ZFC sets.

[Sorcerer]

From wiki: "ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes."

 

I was running through possible sets without subsets and started to think of sets containing only explicitly unique objects. The definition of an explicitly unique object being an object which does not share the same value of its properties with any other object.

 

I then thought about quantifiable properties which define objects uniqueness and eliminated the explicitly unique objects with infinite quantifiable properties from the set and then again eliminated all but the set of explicitly unique objects with only 1 quantifiable property. I then specified a quantifiable property, grams. All other possible quantifiable properties have a value of 0.

 

I then considered the size of the set of all explicitly unique objects with 1 quantifiable property, grams, which weighs 1 gram. This set can only contain 1 object. Let's call it #1.

 

What is its subset? The quote seemed to say ZFC insists it has one.

 

I considered it might contain an empty set as a subset, but there is no quantifiable property of an empty set. Is 0 a quantity? A set lacking objects cannot also have an object with the property of weighing 1 gram. An empty set has no object to weigh one gram.

 

If absence of something 0 is a quantity and all properties are said to be a definition of an object, but with 0 value. Then there can be an infinite amount of empty sets but none are explicity unique, 0 = 0. If an empty set can be defined by no quantity with no properties, we can then add to the definition of #1 having all other possibilities for properties, but with 0 unit value. Comparing 2 or more empty sets shows they all have 0 quantity and 0 value for all properties, thus all identical and not explicity unique, so unable to be the subset.

 

If however 0 value properties make no sense, my definition of the set of unique objects doesn't allow for the empty set to be included as it has no quantifiable property.

 

Is there no subset of this set because it isn't definable using first order logic? Is this set an indivisible unit, part of an infinite series of concentric sets?

 

The problem seems to lie with the dependence of the existence of a quantifiable property on the existence of an object. By insisting objects to be unique it limits the options to only 1 possible object.

 

I can't quite see an error. Is only allowing positive integers restricting ZFC?

I'm off to read more, hopefully it's explained later.

Thinking of fractional subsets and allowing them as objects, how can half of #1 also weigh 1 gram?

Edited February 5, 2016 by Sorcerer

[wtf]

Your ideas are a little tangled up, it would be more confusing than enlightening to reply to your post line by line.

 

Instead, give this a read and see if it helps.

 

http://www.scienceforums.net/topic/92101-what-is-the-minimum-number-of-properties-posessed-by-members-of-a-set/?p=900416

 

All Wiki is saying is that you can't define a set by a predicate alone. Otherwise we'll just use the predicate [math]x \notin x[/math] and derive a contradiction.

 

Rather, you have to start with some existing set, and then apply a predicate to restrict the elements of that set. So in fact we CAN form the set of all real numbers that are not members of themselves. That's just the set of real numbers. No contradiction! (Convince yourself that what I said is true?).

 

The empty set is characterized by the predicate [math]x \neq x[/math]. The empty set is a subset of every set, as you should prove for yourself from the definition in order to clarify your understanding.

 

Grams have nothing at all to do with any of this. There are no sets in the real world. Sets are a purely formal exercise in abstract math. A collection of five apples is not a set in ZFC. ZFC only contains pure sets; that is, sets whose elements are themselves sets.

 

I don't know what you mean by "quantifiable property," but I think you are kind of making that up to try to understand predicates. A predicate is just a sentence with a free variable that has a truth value when the free variable is bound to a constant. In other words if T(x) means "x is tall," then T(Lebron James) is true and T(Tom Thumb) is false.

 

In ZFC you can NOT form the set of all tall things. You CAN form the set of all tall things in some already existing set. (Of course I'm stretching a point here since I just said things in the real world aren't sets.) The point is simply that you can't form a set from a predicate alone; you have to start from an existing set.

 

I was running through possible sets without subsets

There is no such thing. The empty set is a subset of every set.

Edited February 6, 2016 by wtf

[Xerxes]

The point is simply that you can't form a set from a predicate alone; you have to start from an existing set.

Yes. I denied this in another thread, so here is the simple argument I used to convince myself I was wrong.

 

Suppose a proposition [math]P(x)[/math] and that the set [math]A[/math] is such that [math]P(x)[/math] is true for all [math]x\in A[/math].

