2
votes
2answers
90 views

Gödel's incompleteness theorems: where to learn? Is there a straightforward relation between the two?

What would be a good textbook or paper to learn the proofs of the two Gödel's incompleteness theorems from? I would prefer it to be as close to the original proofs as possible. I have not tried to ...
4
votes
2answers
112 views

Modern book on Gödel's incompleteness theorems in all technical details

Is there a modern book on Gödel's incompleteness theorems that goes into each and every technical aspect of the proof of them (a classical one, if such exists)? I'm not interested in popular ...
11
votes
2answers
179 views
+100

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
3
votes
1answer
50 views

Intuitionistic Linear Logic

I am currently going through some papers that use the "intuitionistic version" of Girard's Linear Logic. Problem is, i seem to find very little literature on it. There is a lot done on Linear Logic ...
2
votes
2answers
78 views

I would like some textbook recommendations for model theory

I am a 3rd year undergraduate math student and would like to study model theory. . I have some background with set theory, ordinals etc and also with mathematical logic. This is purely for self study ...
1
vote
1answer
35 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 ...
4
votes
2answers
164 views

Interaction of completeness and second incompleteness theorems

So I was reading the Wikipedia article on Godel's completeness theorem, the section on its relation to completeness. It says that completeness gives the existence of a model of arithmetic $\mathcal M ...
3
votes
0answers
37 views

Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
0
votes
1answer
41 views

Help with positive- and negative-forms in Proof Theory

I need help in understanding a device used bu Kurt Schütte, Proof Theory (1977). In treating classical sentential calculus, he use - in place of truth-tables - the device of positive- and ...
3
votes
3answers
119 views

Some reference for categorical logic?

By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and ...
3
votes
3answers
89 views

Online lectures for a first course in mathematical logic

I have a friend who is interested in learning math. I suggested that he learns mathematical logic. He has never learnt mathematical logic before, however I believe he has all the necessary ...
6
votes
5answers
181 views

The deep structure of logical formulas

A long-standing question to which I never found a concise answer is: Is there something like an unambiguous deep structure of a formula of propositional logic, opposed to its comparingly ...
10
votes
2answers
167 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
4
votes
1answer
120 views

Can one define $\langle x,y\rangle$ in $P(C)$?

I study at course Foundations of Mathematics the below definitions and lemma: $\langle x,y\rangle:=\{\{x\},\{x,y\}\}$ (from Kuratowski 1921) $\langle x,y\rangle:=\{\{\{x\},\varnothing\}\{\{y\}\}\}$ ...
2
votes
0answers
48 views

Certain sequents as inference rules

Fix a signature $\sigma.$ Then a coherent formula is a first-order formula built using only $\{\wedge,\vee,\top,\bot,\exists\}.$ See the link for more information. Furthermore, by a "special" ...
1
vote
1answer
135 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 ...
0
votes
0answers
28 views

T-norms with arity 3

I'm working on some 3D vectors with fuzzy values, and I'd like to devise some operations on them. After my brief search, I haven't been able to find any existing work on t-norms with arity n>2. Is ...
5
votes
0answers
102 views

Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
3
votes
1answer
123 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 ...
4
votes
3answers
194 views

Good textbook for learning Sequent Calculus

There are many modern text books teaching logic using Natural Deduction. There are no books teaching logic using the axiomatic method (see Good book for learning and practising axiomatic logic ) Now ...
2
votes
0answers
30 views

Books/papers on model theory in non-monotonic logic

I am working on a project whose object language is in non-monotonic logic. Since the project involves reasoning about the models, I am thinking of translating a non-monotonic problem into a ...
2
votes
5answers
221 views

Logic and set theory textbook for high school

Do you have any advice for a textbook or a book for high schools students which completely adresses basics of logic (proposition, implication, and, or, quantifiers) and set theory (intersection, ...
4
votes
1answer
121 views

How do we know that certain concrete nonstandard models of the natural numbers satisfy the Peano axioms?

It is easy to come up with objects that do not satisfy the Peano axioms. For example, let $\Bbb{S} = \Bbb N \cup \{Z\}$, and $SZ = S0$. Then this clearly violates the axiom that says that $Sa=Sb\to ...
2
votes
1answer
66 views

The Consistency of Arithmetic

I believe that within ZFC (or maybe even a weaker subset of ZFC) there is a proof of $\mathbb{N}=(\omega,+,.,<,0,1)\models{PA}$. What would be a standard reference for this?
2
votes
1answer
51 views

Algebraic signatures as quivers; is there somewhere I can learn more about these definitions?

In my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying set ...
0
votes
3answers
38 views

Reference for problems without efficient algorithm (in polynomial time)

I'm writing paper and need your help in finding some famous (or not so famous) problems without efficient algorithm, but from logic or computer science. So far, I have: -Boolean satisfiability ...
2
votes
2answers
44 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 ...
2
votes
3answers
164 views

Book request: mathematical logic with a semantical emphasis.

