What is Mathematical logic: Definition and 45 Discussions

Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.
Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

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  1. robertphy

    Suggestions for using Math Puzzles to stimulate my math students

    The question is: my students (K9-12) have received the (sadly) usual routine based Math pseudo-education, before me teaching them. The best way to curb the tide is to use Math puzzles, to re-ignite their love of Math, lost in kindergarten. Now, which book would you suggest to fight the battle...
  2. N

    [Mathematical logic] prenex normal form and skolem normal form

    The attached picture below is the note I solved halfway through. Please tell me the entire process of getting to the correct answer.
  3. S

    Prove the 2nd axiom of mathematical logic using the Deduction Theorem

    prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem
  4. S

    Use mathematical logic to prove this proposition

    Given the following axioms: 1) ##P\implies(Q\implies P)## 2) ##((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))## Where ##P,Q,R## are any formulas 3)##(\neg P\implies\neg Q)\implies (Q\implies P)## then prove: ##\{A\implies B,B\implies C\}|- A\implies C## Without using the...
  5. D

    Is Electromagnetism Explained on Google and Online Forums?

    I am looking for an answer about electromagnetism via google and the forum haw the answer
  6. AndreasC

    Is Yuri Manin's A Course in Mathematical Logic generally unsound?

    A book that has really caught my attention recently is Yuri Manin's A Course in Mathematical Logic for Mathematicians. I am very interested in the foundations of mathematics and mathematical logic, plus I noticed that it had some chapters on quantum logic, so I started skimming through it...
  7. Math Amateur

    Top Undergraduate Mathematical Logic Texts: Recommendations from MHB Members

    What mathematical logic texts do MHB members think are the best at undergraduate level ... that is the best introductory texts ... Peter
  8. Tukhara

    Books on mathematical logic, foundations, and philosophy

    Hello, all. I am looking for some good books to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart of this wide subject which has interested me greatly. Now I have already read Shoenfield and Halmos...
  9. P

    Why does P -> Q hold true when P is false and Q is true in mathematical logic?

    When P -> Q, why is it true when P is false and Q is true, but why is it false when P is true and Q is false? If I suppose P mean "Jon is a guy" and Q mean "Mary is a girl". When both P and Q are true it does make sense that this proposition is true because Jon is a guy and Mary is a girl...
  10. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  11. M

    What are some good books to study mathematical logic?

    I am a CS student and have a very poor understanding of this field of mathematics. I don't properly know the difference in between symbolic logic, first order logic, propositional calculus, model theory and lambda calculus. But, I want to start studying logic formally from the very basic. I'd...
  12. Math Amateur

    Mathematica Mathematical Logic - Dandah symbol for discharging assumptions ....

    I am reading the book Mathematical Logic by Ian Chiswell and Wilfred Hodges (C&H) ... and am currently focused on Chapter 2: Informal Natural Deduction ... I need help with what C&H call the 'dandah' or more specifically symbols with a dandah through them ... The relevant text in C&H...
  13. Math Amateur

    Basic Mathematical Logic - Understanding Sequents

    I am reading the book Mathematical Logic by Ian Chiswell and Wilfred Hodges ... and am currently focused on Chapter 2: Informal Natural Deduction ... I need help with interpreting the notation of an aspect of Exercise 2.1.3 which reads as follows:In the above text after the text: "Possible...
  14. SrVishi

    Expansive Mathematical Logic Text

    Hello, I want to start learning mathematical logic. I was wondering what would be a good "expansive" mathematical logic book that covers as much material as possible. My school has books by both Ebinghaus (et. al) and Monk. Are these good? I've heard good things about Schoenfield, but I was...
  15. Byeonggon Lee

    Good and easy mathematical logic book for computer science

    Hello I recently noticed that mathematical logic is related to computer science. I haven't studied math in university yet I'm not good at math and Since I'm not a native English speaker some English is hard to me. Is there any good and easy book which describes mathematical logic used in...
  16. Byeonggon Lee

    Proof of (p ⇒ q) =(¬p ∨ q) in Mathematical Logic

    Hi :smile: I am studying mathematical logic by a pdf file. But there is no proof about this therorem so I don't understand.. How to prove this?
  17. W

    Please help me refine a dissertation topic in mathematical logic

    I'm looking to write a dissertation in the field of logic (for a philosophy degree). I'm deeply interested in logic, but whenever I consider the material beyond my courses it becomes pretty daunting. I'm reasonably familiar with: *First Order Logic *Set Theory and ZFC *Cantor's Diagonal...
  18. Vahsek

    Is it worth studying mathematical logic?