 

Then unless our proposition is vacuous, there must be some other set for which the proposition is false. Call this set the complement [math]A^c[/math].

 

Then there must exist a third set [math]S=A \cup A^c[/math].

 

But since [math]A[/math] is entirely arbitrary, unless I specify in advance that [math]S \supsetneq A[/math] is well-defined, I may have that [math]A^c = U \setminus A \Rightarrow S=U[/math], the universal "set" whose existence we all abhor as much as Bertie did

 

The empty set is a subset of every set.

Moreover, like any other set,it is a subset of itself! Edited February 7, 2016 by Xerxes

[Sorcerer]

Yes. I denied this in another thread, so here is the simple argument I used to convince myself I was wrong.

 

Suppose a proposition [math]P(x)[/math] and that the set [math]A[/math] is such that [math]P(x)[/math] is true for all [math]x\in A[/math].

 

Then unless our proposition is vacuous, there must be some other set for which the proposition is false. Call this set the complement [math]A^c[/math].

 

Then there must exist a third set [math]S=A \cup A^c[/math].

 

But since [math]A[/math] is entirely arbitrary, unless I specify in advance that [math]S \supsetneq A[/math] is well-defined, I may have that [math]A^c = U \setminus A \Rightarrow S=U[/math], the universal "set" whose existence we all abhor as much as Bertie did

 

Moreover, like any other set,it is a subset of itself!

I'm going to have to learn that language of logic one day. However I can refute that with words only. The set which contains every possible set contains itself and is not contained in any other set.

 

However because it is the ultimate set doesn't mean it is defined purely on predicates, it is actually the set which is reliant on the most sets in order to form its definition.

 

So since that's not right, how about the opposite, the empty set?

 

Well in order for the empty set to be meaningful it is necessary to exist atleast one non empty set.

Edited February 11, 2016 by Sorcerer

[studiot]

 

I'm going to have to learn that language of logic one day. However I can refute that with words only. The set which contains every possible set contains itself and is not contained in any other set.

 

Whatever definition is offered for a set it will not satisfy everyone for every purpose.

 

Various devices have been used to get around this difficulty. These devices usually amount to a restriction on what the theory under discussion applies to.

 

For instance we can avoid the issue in the quote by simply saying at the outset that we are dealing with sets of numbers.

[wtf]

For instance we can avoid the issue in the quote by simply saying at the outset that we are dealing with sets of numbers.

The problem with this approach is that now we have to ask the question, "What is a number?" If we are proposing set theory as the foundation of math, we can't just say, "Well, a set is a thing that contains numbers." Because now we have no definition for numbers. If my professor says, "Consider the set of real numbers," I am now justified in responding: "But how do you know there are any real numbers at all? What are they?" And now we've reverted ourselves back to the state of math in 1870 or so. We've just thrown out 140 years of mathematical progress.

 

The solution in ZFC is to define the real numbers as particular sets. The real numbers are equivalence classes of sets of rationals, the rationals are equivalence classes of pairs of integers, the integers are equivalence classes of pairs of natural numbers, and the naturals are built up out of the empty set and the successor operation. Everything can be built up in terms of sets.

 

In any event, OP asked specifically about ZFC. In ZFC there are only sets. Everything is a set. A set contains other sets and a set must be contained in some other set. In particular, numbers are sets. Sets are logically prior to numbers.

 

The answer to the OP's most recent question is that there is no set of all sets, as demonstrated in the link I provided to a recent thread on that subject. It is true that we may conceptually form the collection of all sets; but this collection is not a set and is not subject to the rules that apply to sets.

Edited February 12, 2016 by wtf

[studiot]

 

The answer to the OP's most recent question is that there is no set of all sets, as demonstrated in the link I provided to a recent thread on that subject. It is true that we may conceptually form the collection of all sets; but this collection is not a set and is not subject to the rules that apply to sets.

 

 

 

So you too have sidestepped the thorny issue of "What is a set ?".

 

You use the word collection, I have seen the words aggregate, type, class, amongst others to park the problem one stage up-the-line.

 

The problem is that no-one has yet come up with a definition of sets that is fit for all purposes we desire to use the concept for.

