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Math Help - ZFC and the foundation of mathematics

  1. #1
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    ZFC and the foundation of mathematics

    Today I've been studying ZFC:

    Zermelo

    and I have one particular issue I'm not clear on. It seems that ZFC by itself only allows for sets as the mathematical objects that exist. On the other hand, it is said that ZFC can serve as a foundation for most of mathematics. I'm curious, how then can ZFC be used to talk about say, Calculus and the real numbers?

    I have read a bit about "ur-element" but they seem to be peripheral and not central to ZFC itself. Could somebody explain?
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    It seems that ZFC by itself only allows for sets as the mathematical objects that exist... I'm curious, how then can ZFC be used to talk about say, Calculus and the real numbers?
    I am not sure about the first sentence. What do you mean when you say that a mathematical object exists? Specifically, do you mean that real numbers don't exist?
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Ontolog View Post
    Today I've been studying ZFC:

    Zermelo

    and I have one particular issue I'm not clear on. It seems that ZFC by itself only allows for sets as the mathematical objects that exist. On the other hand, it is said that ZFC can serve as a foundation for most of mathematics. I'm curious, how then can ZFC be used to talk about say, Calculus and the real numbers?

    I have read a bit about "ur-element" but they seem to be peripheral and not central to ZFC itself. Could somebody explain?
    You can construct the natural numbers in a relatively easy way using ZFC,

    0:=\{\}
    1:=\{0\}
    2:=\{0, 1\}
    3:=\{0, 1, 2\}
    etc.

    With \mathbb{N} defined to be the smallest set containing the the empty set, and closed under the operation n\cup\{n\}.

    Does that make sense?

    One can define the real numbers in a much more hideous way - look up Dedekind cuts if you want to learn more...

    So one can define numbers using set theory. However, as Goedel pointed out, such a formal system cannot prove all true statements about numbers.

    Anyway, to calculus! Calculus is, in some ways, the study of functions between sets. However, one can define ZFC using only functions between sets! That is, there is a set of axioms which can be derived from ZFC, and they in turn derive ZFC, and all the axioms talk about functions between sets.

    I can't actually find a link to this set of axioms, because I can't remember what they're called. I have it written down somewhere, so I'll try and find it...

    An example, however, would be the axiom of choice (given some (non-empty) sweety jars, you can pick a sweet from each jar). The following are equivalent,

    -For every family of (nonempty) sets (S_i)_{i \in I} there exists a family (x_i)_{i\in I} of elements with x_i\in S_i for i\in I.

    -Every surjective function has a right inverse.

    EDIT: This is what I was looking for - it is called `The Elementary Theory of the Category of Sets', by F. William Lawvere (who was tragically born with only a first initial).

    You have things called `sets', things called `functions', and you can compose functions. The axioms are,

    -compositions is associative, unital and has identities
    -there exists a set with exactly one element
    -there exists a set with no elements
    -a function is determined by its effect on elements (f(a)=g(a) for all a then f=g)
    -can form cross products of sets, AxB
    -we can form the set of functions from A to B
    -can form the inverse image of a function
    -given a set A, the subsets correspond to functions from A into {0, 1}
    -the natural numbers form a set
    -every surjection has a right-inverse
    -cocompleteness (don't ask).

    So, basically, functions `work'; they make sense. So studying calculus makes sense, and is possible.
    Last edited by Swlabr; May 12th 2011 at 04:35 AM.
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    anytime you make a statement like: let A = {x in R: x > 0}, you are making an implicit appeal to ZF(C) that A is well-defined and unambiguous.

    that is, you are saying the the property P(x) (in this case P = "greater than 0") applies to some subset of R (which might be all of R), and A is the (unique)

    set defined by (P(x)) & (x∈R) (i think this is the axiom of (restricted) comprehension, with uniqueness provided by the axiom of extensionality).

    if one shows later, that A has the same members as the open interval (0,∞), another appeal to ZF(C) lets us say A = (0,∞).

    so ZF(C) is used all the time in Calculus, without it being all that obvious. an example might be a problem like so:

    "find all critical points of f(x) = x^3 - x."

    well that is some set or another (it might be the empty set if f has no critical points), and the task is usually to find an explicit

    description of such a set in terms of actual real numbers x.

    the ZFC axioms basically give us "enough sets to work with": the natural numbers, finite sets of specific objects, unions, intersections, cartesian products.

    from these, we can build other sets which are often the things we're really interested in. for example, we can define a function as:

    f = {(a,b) in AxB : (a1,b1) & (a1,b2) in f --> b1 = b2} (more formally, and less circularly, a relation ~ on AxB such that (a1,b1) ~ (a1,b2) iff b1 = b2).

    the unique b1 corresponding to a1 is usually denoted f(a1). note this says nothing about pairs (of pairs) (a1,b1) and (a2,b1), which may, or may not exist.

    one sees various notations for a function: f:A-->B, a-->b, a-->f(a), etc., but the key notion is that a function is defined by its graph (and not, as one often

    sees people mistakenly think in calculus, by its range of values. this is why it's not such a good idea to talk about "the function x^2", although it is a common abuse).
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  5. #5
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    Quote Originally Posted by Swlabr View Post
    You can construct the natural numbers in a relatively easy way using ZFC,

    0:=\{\}
    1:=\{0\}
    2:=\{0, 1\}
    3:=\{0, 1, 2\}
    etc.

    With \mathbb{N} defined to be the smallest set containing the the empty set, and closed under the operation n\cup\{n\}.

    Does that make sense?

    One can define the real numbers in a much more hideous way - look up Dedekind cuts if you want to learn more...
    Yes, that makes sense on a certain level. There are infinitely many sets so you can come up with ways to map them to numbers. But what does this achieve, exactly? I haven't taken a course on it but I think it's 'Abstract Algebra' that explains elementary things about numbers like why 1 + 1 = 2 and why we can be sure that if we add any two numbers that we will get another number as a result. If we map these numbers to sets, and we then go on to derive properties about these numbers, aren't we really deriving properties about sets (or the mapping of sets)?

    I think this is mentioned in the article you linked me to:

    Benacerraf wanted numbers to be elements of abstract structures which differ from ZF sets this way:

    in giving the properties (that is, necessary and sufficient) of numbers you
    merely characterize an abstract structure—and the distinction lies in the fact
    that the “elements” of the structure have no properties other than those relating
    them to other “elements” of the same structure. (Benacerraf 1965, p.70)
    I've only read the introductory chapters of a book on Category Theory ("Conceptual Mathematics", 2nd Ed., Lawvere & Schanuel). (Just noticed it's the same guy who wrote the article you linked me to! lol). But it seems so far that Category Theory makes more sense to me intuitively than Set Theory.

    Quote Originally Posted by Swlabr View Post
    So one can define numbers using set theory. However, as Goedel pointed out, such a formal system cannot prove all true statements about numbers.

    Anyway, to calculus! Calculus is, in some ways, the study of functions between sets. However, one can define ZFC using only functions between sets! That is, there is a set of axioms which can be derived from ZFC, and they in turn derive ZFC, and all the axioms talk about functions between sets.

    I can't actually find a link to this set of axioms, because I can't remember what they're called. I have it written down somewhere, so I'll try and find it...

    An example, however, would be the axiom of choice (given some (non-empty) sweety jars, you can pick a sweet from each jar). The following are equivalent,

    -For every family of (nonempty) sets (S_i)_{i \in I} there exists a family (x_i)_{i\in I} of elements with x_i\in S_i for i\in I.

    -Every surjective function has a right inverse.
    I'de like to say "I see" but it looks like I'll have to study a bit more.

    Quote Originally Posted by Swlabr View Post
    EDIT: This is what I was looking for - it is called `The Elementary Theory of the Category of Sets', by F. William Lawvere (who was tragically born with only a first initial).

    You have things called `sets', things called `functions', and you can compose functions. The axioms are,

    -compositions is associative, unital and has identities
    -there exists a set with exactly one element
    -there exists a set with no elements
    -a function is determined by its effect on elements (f(a)=g(a) for all a then f=g)
    -can form cross products of sets, AxB
    -we can form the set of functions from A to B
    -can form the inverse image of a function
    -given a set A, the subsets correspond to functions from A into {0, 1}
    -the natural numbers form a set
    -every surjection has a right-inverse
    -cocompleteness (don't ask).

    So, basically, functions `work'; they make sense. So studying calculus makes sense, and is possible.
    Am I correct in saying that the "Category of Sets" differs from ZFC in that elements of a set in the Categorical sense do not have to be themselves sets, while in ZFC all elements of sets are also sets? If so, do you think this distinction is important?
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  6. #6
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    But it seems so far that Category Theory makes more sense to me intuitively than Set Theory.
    You must be really smart... Like one of those people who think that the fifth Euclid's postulate is nonobvious or that the law of excluded middle is problematic. It's not sarcasm, either.

    Am I correct in saying that the "Category of Sets" differs from ZFC in that elements of a set in the Categorical sense do not have to be themselves sets, while in ZFC all elements of sets are also sets? If so, do you think this distinction is important?
    In the category of sets, one does not have elements, only sets and functions between them. An element of A is defined as a function from a singleton to A. A singleton is in turn defined as a set such that every set has exactly one function into it.
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    Quote Originally Posted by emakarov View Post
    You must be really smart... Like one of those people who think that the fifth Euclid's postulate is nonobvious or that the law of excluded middle is problematic. It's not sarcasm, either.
    Absolutely TRUE.