    I want to address this post primarily to people who have already studied mathematical logic, or are currently studying it. Since a while I've immersed myself in studying some math, and I must say that I started to enjoy pure mathematics. However, sometimes I don't really feel comfortable...
  19. R

    Mathematical Logic and Structures

    URGENT ! Mathematical Logic and Structures Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something. Part 1...
  20. R

    Mathematical Logic and Structures

    URGENT ! Mathematical Logic and Structures Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something. Part 1...
  21. T

    Which Implications are True in Mathematical Logic?

    Hi This is a question from my self study of ch.2 of Alfred Tarski's Introduction to Logic Which of the following implications are true from the perspective of mathematical logic? a) If a number x (assuming x is an integer) is divisible by 2 or by 6, then it is divisible by 12 b) if 18 is...
  22. C

    Mathematical logic and statistics

    Let's say that a treatment A has been proven to have an impact on the levels of B with a given confidence interval. Let's also say that we know that the treatment C causes the treatment A to be imposed on our sample. Before the testing on the effects of C has been done, which statistical models...
  23. K

    Using induction technique in mathematical logic

    Homework Statement Prove the following LEMMA: For every proposition A[P_{1}, \dots, P_{n}] and any two interpretations v and v', if v(P_{i})=v'(P_{i}) for all i=1, \dots,n, then v^{*}(A)=v'^{*}(A). Homework Equations The Attempt at a Solution Sure this is obviously an...
  24. C

    Mathematical logic & its branches

    What branches of mathematical logic are there? I've taken formal logic (that is, the logic where one has various operators, like conjunction, disjunction, etc). What other fields of logic should I take to become better at mathematical logic? Which fields of logic are the most useful and essential?
  25. H

    Best place to study mathematical logic uk

    I was wondering where the best place to study logic at postgraduate level in the UK. As far as I know it's Manchester but I could be wrong? Thanks for any help
  26. F

    Good Mathematical Logic Textbook?

    I've been trying to decide on a mathematical logic textbook to teach myself a bit. I'm taking a course on it next semester, but I have never had a logic course before (I've had some CS courses though and proof-y math courses). I'm also taking a modal logic course the semester after math logic...
  27. A

    Mathematical Logic, Interpretation, Satisfiable, Consequence relation

    Theorem : let A be a set of formulas, a be a formula For all A and all a, Every interpretation which is a model of A is also a model of a iff not (Sat A) U {~a} Proof Every interpretation which is a model of A is also a model of a iff(1) there is no interpretation which is a model of A but...
  28. K

    What are some recommended books on mathematical logic and set theory?

    Hello, I would like to know about a good introductory book on mathematical logic. It should start from set theory , include ZFC axioms and also touch on Godel's theorems.
  29. O

    Mathematical Logic by Cori and Lascar: Incomplete proof of Lemma 1.9?

    "Mathematical Logic" by Cori and Lascar: Incomplete proof of Lemma 1.9? I have a question on the book "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar. Proof of Lemma 1.9 given on...
  30. O

    Mathematical Logic by Cori and Lascar : Possible typo?

    I have a question on the textbook "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar. This is not about an exercise but about the conceptual content of the book. So I did not post this in the "Coursework and Homework questions"...
  31. J

    Why do some propositions defy logical proof?