 

So we have a 'bunch' of 'thingies' and restrict these to particular cases with particular properties, that serve our purpose of the moment, not caring that some excluded thingies break our rules.

 

In fact we are so enamoured of this ruse that we construct a whole heirarchy of bunches and sub-bunches, each with a new name and more restricted properties.

But hey, it is a good ruse and serves us well.

 

How do you get on with Russell's shoes and socks paradox?

Edited February 12, 2016 by studiot

[wtf]

So you too have sidestepped the thorny issue of "What is a set ?".

Since the subject is ZFC, a set is an undefined term.

 

The problem is that no-one has yet come up with a definition of sets that is fit for all purposes we desire to use the concept for.

That may well be a problem for some philosophers. But it's not a problem in ZFC, since "set" is an undefined term in ZFC, in the exact same way that "point" and "line" are undefined terms in Euclidean geometry.

 

In fact we are so enamoured of this ruse that we construct a whole heirarchy of bunches and sub-bunches, each with a new name and more restricted properties.

But hey, it is a good ruse and serves us well.

To be frank I do not understand where you're coming from. Set theory has served mathematics well for a century. I don't understand why you seem to be arguing against standard set theory when -- you'll pardon my directness -- you clearly don't have enough knowledge of set theory or philosophy to have an informed opinion. That's not a slur or an attack. If you are an expert at tennis and I've never played before, my skill level will be immediately apparent to you the moment you see me play. I don't understand why you're pretending so much knowledge that you don't actually have. Like I say that is not a personal remark. It's something I'm very curious about.

 

How do you get on with Russell's shoes and socks paradox?

That's a perfect illustration of what I mean. Russell's shoe and sock example illustrates why we need the Axiom of Choice. If we have infinitely many pairs of shoes, we may say, "From each pair choose the left shoe." There is a first-order definable property that tells you exactly which choice to make.

 

In the case of socks, there is no such property and therefore we need a whole new axiom to allow us to choose an element from each set. AC is a pure statement of existence and gives no clue as to the nature of the elements being selected.

 

Do you know of some different context or meaning for this example? I'm curious as to why you brought it up, since (as far as I know) it does not bear on the question at hand.

Edited February 12, 2016 by wtf

[studiot]

 

To be frank I do not understand where you're coming from

 

In that case your subsequent remaks are not founded on any justification.

 

 

That may well be a problem for some philosophers. But it's not a problem in ZFC, since "set" is an undefined term in ZFC, in the exact same way that "point" and "line" are undefined terms in Euclidean geometry.

 

To quote your later comment

 

 

That's a perfect illustration of what I mean.

 

I'm sorry I didn't phrase it in such a way that you could understand it.

 

ZFC theory is a restriction of a more general theory for a specific purpose (which I loosely associated with numbers).

 

You say that ZFC does not define sets and said (or implied) last time that ZFC also does not define numbers.

 

So why is the first OK but the second verboten?

 

Finally in my study of Euclid,

 

Definition 1 concerned a point, definition 2 concerned a line.

 

Euclid puts the definitions where they belong, preceding the axioms which tell us what we can and can't do with the material of the definitions.

[Sorcerer]

If ZFC doesn't allow a set of all sets, is there an alternative which does. I find this quite important considering if this is the leading theory on which all subsequent math is based and we use that math to model reality.

 

It seems that since a set of all sets is a possible thing, that we must begin with a theory which can include it. If we exclude the possible or subsequent descriptions of reality may also exclude crucial prices.

 

By removing the paradoxes in naive set theory, and introducing a restriction on the existence of a universal set, we still leave a problem, is there another step to fix this, or did we step the wrong way before?

[wtf]

If ZFC doesn't allow a set of all sets, is there an alternative which does. I find this quite important considering if this is the leading theory on which all subsequent math is based and we use that math to model reality.

 

It seems that since a set of all sets is a possible thing, that we must begin with a theory which can include it. If we exclude the possible or subsequent descriptions of reality may also exclude crucial prices.

 

By removing the paradoxes in naive set theory, and introducing a restriction on the existence of a universal set, we still leave a problem, is there another step to fix this, or did we step the wrong way before?