Suppose I am interested in the semantical aspect of logic; especially the satisfaction $\models$ relation between models and sentences, and the induced semantic consequence relation $\implies,$ ...
3
votes
1answer
61 views

Construction of a Kurtz random sequence that's not Martin-Löf random

How can one construct a Kurtz random sequence that's not Martin-Löf random? I'm also interested in the paper that included the first of such constructions. I suspect it was in Kurtz's dissertation, ...
3
votes
2answers
62 views

Textbook on Basics of Formal Systems

Whilst trying to learn more about logic I came across Smullyan's Theory of Formal Systems on Google Books. What I liked about the book was how clearly it managed to describe (on pages 3-5 in chapter ...
0
votes
0answers
54 views

Book for teaching mathematic and IQ for kid.

I'm finding some books talk about teaching mathematic for kid ,logic and IQ for kid. Can you present for me some books?
3
votes
1answer
110 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 ...
1
vote
1answer
44 views

Conservativity theorem

There is an article in Wikipedia titled "Conservativity theorem" - see http://en.wikipedia.org/wiki/Conservativity_theorem I have looked through a dozen of mathematical logic textbooks, but could ...
5
votes
1answer
95 views

Advice regarding best-practice mathematics / categorial logic.

A good heuristic is: If it doesn't cost anything, generalize. In particular, if we have a theorem, and a proof thereof, then we ought to look for a maximal generalization of this theorem, ...
0
votes
2answers
89 views

The Adjunction $\_\times A\dashv (\_ )^A$ for Preorders: The Deduction Theorem.

The following is from Turi's Category Theory Lecture Notes. Definition 11.11 Let $A$ be an object of a category $\mathbb{C}$ with binary products. The right adjoint of $\_\times ...
4
votes
0answers
91 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 ...
6
votes
1answer
93 views

Have mathematical structures equipped with “generalized relations” been considered in a systematic way?

A binary relation on $X$ is basically just a function $X^2 \rightarrow \mathbb{B}$, where $\mathbb{B}$ is the prototypical Boolean algebra $\{0,1\}.$ We can generalize by replacing $\mathbb{B}$ with a ...
9
votes
1answer
100 views

Properties of the internal language of the category of sheaves.

Consider a simple case of sheaves on some topological space $X$, $\operatorname{Sh}(X)$ (recall that a sieve on $U$ is covering iff its $\operatorname{sup}$ is $U$). All of these are Grothendieck ...
1
vote
1answer
77 views

How to prove “basic” identities in first order logic?

On the Wikipedia page for First-order logic, there is a list of Provable Identities. Although they seem very basic, I can't find anyone giving a formal proof of them. In particular, consider one ...
5
votes
1answer
126 views

where to start reading theory of logics?

I am a student who is working Lie Theory. I want to start read theory of logics. I just need some reference and I have few questions regarding this, i) will studying theory of logics will improve my ...
2
votes
2answers
181 views

Book that is more accessible than Shoenfield

My logic course is based on my Computer Science education and on some random Internet pages (mostly Wiki). I want to make my knowledge of logic more coherent and fill in missing gaps. Thus I started ...
2
votes
1answer
64 views

Genericity and category

This paper by Ambos-Spies and Mayordomo on the theory of algorithmic randomness introduces the notion of genericity saying that it is based on Baire category while the usual notion of randomness is ...
1
vote
0answers
54 views

Fragments of first-order logic and the functions that preserve them - reference request.

Is there a good resource for learning about different fragments of first-order logic? At this point, I'm mainly just interested in the basic facts, nothing too deep, but preferably presented in a ...
4
votes
2answers
103 views

Reading on Mathematical Logic

I am looking for books to read, so as to dive into mathematical logical and related disciplines like set theory, model theory, and topos theory. I have a decent background in category theory and ...
4
votes
2answers
200 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
4answers
107 views

Boolean prime ideal theorem and the axiom of choice

The Boolean prime ideal theorem is strictly stronger than ZF, and strictly weaker than ZFC. I'm looking for nice examples (like the existence of non-measurable set) that request at least that theorem ...
1
vote
2answers
104 views

Partial functions - where can I learn more about this (heuristic, informal) system of conventions?

Is there a name for the following (heuristic, informal) system of conventions for dealing with partial functions and undefined expressions? I'd like to know whether it has any undesirable quirks that ...
3
votes
1answer
53 views

Con ZF implies Con ZFC using set sized models

Can we use forcing to construct models of ZFC and ZFC + GCH starting from c.t.m s of ZF? The usual way to obtain the associated relative consistency results (Con ZF implies Con ZFC and Con ZF implies ...
8
votes
2answers
164 views

Set theory based on inclusion

There are several axiomatizations of set theory based on inclusion rather than membership. I found only two papers, but they are both in German, and I could not read them even using a disctionary. Can ...
5
votes
3answers
173 views

Foundations of Forcing

I am currently studying Forcing methods in order to understand some independence results and model's constructions. Now I am interested on formalizing the main notions around forcing such as ...