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Math Help - On a discusssion of the Peano axioms

  1. #1
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    On a discusssion of the Peano axioms

    Quote Originally Posted by Hartlw View Post
    so 1+1 = 1+1?

    You haven't defined the successor function. You only defined it for 1 and 2. What is your definiton of the successor function. All you have said so far is 2 is the successor of 1 by definition.

    1 and 2 is a notation convention for the natural numbers which is independent of the Peano Postulates.
    Having defined the natuaral numbers, you can then define a notation convention such as binary, decimal, hexadecimal, etc. But informally it's ok.

    Edit: To save some back and forth, my definition of the successor function for the natural numbers is S(n)=n+1. What's yours?
    Edit: Peano's potulates don't define addition. But if you want to assume it's been defined somewhere, informally that's ok.
    Correct, I *haven't* defined the successor function. There are various variants of the Peano axioms, which take different basic assumptions as starting points. This is *my* version:

    We start with a set K, such that 0 ∈ K. We also start with an INJECTIVE function S:K→K (we have no idea as yet what the function might do, or what its values specifically are).

    First limiting property of (K,S):

    0 is not an element of S(K) (in other words, S is not surjective in a specific way). Although we will not use this property much, it prevents K from being some sort of "cyclical" structure.

    Second limiting property of (K,S):

    If, for a subset T of K, we have: 0 is in T, and S(t) is in T whenever t is in T, then T = K (this means we can do induction on K by using S). (*)

    Clearly, we obviously have: {0,S(0),S(S(0)),.....} ⊆ K. But by (*) we have something much more: {0,S(0),S(S(0)),....} = K.

    We would like to say, of course, S(k) = k+1, but we haven't even said what "+" means, yet. Let's fix this:

    DEFINITION:

    We define a+b, for a,b in K as follows:

    a+0 = a
    a+S(k) = S(a+k). For example:

    S(S(S(0))) + S(S(0)) = S[S(S(S(0))) + S(0)] = S[S{S(S(S(0))) + 0}] = S(S(S(S(S(0))))).

    Let's take a moment to be sure we can actually evaluate this for any a,b in K. Since b is in K, we have either:

    1)b = 0, in which case we have a+0 = a, or:
    2)b = S(k), for some k, in which case we have a+b = a+S(k) = S(a+k).

    Of course, to evaluate THIS, we either have:

    1A) k = 0, in which case S(a+k) = S(a) in K
    2A) k = S(k'), in which case S(a+k) = S(a+S(k')) = S(S(a+k')).

    It seems "obvious" this process should terminate in a finite number of steps. Let's try to avoid this by using (*):

    Theorem: Let fa, for a given a in K be defined by: fa(b) = a+b. Then fa(b) is in K for all b in K.

    Proof: If b = 0, we have fa(b) = a+0 = a, which is in K. Suppose then fa(k) is in K.

    We have: fa(S(k)) = a+S(k) = S(a+k) = S(fa(k)), which is in K, since S is a function on K with S(K) ⊆ K. Thus by (*), fa is defined on all of K, and fa(b) ∈ K for all b in K.

    DEFINITION:

    1 = S(0).

    Now we can PROVE S(k) = k+1:

    k+1 = k+S(0) = S(k+0) = S(k).

    If we make the subsequent definitions:

    S(1) = 2, S(2) = 3, etc. we have:

    K = {0,S(0),S(S(0)),S(S(S(0))),...} = {0,1,S(1),S(S(1)),....} = {0,1,2,S(2),....} = {0,1,2,3,.....} =.....= N

    *****
    this "roundabout" way of explicitly defining S:N→N as S(n) = n+1 has the advantage of logically placing the functional concepts before the "operational" concepts (we really need functions before we can even say what a binary operation IS).

    The property (*) is KEY: and there is no point in trying to "prove" it, it's not provable (although it is "justifiable" by an appeal to ordinary objects, and our experience with counting things).

    (This treatment is based on a development given by Jacobson in Basic Algebra I, and the Wikipedia page on the Peano axioms).
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    Re: On a discusssion of the Peano axioms

    Hi Deveno,
    I first met the Peano axioms many years ago in Landau's book Foundations of Analysis -- I still believe this is about the best development of the reals from the Peano axioms. I remember a little problem glossed over by Landau. Namely recursive definition. Landau defines addition exactly as you do in your post.

    I think one needs a theorem that validates the recursive definition.

    Theorem. Let X be a set, x\in X and f:X\rightarrow X. Then there exists a unique function g:K\rightarrow X such that g(0)=x and \forall n\in K, g(S(n))=f(g(n)).