There are non-wellfounded set theories, in which sets can be members of themselves. I don't know much about them. Perhaps they would be of interest to you. https://en.wikipedia.org/wiki/Non-well-founded_set_theory

In that case your subsequent remaks are not founded on any justification.

So then why not just tell me where you're coming from?

 

ZFC theory is a restriction of a more general theory for a specific purpose (which I loosely associated with numbers).

Can you say more about what you mean? What general theory is ZFC a restriction of?

 

You say that ZFC does not define sets and said (or implied) last time that ZFC also does not define numbers.

On the contrary, I explicitly outlined exactly how numbers are defined as sets within ZFC.

 

 

Finally in my study of Euclid,

 

Definition 1 concerned a point, definition 2 concerned a line.

 

Euclid puts the definitions where they belong, preceding the axioms which tell us what we can and can't do with the material of the definitions.

Not how I learned geometry, and certainly not the modern view. So at best, your knowledge of what Euclid originally said would be of historical interest. It doesn't actually bear at all on the fact that in ZFC, both "set" and "element of" are undefined terms.

[Xerxes]

If ZFC doesn't allow a set of all sets, is there an alternative which does.

Yes,it is called "category theory". Here, one defines a category of objects Set which includes all sets. This category is not, for the reason you have been given, a set itself.

 

Similarly, one may define (for example) the category of rings Rng, the category of fields Fld and so on.

 

The important point about this category theory is that it extracts only those properties of mappings between sets, rings, fields etc that they have in common. Which leads to the (to me at least) nice abstraction that the exact nature of the objects in a category are of less importance that themapings between them.

 

Even more interesting is the fact that one can define a mapping between categories, called a functor and a lot more besides.

 

It leads to some very nice mathematics which is difficult to get one's head around, especially if one is classically trained.

 

I know of only a handful of people who dare to use this theory in applications. The excellent John Baez is one of them. Look him up.

[studiot]

 

Xerxes

Yes,it is called "category theory". Here, one defines a category of objects Set which includes all sets. This category is not, for the reason you have been given, a set itself.

 

 

 

What would happen if you interchanged the words set and category?

 

One of my beefs with the system is that different, well respected, authorities use different words for 'category'.

Russell developed a system of 'types'

Graves, Simmons and Phillips use Classes

Hobson and Kestleman use aggregate.

Some (eg Borowski and Borwein) have categories as particular restrictions of classes, more general than sets, but less so than classes

 

It is small wonder that outsiders of this clique are confused.

 

The rest of your post simple demonstrates my point that there is a heirachy of different types of collected objects for different purposes, because some purposes are incompatiable with each other.

 

For instance if we want to include the set of all sets or even a set which can include other sets we have to abandon a very important property of some sets.

Namely that every member of the set is the same type of object and that defined operations on a set will apply to every member and always be guaranteed to be the only way to produce another member of the set and to always produce one. this is particularly desirable in numbers and arithmetic.

 

Another oft required property is that each instance of a member is counted only once. But that cannot be resolve Russell's sock paradox.

 

Sorcerer,

I am trying to offer you a practical man's balanced view to make sense of all the confusion.

I don't think you can start with some glorious fundamental set or collection and refine it to suit all purposes because of the incompatabilities.

You need to start with what you want your collection to do for you.

[Sorcerer]

Well, I just blew a brain gasket visualizing the set of all sets, which contains itself ad infinitum. The set never reaches a conclusive boundary, since infinite progression would end in the creation of a set which is included in the set of all sets.

 

WTF you mentioned a link in another thread on the topic, showing this set doesn't exist, I'll try to find it later, but maybe you could find it first?

 

studiot : I'm actually just getting my head around sets, I glanced over this topic at high school and always considered it to be childish, I remember doing set diagrams at primary school and putting counting blocks in them, it never really seemed to be of any deeper level than that. I'm not actually going to attempt to do any kind of maths with sets and I don't see myself ever using them practically.

 

So I don't have a collection or a want. I'm just playing with ideas and learning. Actually this other thread, maybe you could show me how to use sets here, I guess it starts with the collection of every possible concept. http://www.scienceforums.net/topic/89662-question-about-nothing/?p=905563

Edited February 13, 2016 by Sorcerer

[wtf]

Well, I just blew a brain gasket visualizing the set of all sets, which contains itself ad infinitum.

