votes
0answers
89 views

Dividing line between useful ( for non-foudational Math ) and unnecesary, in Foudational Math.

I started studying mathematical logic because I was curious about the behind-the-scenes of proofs, theorems and axiom systems of math. I'm interested in understanding the big picture that ...
5
votes
2answers
125 views

Finite Model Theory

It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more ...
4
votes
2answers
70 views

Theorems which are proven by proving the existence of a formal proof without knowing the formal proof

Let $L$ be a first order language. Let $T$ be a set of sentences in $L$ and $S$ a sentence in $L$. Let's define a meta-proof to be a proof that there exists a formal proof of $S$ from $T$. Question: ...
3
votes
1answer
134 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
1
vote
2answers
77 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
1
vote
3answers
118 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
2
votes
4answers
528 views

Conjectures which cant be right or wrong

Recently I was talking with some of my non-mathematician friends. On some very unrelated subject in order to make my point I said: "There are some conjectures in mathematics which are proven to be ...
0
votes
1answer
38 views

Other conditions than necessary and sufficient conditions, $\Rightarrow$, $\Leftrightarrow$?

I know that $$A\Rightarrow B$$ means that $A$ is a necessary condition for $B$ and $B$ is a sufficient condition for $A$. Also, $$A\Leftrightarrow B$$ means that $A$ is necessary and sufficient ...
1
vote
1answer
54 views

Suggestion for independent study of mathematical logic

Hello I'm looking for advice on mathematical logic books that are good for self-study. I would really like a text that has some if not all of the answers to exercises so I can check my progress as I ...
3
votes
2answers
239 views

De Morgan's laws in logic and set theory

In logic De Morgan's law means $\lnot (A \land B) \Leftrightarrow \lnot A \lor \lnot B$ In set theory De Morgan's law means $(A \cap B)^C = A^C \cup B^C$ I'm surprised that the same idea is true in ...
5
votes
2answers
515 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
4
votes
1answer
139 views

The method of “Mathematical Induction” as explained in my book.

I am reading the book "A Treatise on Advanced Calculus" by Philip Franklin. I found this book in our city's central library and liked it at the first reading, am continuing this book as a reference. ...
0
votes
3answers
196 views

What are the prerequisites to learn mathematical logic?

I have no background in mathematical logic.
14
votes
3answers
314 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
1
vote
1answer
199 views

Recommendation on a rigorous and deep introductory logic textbook

In this post, I don't mean any word by its somewhat "mathematical or logical" meaning but just "literally". It's been three years since I started "formal" mathematics, and now I'm familiar with set ...
3
votes
1answer
247 views

Mathematical logic book with answers to exercises

I'm sure a question similar to mine has been asked before, but I am looking for a mathematical logic book with answers to the exercises. I am studying independently and although I have good logic ...
6
votes
6answers
172 views

Any ideas how I can rewire my brain such that $\varphi \leq \psi$ “obviously” means that $\varphi$ implies $\psi$?

The Boolean domain $B=\{\mathrm{False},\mathrm{True}\},$ can be viewed as a partially ordered set in two different ways. In the best approach, $\mathrm{False}$ is the least element and $\mathrm{True}$ ...
14
votes
6answers
490 views

When to use the contrapositive to prove a statment

My question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the form ...
3
votes
3answers
197 views

Paradoxes in Logic

What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about ...
2
votes
2answers
49 views

Metatheoretical terms for logic

When we study logic we define various metatheoretic properties for logical systems and first-order theories, and then ask whether particular systems or theories have these properties. "Consistent" and ...
1
vote
1answer
99 views

Soft Question:Is the following a Paradox?

Can the statement: "I swear by God that I will never swear" be regarded as a variant of the Paradox of Self-Reference like the one "I am a liar"?
4
votes
4answers
459 views

If we accept a false statement, can we prove anything? [duplicate]

I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$) to ...
5
votes
1answer
185 views

Was Fermat's last theorem proved based on Peano's postulates?

Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math ...
3
votes
1answer
147 views

Good books for building number/math intuition

I'm wondering if there are some good book/textbooks that were designed with algebraic logic in mind (ie. building intuition rather than rote learning). As an example of what I mean, consider this ...
12
votes
1answer
366 views

What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
2
votes
1answer
77 views

Can axioms of the Euclidean space be proven in the Real space?

I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean ...
3
votes
1answer
286 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
4
votes
4answers
297 views

The set of all things. A thing itself?

If the universe is the set of all things. Does it contain itself? In other words is it a thing itself? I know its a stupid question, but it really grinds my gears. Thanks! Edit 8.12 Okey, someone ...