    The proof of uniqueness is trivial from the induction axiom, but the existence proof requires proving the intersection of all subsets T of K\times X with the property (0,x)\in T \text{ and for any n in K if }(n,y)\in T\text{, then }(S(n),f(y))\in T. is such a function g. This last statement is straight forward using the axioms.
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    Re: On a discusssion of the Peano axioms

    Deveno,

    Personally, I see no point in rewording the Peano Postulates as simply given in “Landau, Foundations of Analysis” unless it leads to something more, for ex, a proof that they are consistent, which Landau says can’t be done, or as an exercise in abstraction or alternate notation, in which case that is the topic of your post, not the Peano Axioms (Postulates).

    You write: “We would like to say, of course, S(k) = k+1, but we haven't even said what "+" means, yet. Let's fix this:”

    You don’t have to say what + is until you come to the definition of addition. k+1 simply stands for the value of a function of k. If you like, call it k*. So you see, there is nothing to fix.

    Then successive application of S gives the natural numbers sequentially in the form:
    1
    1+1
    1+1+1
    1+1+1+1
    .
    .
    Addition has nothing to do with this.
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    Re: On a discusssion of the Peano axioms

    If one regards +1 as simply a formal symbol, with no meaning attached to our (usual) defintion of +, sure. in fact, you can even omit the + signs and simply use:

    1
    11
    111

    (or any symbol you like, such as a vertical stroke (tally marks) or a letter such as A).

    These are all perfectly acceptable, as long as one is CLEAR as to what one is doing. Landau's treatment is fine, I have no quarrel with his level of sophistication at all (and I appreciate johng's commentary). It makes no real difference to me if we express:

    1+1 = 2 as:

    (0*)* = 0** or:

    A and A is AA

    or any of a myriad of equivalent ways of saying the same thing: what people usually mean when they say: "one orange and another orange make two oranges". There is not, in particular, any PREFERRED from of a natural number object, only a host of isomorphic ways of describing things that "all follow the same rules" (although the language in which we describe these rules may seem dissimilar).

    You asked me to define what I meant by successor. I tried to give a reasonably coherent answer. I have little doubt that what you meant, and what I meant are on some level, "the same". In fact, I would be very dismayed if they WEREN'T.

    On a different level, I suppose that philosophically, I am a structuralist...the exact construction of the natural numbers is unimportant to me, as long as they satisfy the Peano axioms. I am far more interested in mathematical VERBS, than in mathematical NOUNS. The action, the interplay is what creates the essential "identity" of some mathematical object for me, the clothes it wears are immaterial (I can think of at least half a dozen distinct concrete realizations of the klein 4-group, for example).

    I suspect (but I could be wrong, I admit this is conjecture on my part) you are more rooted in a more traditional and intuitive approach. There is nothing whatsoever wrong with this, it's largely a matter of style.

    Are these discussions pointless? Well, sure, we're not "changing the world" with them, but I, for one, actually enjoy these discussions, which is some small raison d'etre in and of itself.
    Last edited by Deveno; September 9th 2013 at 06:55 AM.
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    Re: On a discusssion of the Peano axioms

    The use of other conventions for the value of S has already been discussed. The cave-man analogy is obvious.

    The advantage of + is that it becomes meaningful when you define addition, with-out extra baggage.
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    Re: On a discusssion of the Peano axioms

    Personally, I like the characterization of the natural numbers as the free monoid generated by a one-element set, which carries very little baggage at all (not even a plus sign). Historically, the use of "+" and "1" as the "alphabet" to begin a description of the natural numbers is a mixture of the hindo-arabic numeral system and a shorthand for the Latin word "et" (and), and is rooted in our cultural traditional way of expressing mathematics. While there is something to be said for embedding something of our cultural history in our symbolism, it is by no means necessary, nor even desirable in discussing these concepts on, say, a global level (or perhaps, fantastically, with intelligent aliens we may someday meet).

    But I want to emphasize an additional point:

    The statement "1+1 = 2" CAN be viewed, in a certain light as: "true by definition". As someone famously remarked: it depends on what you mean by "1", "+", "=" and "2". Part of the motivation behind Peano developing his axioms, was to put them on a basis of (second-order) logic (although one can use first-order logic if one uses an axiom schema for the axiom of induction). Arithmetic, then, comes LATER. If YOU prefer to "jump ahead" to the fore-shadowing of the later introduction of addition by using "+1" as a (at first) purely symbolic denotation of succesorship, fine. I do not. Although my reasons may not seem amenable to you, I'm perfectly comfortable with them.
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    Re: On a discusssion of the Peano axioms

    "1+1=2" has nothing to do with the Peano axioms.

    It is part of a defined identification scheme for the natural numbers, one of many, binary for example, which is another topic.

    2 never appears in Landaus book, only "2," ie, as part of an undefined symbol scheme for axioms, definitions, theorems, subscripts, and page numbers.
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    Re: On a discusssion of the Peano axioms

    Wolfram Math World gives the Peano Axioms as:

    1. Zero is a number.

    2. If a is a number, the successor of a is a number.

    3. zero is not the successor of a number.

