Notices
Results 1 to 7 of 7

Thread: Quick question...

  1. #1 Quick question... 
    Forum Junior epidecus's Avatar
    Join Date
    Aug 2012
    Posts
    268
    How does one go about rigorously proving that the sum of two integers is an integer?

    The answer is likely simple, but I ask this because I have a feeling the subject is axiomatic in nature.


    Dis muthufukka go hard. -Quote
    Reply With Quote  
     

  2.  
     

  3. #2  
    Forum Professor
    Join Date
    Jul 2008
    Location
    New York State
    Posts
    1,238
    Start with a precise definition of integer.


    Reply With Quote  
     

  4. #3  
    Forum Junior epidecus's Avatar
    Join Date
    Aug 2012
    Posts
    268
    Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

    Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?
    Dis muthufukka go hard. -Quote
    Reply With Quote  
     

  5. #4  
    Forum Professor
    Join Date
    Jul 2008
    Location
    New York State
    Posts
    1,238
    Quote Originally Posted by epidecus View Post
    Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

    Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?
    Now you need a precise definition of natural number.
    Reply With Quote  
     

  6. #5  
    Forum Professor river_rat's Avatar
    Join Date
    Jun 2006
    Location
    South Africa
    Posts
    1,517
    Hi epidicus

    The way the integers are defined axiomatically is that they are closed under all the basic arithmetic operations. So there is nothing to prove in that space, as if you have something that isn't closed under addition it can't be the integers.

    If you want to prove that something like the integers actually is constructable in set theory, you have to start by defining the natural numbers as some set construction and then slowly pulling yourself up the ladder, showing at each step that all the axioms we want to hold are true.

    Assuming you have defined the natural numbers , one way of constructing the integers is as an equivalence class on where if .
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
    Reply With Quote  
     

  7. #6  
    Forum Junior epidecus's Avatar
    Join Date
    Aug 2012
    Posts
    268
    Quote Originally Posted by mathman View Post
    Quote Originally Posted by epidecus View Post
    Um... I can't really find a mathematically-precise definition via search engine, as all of the ones I find are by intuitive approach, even in Wikipedia.

    Is it fair to say that an integer is a number that is either a natural number (starting at 0) or its negative?
    Now you need a precise definition of natural number.
    And this is where the axioms come in, I think. I'm gonna have to get used to all this axiological proofing.
    Dis muthufukka go hard. -Quote
    Reply With Quote  
     

  8. #7  
    Forum Junior epidecus's Avatar
    Join Date
    Aug 2012
    Posts
    268
    Quote Originally Posted by river_rat View Post
    Hi epidicus

    The way the integers are defined axiomatically is that they are closed under all the basic arithmetic operations. So there is nothing to prove in that space, as if you have something that isn't closed under addition it can't be the integers.
    Ah, thanks. I found "axiom of closure". So if I ever have to prove that the sum of two integers is an integer, I can just cite the axiom of closure (which I guess, as you said, wouldn't really be "proving").

    What I find confusing is this... Several axioms, including this one, don't really sound like axioms to me, at least not at first. Not that I'm doubting the system in anyway; I know this is just how it is. But I can't really get myself to understand why. To say "the set of integers is defined to be closed under these operations" sounds more like a property that you would observe after defining the integers themselves. In other words, I'm taking this more like an observational postulate, rather than a laid-out axiom. But it's probably just because I'm not that well-versed at all in the foundational mathematics. Do you get what I'm saying?
    Dis muthufukka go hard. -Quote
    Reply With Quote  
     

Similar Threads

  1. Just a quick question. PLEASE HELP!!
    By DAComputerScience in forum Computer Science
    Replies: 4
    Last Post: March 15th, 2012, 03:47 PM
  2. quick question...
    By haim876 in forum Chemistry
    Replies: 3
    Last Post: December 15th, 2009, 10:16 AM
  3. quick question
    By Raymond K in forum Chemistry
    Replies: 4
    Last Post: August 4th, 2008, 11:46 PM
  4. Quick Question?
    By ajg624 in forum Astronomy & Cosmology
    Replies: 5
    Last Post: April 3rd, 2008, 04:30 AM
  5. just a quick question on books
    By melting_clocks in forum Astronomy & Cosmology
    Replies: 3
    Last Post: March 6th, 2008, 08:36 PM
Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •