Arbitrary definition [closed]

I find it most perplexing for how the definition of say, a set, defines itself. How does it become tangible? Where does it come from? Because, arbitrary definition seems more so like just saying "it just is", or "let it be so" without any axiom to ground it from which it may become literal.

Apologies, I don't know a lot of Math (and I'm not good at reading it either, so algebra eludes me visually) but it's troubled me of recent with regards to defining "something". All I keep finding is that "something", however arbitrary and abstract it may be, invariably requires an axiom to connect it with something else from which it may become real (literal).

I hope this question makes sense. I mean -- from what I understand, there should be an all defining axiom from which anything may occur (perhaps which one may find by traversing through other axioms, so on and so forth). Perhaps likewise for numbers too I suppose, I mean I assume that for them to simply be - for abstract quantities to be - that there must be some defining axiom, from which numbers just become.

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I’m afraid that I can’t make much sense of this. Whatever content it has appears to be philosophical rather than mathematical. –  Brian M. Scott Feb 7 '13 at 22:50
Well, I thought Math was regarding the means by which one defines 'something'. I was reading about sets before and it troubled me of how something can possibly define itself, and any subsequent application of it. I mean -- where does that definition come from? I assume that it must be from a prior axiom which thereby enables it to be, and then that too seems to require subsequent axioms from which it may be. Thus, I assume that there must be an all defining axiom. –  user784446 Feb 7 '13 at 22:52
@user784446 Are you aware of what a primitive concept is? –  Git Gud Feb 7 '13 at 22:53
I find the definition of "apple" most perplexing. I look it up in the dictionary, and it's defined in terms of other words, which I look up, and they're defined in terms of other words, which I look up, and so on, and pretty soon I'm back to some of the words I already looked up. Where is the ground for the definition of "apple"? When you figure that one out, you'll be ready to tackle "set". –  Gerry Myerson Feb 7 '13 at 22:57
I don't get why this question is closed. It is natural for beginners to be confused about how the notion of definition is used, when one only knows the kind of "defnitions" found in lexika. Also, the concept of implict definitions can be even more subtle in the beginning. –  Michael Greinecker Feb 8 '13 at 0:47

closed as not a real question by Brian M. Scott, Ittay Weiss, Micah, Peter Smith, 5PM Feb 7 '13 at 23:39

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Sets don't define themselves.

We have an idea of what properties sets should have. For example sets with similar elements should be equal. We formalize these properties into the form of axioms, and we declare the collection of axioms a "theory", and since it's about sets we call it set theory.

Sets, if so, are mysterious objects which are the elements in the universe of set theory. Which set theory? It could be one of many.

Are sets real? In what sense mathematical objects are real? Sure you can hold "one" stone, or drink "five" beers. But it's not really "one" or "five", it's a realization of the abstract notion of $1$ and $5$. Mathematical objects don't always have such realizations, in fact as we can only perceive finite information, it means that almost all the natural numbers cannot be realized.

But mathematics is not about things we can perceive in real life, it's about mathematical objects, which may or may not exist somewhere. These are ideal objects which have particular rules to which they are subject in a most perfect way.

Indeed it is sometimes the case that we thought that some properties which describe a certain object turn out to be inconsistent, for example if we allow unrestricted comprehension then we run into the problem of having a self-contradictory "set of all sets which do not contain themselves". So clearly sometimes we go too far in our attempts to describe what sets are, but this is good because it tells us something about mathematical objects, wherever they might be. For example it tells us that not every collection can be a set. But I digress.

The main point, if so, is that mathematical objects are ideal objects living in the mathematical universe which may or may not exist - but we certainly can't perceive it. There is no circularity in the definition of sets, we just described properties and decided that objects which obey these rules are fit to be called "sets".

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I appreciate the notion of sets. It's been most helpful with understanding other topics and various problems. However, I certainly think a "set of all sets" can be valid - a set containing itself, which contains all others, containing itself. The trouble of this however seems more so tantamount to the need for it to be. Thus, something must always reside outside the box, enabling it to be (i.e. such that the box isn't a box without an outside). –  user784446 Feb 7 '13 at 23:28
@user784446: You said that you are not a mathematician, let me assure you that whatever intuition you have is almost certainly missing a lot of fine points. Mathematical intuition is very different, especially with regards to infinite objects. It is possible to have set of all sets, but this requires a very delicate handling in how we do this. For example, in the common set theory ZFC, this is flat out impossible. –  Asaf Karagila Feb 7 '13 at 23:34
I appreciate your response. This is what I've been searching for, whatever else is necessary for comprehension. What I mean is, well... with regards to what you've said, what is the exception of Math amidst logic? I.e. how does Math differ from logic? How does it differ from philosophy? All require the axiom of thought to be, yet they are somehow different. Does one not at the very least give birth to the rest? –  user784446 Feb 7 '13 at 23:40
Well, this is really going beyond the scope of what this site is about, and what the comments are for. It also slides deeply into philosophy, and I have neither the powers or the time to spread my entire philosophical approach to mathematics (which is saturated with my own personal beliefs about life, existence, and reality). As I wrote in that other comment, you will have to study mathematics for yourself for a few years and then come by that answer on your own. –  Asaf Karagila Feb 7 '13 at 23:50
okay, but if we could discuss that via some other channel then I would be most appreciative of your response. I'll invariably need to learn more Math, but intuition seems to be the means by which it becomes. –  user784446 Feb 7 '13 at 23:57