Forgive my pedantry here but there is no set of all sets and this is a semantic point of importance to me.

 

The reason I insist on this is because in order to get anywhere at all, we have to agree on the meaning of "set". In this thread we are discussing sets in ZFC. In ZFC there is no set of all sets.

 

It's perfectly ok to think about the collection of all sets, the aggregate of all sets, the proper class of all sets, and as Xerxes notes, the category of all sets.

 

But in math the word "set" has a very specific meaning and according to this universally accepted meaning, there is no set of all sets.

 

As far as mind-blowingness, that's what we love about set theory! Transfinite set theory is a triumph of the human imagination. It allows us to reason rationally about infinite sets. It's a brand new discovery, only 140 years old. That's pretty recent, considering that Aristotle thought about these same issues. So it takes some work to get one's mind around it. You're in good company there.

 

 

The set never reaches a conclusive boundary, since infinite progression would end in the creation of a set which is included in the set of all sets.

That's true. But lots of simpler things in math "never reach a conclusive boundary." For example say we have the set of counting numbers 1, 2, 3, 4, ...

 

Then we can make the set of rational numbers 1/2, 1/4, 1/8, ..., 1/2^n, ... and this also is a set that "never reaches a conclusive boundary." Lots of things don't have boundaries. So we should not be unduly impressed. For example, we could think about the class of all sets as "being the limit" of larger and larger sets of sets in the same way that 1/2^n has the limit 0 but no individual term is zero. [This is an idle thought on my part, I don't know if set theorists think that way or not.]

 

 

 

WTF you mentioned a link in another thread on the topic, showing this set doesn't exist, I'll try to find it later, but maybe you could find it first?

First, here is Russell's paradox. It shows that

 

1) You cannot form a set out of a property alone. For example "is a set" is a property. Russell's paradox shows that we MAY NOT form a set by saying, "The collection of all things that satisfy some property." If we allow that, we get a contradiction.

 

2) It follows as a corollary of (1) that there is no set of all sets.

 

I don't want to get bogged down in the details of this proof in this thread right now.

 

https://en.wikipedia.org/wiki/Russell%27s_paradox

 

The post I referred to earlier is this one:

 

http://www.scienceforums.net/topic/92101-what-is-the-minimum-number-of-properties-posessed-by-members-of-a-set/?p=900416

 

That's a long post in the middle of a long thread but inside there is an explanation of why Russell's paradox implies there's no set of all sets. Read the Wiki article on Russell's paradox first.

 

I glanced over this topic at high school and always considered it to be childish, I remember doing set diagrams at primary school and putting counting blocks in them, it never really seemed to be of any deeper level than that.

My own opinion is that set theory doesn't get interesting until you consider infinite sets. But then I'd be insulting the combinatorialists, who do amazingly clever things with finite sets!

 

 

But as far as transfinite set theory being "practical," there's no known use for it at the moment. If tomorrow morning some physicist finds evidence of an infinite collection of things in the universe, then transfinite set theory will become important in the real world. Till then, this is purely an abstract intellectual pursuit.

 

Although I would point out that there is a connection between transfinite set theory and computer science. Turing and Gödel were all over these connections in the 1930's. So one should not discount the theoretical importance of infinite sets even in the real world.

 

 

So I don't have a collection or a want. I'm just playing with ideas and learning. Actually this other thread, maybe you could show me how to use sets here, I guess it starts with the collection of every possible concept. http://www.scienceforums.net/topic/89662-question-about-nothing/?p=905563

I gave that a very brief glance, is it mostly philosophical? It think it's helpful if we realize that we have two different concerns: (1) learning about elementary set theory the way modern mathematicians understand it; and (2) arguing about philosophy.

 

I'm attempting (1) and perhaps my friend studiot is coming from a perspective of (2). We are in the math forum after all, so why not stick to math and open a thread in the philosophy forum for philosophical concerns. My two cents on that. I happen to love math philosophy but I do not confuse it with doing math.

Edited February 13, 2016 by wtf

[Sorcerer]

well said