    Some twenty years ago I was part of NHF summer group on the philosophy of mathematics with Colin McLarty. He has a well respected book on Category Theory
    But I never got the point.
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    Quote Originally Posted by Swlabr View Post

    EDIT: This is what I was looking for - it is called `The Elementary Theory of the Category of Sets', by F. William Lawvere (who was tragically born with only a first initial).

    You have things called `sets', things called `functions', and you can compose functions. The axioms are,

    -compositions is associative, unital and has identities
    -there exists a set with exactly one element
    -there exists a set with no elements
    -a function is determined by its effect on elements (f(a)=g(a) for all a then f=g)
    -can form cross products of sets, AxB
    -we can form the set of functions from A to B
    -can form the inverse image of a function
    -given a set A, the subsets correspond to functions from A into {0, 1}
    -the natural numbers form a set
    -every surjection has a right-inverse
    -cocompleteness (don't ask).
    i'm sorry, i couldn't resist. cocompleteness. explain.
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    When I took a course in Category theory, I was told that Category theory was also known as "Abstract Nonsense". But the teacher had an unfortunate sense of humor. He also maintained that general Category theory required "classes" rather than sets. If you used only sets, that was "Kittygory" theory.
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    well, the atomic notions of a category (objects, arrows, identity and composition) makes sense if one takes objects to be categories, and arrows to be functors. but one can only define this "locally" because the category of all categories runs into the same logical dilemma as the set of all sets. in fact, one need classes to even talk about the category Set, because the collection of all sets is not a set. and the collection of all classes (which is the type of thing most "large" categories are) is "too big" to even be a class (and the category Cat is "bigger than that"). so one starts looking at "local bits" of categories (in particular, hom-sets like Hom_C(A,B), for categories "small" enough for these to be sets).

    something like "restricted comprehension" rears its ugly head no matter where we go. apparently, the word "everything" is insufficient to describe everything.
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Deveno View Post
    i'm sorry, i couldn't resist. cocompleteness. explain.
    So, forgot I mentioned cocompleteness. Then what you have would be weaker than ZFC. However, it would be weaker in a silly sense - you can't construct, the disjoint union,

    \mathbb{N} \coprod \mathcal{P}(\mathbb{N}) \coprod \mathcal{P}(\mathcal{P}(\mathbb{N}))\coprod\ldots

    which is something most people wouldn't really want to do...

    Cocompleteness makes your theories equivalent. However, I said `don't ask' because I amn't actually sure myself...according to Wiki, "A category C is complete if every diagram from a small category to C has a limit; it is cocomplete if every such functor has a colimit". Which makes sense. So I suppose that is your answer...
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    Quote Originally Posted by Deveno View Post
    one need classes to even talk about the category Set, because the collection of all sets is not a set.
    You mean, I gather, that we need proper classes to prove the existence of the category of sets. I'm not well versed in category theory, but am I not correct that the category of sets doesn't depend on having a set of all sets? I thought that when we add an axiom "every set is a member of a nonempty Grothendieck universe" to ZF we do get category theory and, with it, the category of sets. Also, that we can instead get all that by adding to ZF an axiom "every cardinal is in an inaccessible cardinal".

    I'm interested in what particular book or article you find that proper classes are required for the category of sets.
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    it depends on whether or not you want to take categories as the building blocks, or sets as the building blocks. if you want to consider categories as something built out of structures we already have at hand, then you need something "bigger than sets" to describe what Set IS. not every text on categories takes this approach. using Grothendieck universes is a viable alternative.

    this uderscores the fact that there is "more than one flavor of category theory" just as there is more than "one flavor of set theory".

    not every mathematican accepts Grothendieck universes, not every mathematician accepts the hierarchy of sets, classes, ensembles, etc., and not every mathematician accepts that "strongly inaccessible cardinals exist".

    @Swlabr: yes, i read about co-completeness, and co-limits. i'm still a bit vague about what this MEANS, on a level i can relate to. from what i gather, a category is co-complete, if it has pushouts and an initial object, or if it has co-equalizers, co-products and an initial object. and what this appears to mean to me, is that we want to be able to form disjoint unions, and refinements of equivalence partitions.
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    Quote Originally Posted by HallsofIvy View Post
    When I took a course in Category theory, I was told that Category theory was also known as "Abstract Nonsense". But the teacher had an unfortunate sense of humor.
    Some theories are so general that have no particular cases.
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    Quote Originally Posted by Deveno View Post
    if you want to consider categories as something built out of structures we already have at hand, then you need something "bigger than sets" to describe what Set IS.
    I don't know what your criterion is for "already at hand", so I still don't see why you cliam we must have something "bigger than" sets. With an axiom (that, as far as I know, is consistent with ZF) that there exists a certain kind of set (viz. a Grothendieck universe), we obtain the category of sets, right? So nothing "bigger than" sets is involved, right? So we don't need to have the existence of proper classes to have the category set, but rather it suffices to have the existence of a certain kind of set.
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