    I recently read about the unexpected hanging problem and I was so surprised that logic actually failed in determining the solution!:( Is this just an isolated exception, or are there more paradoxes like this? And more importantly, why does logic fail? Isn't there any way around this? I just...
  32. I

    Undergraduate mathematical logic questions

    Hello I'm reading Y. Manin's http://books.google.co.il/books?id=8NTWRFD5lZ8C&printsec=frontcover&dq=yuri+manin+introduction+to+mathematical+logic&hl=en&ei=cfp_TJ2vJ8KSjAeB-6xl&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false" and I've been having some difficulties. So...
  33. M

    Mathematical logic- need some suggestion

    Mathematical logic-- need some suggestion I'm planning to start self-study of mathematical logic and axiomatic set theory. In fact I have already started and but facing a lot of problems to grasp the conception and formalism used there. After studying Hilbert's program and Godel 's...
  34. H

    How to Prove the Logical Implication from Church's Mathematical Logic?

    This from Alonzo Church's Mathematical Logic, been stuck on it for a week =(. Homework Statement 14.3 Present a Formal Proof: p \Rightarrow (q \Rightarrow r) \Rightarrow ((p \Rightarrow q) \Rightarrow r)Homework Equations The Attempt at a Solution A truth table has shown that the previous...
  35. B

    What are some recommended books on mathematical logic?

    Sorry to have two threads up at the top of the Science Book Discussion forum, but I couldn't find a thread for this. I'm interested in learning some mathematical logic. Here are the books I'm considering, please tell me what you think of them or suggest better alternatives. Mathematical Logic...
  36. S

    Ask for Mathematical Logic Solutions

    Homework Statement 1.Assume the language has equality and a two-place predicate symbol. Given two structures (N;<) and (R;<), find a sentence true in one structure and false in the other. Can these two structures be elementarily equivalent? Can they be isomorphic? Why or why not? 2.The...
  37. B

    Top PhD Programs in Mathematical Logic in the US

    I cannot find any sort of comprehensive list of top schools in mathematical logic. It seems that U Wisconsin should be good, though I don't know if that is the case now that Barwise has passed. UC Berkeley is clearly a good school for logic. All of the places that show up are in Europe. I would...
  38. R

    Discover the Best Books on Symbolic & Pure Mathematical Logic

    I'm trying to find a good book on Symbolic, Pure, Mathematical Logic. Anyone know such a book? I'd prefer it have no mention to number theory, set theory, etc. since I have books on those already and I find that they detract value from the books since it's just less time spent on the main topics.
  39. B

    Good books in Set theory and Mathematical Logic

    I am more precisely looking for a book on mathematical logic which presupposes only minimal exposure to set theory. Preferably something which includes an introductory chapter delineating relevant set theoretic principals. I am familiar with only basic set theory. More precisely this means...
  40. G

    Mathematica Mathematical Logic: For all and There exists

    Mathematical Logic: "For all" and "There exists" I need to show that \vdash (\forall x)(A \rightarrow (B \equiv C)) \rightarrow ((\forall x)(A \rightarrow B) \equiv (\forall x)(A \rightarrow C)) My question to you, how does the (\forall x) affect this equation? If they weren't there, I...
  41. W

    Mathematica Question about mathematical logic

    Is an assertion that could be true but isn't always considered to be false or just lacking information?
  42. E

    Mathematica Excellent mathematical logic books

    I am looking for excellent mathematical logic books that start from the beginning and go to the (what we think is) the end. Hopefully something with lots of editions so I can pick up a 2nd edition for 3 bucks on Amazon used. Ideas?
  43. H

    Mathematica Good book on mathematical logic

    What are your recommedations, all I could find in the local library is Introduction to Mathematical Logic by Mendelson, it is dated 1963, is that still ok? Is there any more state-of-the-art book on mathematical logic? I am interested in self-learning of mathematical logic. I would like to know...
  44. honestrosewater

    Mathematica Mathematical Logic by Joseph R. Shoenfield

    "Mathematical Logic" by Joseph R. Shoenfield I started reading this book an Amazon, and I can't stop. Has anyone else read it? Is there some reason I haven't seen anyone recommend it? -because it's absolutely amazing so far.
  45. C

    Mathematica What are some recommended books on mathematical logic and set theory?

    Hi everybody, I am looking for books about Logic and Set Theory. In particular, I am looking for not very advanced books. What are axioms, how do theorems connect to the axioms, how are we sure that some methods of proving give always correct and general results-these are some of the questions...