    4. Two numbers of which the successors are equal are themselves equal.

    5. (induction axiom.) If a set of numbers S contains zero and also the successor of every number in
    S, then every number is in S.

    That suits me fine, it's all I can do to remember that. To each his own. I always wished I had a better memory.

    I note above starts with 0 while Landau starts with 1.
    Last edited by Hartlw; September 9th 2013 at 09:22 AM. Reason: copy fro mweb missed something.
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    Re: On a discusssion of the Peano axioms

    Quote Originally Posted by Hartlw View Post
    Wolfram Math World gives the Peano Axioms as:

    1. Zero is a number. (This is equivalent to my stipulation that 0 is in K).

    2. If a is a number, the successor of a is a number. (This is equivalent to saying that if a is in K, S(a) is in K).

    3. zero is not the successor of a number. (Limiting property #1 of (K,S)).

    4. Two numbers of which the successors are equal are themselves equal. (Equivalent to the injectiveness of S).

    5. (induction axiom.) If a set of numbers S contains zero and also the successor of every number in
    S, then every number is in S. (Equivalent to limiting property #2).

    That suits me fine, it's all I can do to remember that. To each his own. I always wished I had a better memory.

    I note above starts with 0 while Landau starts with 1.
    One can always create a monoid by starting with a semi-group and "adjoining an identity". It's not a big deal (the basic idea here is that 0 "doesn't do anything", it's pretty non-number-ish for a number).

    While I denoted my function by S (to stand for "successor"), one need not use the term successor at all, nor the term "number". From a logical standpoint, it is preferrable to me to be able to do this...one should in mathematics try to avoid using undefined terms as much as possible. The editors at Wolfram Math World may feel that "what a number is" is "well-known", but actually, defining "number" is sort of hard to do. Similarly, the notion of "successor" invokes an intuition of a (linear) order, again, at this stage of the game, an undefined concept. The concept of sets and functions are, for most of mathematics, more basic and "primitive" notions.

    While in our actual lives, we tend to start with the concrete and move towards the abstract (induction itself (in both the mathematical sense, and the usual English sense) is a perfect example of this), in mathematics, it often is more streamlined to do the opposite: start with very general limiting principles and "instantiate" them to specific applications. For example, a knowledge of Abstract Algebra makes Linear Algebra much easier, which in turn makes Vector Calculus much easier (although these are usually taught in the reverse order).

    Ironically, set theory and mathematical logic, the very underpinnings of all this other stuff, are typically reserved for upper-level undergraduate, or even graduate level study, meaning that for the vast majority of math students (and almost ALL non-mathematicians who employ math) mathematics is something taken purely on faith, or the authority of others. How ionic is it that Landau (for example), who writes for a very sophisticated audience, proves things we have used since we were small children?
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    0, zero

    Looks lik louy vitone is intent on shurring down the site. Anyway, in reply to part of your last post:

    You write “the basic idea here is that 0 "doesn't do anything", it's pretty non-number-ish for a number”
    0 is important conceptually (nothing) in terms of what a number represents, and essential for identifying numbers.
    You need a way to identify 1,11,111,1111,…, the basic number system since the cave man, for large numbers. For ex, for the decimal number system:
    Assign names (symbols) as follows:
    0,1,11,……,111111111: 0,1,2,3,4,5,6,7,8,9
    Successor of 9: 10
    The next 9 (now defined) numbers: 11,12,13,……….19
    Successor of 19: 20
    The next 9 numbers: 21,22,23………..29
    Successor of 29: 30
    The next 9 numbers: 31,32,33,…….39
    etc.
    You don’t need Peano axioms, or abstract rewording of them, to do this. The basic concept of number and successor resides in, and is crystal clear from, 1,11,111,.. since the cave people. Even 0 (I have no stones). It can be taught in kindergarden.
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    Re: On a discusssion of the Peano axioms

    At one point I asked myself, what are the foundations of mathematics. I bought some books on Logic, Set theory, and Functional Analysis, futileley broke my teeth on them, and found them to be absolutely useless to me for understanding mathematics, or anything else for that matter. I can hardly remember any of it.

    To call “this page intentionally left blank” a paradox is oviously nonsense, it is simply the result of economy of lanquage to save the obvious “except for this sentence.” It is akin to “the set of all sets” which permeates all of “logic.”

    If a group of people decided to create a club for themselves in which they created symbols and a set of rules for manipulating them, and then manipulated them, I could care less, and might be curious as to what they are doing.

    It only becomes insidious as a weapon of intimidation and denigration (politics). To introduce mathematical logic and abstract mathematics to science college students becomes a filter to sort out students with the ability to think from those with the ability to memorize and accept without thinking. “But what does it mean?” is the first step to dropping out. I have known many intelligent promising young Engineers to whom that has happened.

    EDIT: In a sense it's like responding to a post in Latin when English works fine and then suggesting you learn Latin.
    Last edited by Hartlw; September 10th 2013 at 07:37 AM.
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