0
votes
3answers
152 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
95 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, ...
6
votes
2answers
235 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
18 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
53 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
139 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
44 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
88 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
110 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
68 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
106 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
69 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
110 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
512 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
47 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
150 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
5answers
348 views

Are there finitely many interesting theorems? [closed]

I'm not a logician, I read, a long time ago, about Gödel etc... A theorem is a provable proposition under a system of axiom. Let's take the usual system of ZFC. Of course, the notion interesting is ...
5
votes
2answers
438 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
132 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
132 views

What are the prerequisites to learn mathematical logic?

I have no background in mathematical logic.
14
votes
3answers
283 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
162 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
201 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
163 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
417 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
159 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
47 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
419 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 ...
4
votes
1answer
172 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
130 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
348 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
70 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
251 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
286 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 ...
4
votes
0answers
106 views

Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized ...
1
vote
2answers
105 views

Virtues of Presentation of FO Logic in Kleene's Mathematical Logic

I refer to Stephen Cole Kleene, Mathematical Logic (1967 - Dover reprint : 2002). What are the "pedagogical benefits" (if any) of the presentation chosen by Kleene, mixing Natural Deduction and ...
7
votes
1answer
147 views

Is there any point in a logician studying $\infty$-categories?

My primary areas of interest lately have been set theory, logic, and category theory, so naturally topos theory has been a large part of what I'm learning (in between getting caught up on some other ...
4
votes
2answers
87 views

Crowding the boundary of non-constructivity without crossing it?

Can any sense be made out of my vague feeling that some proofs in Ramsey theory are as close as you can get to non-constructive proofs without crossing the line? Is there any way to make this ...
4
votes
2answers
224 views

How would I know if I'm good in logic?

I've always been interested in logic, but unfortunately my school contains no logicians. What are some good logic puzzles/books and how would I know if logic is right for me? Also, what can I do with ...
2
votes
1answer
88 views

“Generalization” of Gödel's Theorem

One question came to my mind while arguing with a friend about the necessity of Judges in society (I will explain...) In the reasoning, we came across the following argument: "If we put up a good ...
4
votes
2answers
154 views

Are there any foundations in which the universe itself gets dynamically extended?

In the foundations I'm familiar with, you don't ever "create" anything. Take ZFC for example. If we need the real numbers, we search in our static universe for a set $\mathbb{R}$ of the correct ...
4
votes
4answers
356 views

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a ...
5
votes
0answers
130 views

which branch of maths studies Standard Logical Matrices

In classical logic you have truthtables like: & | T | F ---|---|--- T | T | F F | F | F In many valued logic you have tables like: (this one is ...
5
votes
0answers
268 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
5
votes
7answers
322 views

Is there a name for this type of logical fallacy?

Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$. Example: As $3$ is odd, $3$ is prime. In this case, it is true that $3$ is odd, and ...
3
votes
4answers
145 views

Applications of logic

What are some applications of symbolic logic? I tried using Google and Bing but just got a bunch of book recommendations, and links to articles I did not understand.
3
votes
1answer
134 views

Can we use proper classes in this way, to define a new infinity larger than |Ord|?

I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...