What is the minimum number of properties posessed by members of a set?

[studiot]

What is the minimum number of properties posessed by members of a set?

[John Cuthber]

Interesting question. How many members has the set of {things that have no properties} got?

[studiot]

At least one.

 

I was thinking that either we have to do away with the ability of specifying set membership by property or accept two as a minimum number of properties.

Edited November 10, 2015 by studiot

[John Cuthber]

Interesting question.

There is a defined set of {my brother} which has just one defining property but that also identifies him as human, male etc. which are other properties

Can you have a set that is { the number 1} without it having at least two properties; being a number and that number being 1

[studiot]

Interesting question.

There is a defined set of {my brother} which has just one defining property but that also identifies him as human, male etc. which are other properties

Can you have a set that is { the number 1} without it having at least two properties; being a number and that number being 1

 

 

I really started by thinking about the two ways of defining a set.

 

1) List all the members

 

2) Specify a common property possessed by all members.

 

Now every member of every set has one property. The property of being on the list of members of that set.

 

Your {my brother} example identifies one aspect.

 

It does not include all men, or even maybe all your brothers.

So there are potential male members that are excluded.

 

But let us move on to numbers to make this point clearer.

 

Suppose we make our set

 

The set of all numbers greater than zero.

 

1) The membership list property includes all the positive numbers

 

2) Every number on the list has a second property that has to be stated, that of exceeding one.

 

We also know that there are more numbers that are not on the membership list.

 

So we have at least two properties, and we don't need to mention any more, even though we could easily propose many.

 

But what about the set {3, 26, goats}

 

Do we need a second property?

[ajb]

I think that all we basically need is some equivalence relation (or relations?) on the collection of objects that we use to build a given set. That is we must have some clear notion of distinct objects: this is used in the axiom of extensionality in the ZF-axioms.

 

For example {3,26, goat} = {3,3, 26, goat} = {3,26,26, goat} = {3, 26, goat, goat}. But we know that a goat is not equivalent to any number and for sure goat does not equal 3 or 26.

 

I guess we should read the ZF-axioms very very carefully and see if they tell us more.

[fiveworlds]

At least one.

 

I was thinking that either we have to do away with the ability of specifying set membership by property or accept two as a minimum number of properties.

 

 

What about the null set?

[studiot]

 

What about the null set?

 

 

It has no members, so how many properties do its members have?

 

In a way this a quite a good question since there are zero members on the membership list, there are zero member properties.

I don't know if this is a valid answer though, according to ZF axioms.

 

Thank you , ajb, for introducing those.

 

How would the null set play with axiom3, the exclusion or russian doll axiom?

That would suggest that no (zero) properties can be associated with the null set.

 

Axiokm3 would allow (demand?) the formation of a subset of {3, 26, goats} which contains only numbers

[fiveworlds]

It has no members, so how many properties do its members have?

 

 

Surely we assign the properties of each set lets say I have 3 sets.

 

A->People who visit scienceforums.net but are not members and haven't registered

B->People who have registered to become a member but are not yet members

C->People who are members

 

At any one point in time set A,B or C could be the empty set. Does the set cease to exist if it has no members?

[studiot]

 

At any one point in time set A,B or C could be the empty set. Does the set cease to exist if it has no members?

 

 

Somewhere, buried in set theory, you will find a theorem that the empty set is a subset of every set.

In other words there is one and only one empty set.

 

 

Surely we assign the properties of each set lets say I have 3 sets.

 

Please remember ( I admit to difficulties here as well), this thread is about properties of the members, not the set.

[fiveworlds]

In other words there is one and only one empty set.

 

This seems a bit strange given

 

The axiom of the empty set stating the existence of the empty set

 

 

and the axiom of extensionality stating that if two sets have the same elements they are equal

 

[studiot]

 

and the axiom of extensionality stating that if two sets have the same elements they are equal

 

 

 

Does this not prove there is only one empty set?

[John Cuthber]

Is the set of [my hit records] the same as the set of [whales on Mars] and the set of [even primes bigger than 2]?

I think it is, but what would I know; I'm a chemist.

[fiveworlds]

Does this not prove there is only one empty set?

 

You can say that yeah. Simply because both sets are the same. But it is the equivalent of saying only one set exists which contains only the number 1.

[studiot]

^{You can say that yeah. Simply because both sets are the same. But it is the equivalent of saying only one set exists which contains only the number 1.}

 

 

Why would that be?

 

Please let's not go too far down this side road, about sets.

 

The thread is about properties of members of sets.

Edited November 11, 2015 by studiot

[ajb]

I have not thought much about such fundamental issues in mathematics, but plenty of people have and continue to do so. I would not be at all surprised if there was no universally agreed upon notion of a set: the ZF-axioms together with the axiom of choice is often understood as 'set theory'. However, people do ask if all the ZF-axioms are needed, what can replace them, what really is the minimum we need and so on. People have written volumes of books on seeming simple questions!

[TheGeckomancer]

I don't have to be able to explicitly list a set do I? So long as I can adequately describe it.

 

How about a set who's defining property is having no members with properties in common?

 

This makes a 1 definition set. I mean theoretically it could even have 0 members couldn't it?

 

Or if you wanted to be technical. You could say a minimum of 2. Because you have to say that this set exists and that it's members have no properties in common, but to me it makes sense to ignore the exists definition as that's implied in every single set you state.

Edited November 24, 2015 by TheGeckomancer

[ajb]

I don't have to be able to explicitly list a set do I? So long as I can adequately describe it.

You don't need to explicitly write out all members of a set. However, for finite sets you can, in principle do this. Not so for sets of infinite cardinality.

 

How about a set who's defining property is having no members with properties in common?

For finite sets this seems absolutely okay. The only thing the elements have in common is that they belong to the given set. I am wondering if there is any real difference here when we encounter sets of infinite cardinality. I am not sure if there is any real problem here or not. For sure we cannot write out explicitly all the members of such a set.

 

I mean theoretically it could even have 0 members couldn't it?

We have the null set.

 

Or if you wanted to be technical. You could say a minimum of 2. Because you have to say that this set exists and that it's members have no properties in common, but to me it makes sense to ignore the exists definition as that's implied in every single set you state.

In naive set theory you do not suppose much. Just that the objects are well defined and distinct. As I said earlier, distinct implies you have some way of comparing the objects.

[TheGeckomancer]

So wouldn't the minimum definitions be 1 or 2 depending on how you look at it? I can't think or any other definitions that can be applied to that set besides exists and shares no properties.

[ajb]

So wouldn't the minimum definitions be 1 or 2 depending on how you look at it? I can't think or any other definitions that can be applied to that set besides exists and shares no properties.

 

In naive set theory there seems to be only two requirements. But then we know naive set theory is not so well founded. You need to pass to something like the ZF-axioms (usually with the axiom of choice).

 

The main thing that I see, and this is hidden in naive set theory is that I must be able to say if objects are the same or not. This requires an equivalence relation. This I suppose is the same as your 'shares no properties' as I can always apply not to the statements.

 

So to me it looks like two requirements: 1. the objects are well defined; 2. I must be able to say if the elements of sets are equivalent or not.

 

But this maybe too naive. I don't worry too much about foundational issues myself.

Edited November 24, 2015 by ajb

[studiot]

Think of lightburst at some arbitrary point in the Universe.

 

Now consider the set of all points illuminated by that lightburst some arbitrary time later.

 

Do the the points included or excluded not depend upon the observer and his velocity relative to the initial point?

[ajb]

You mean something like the forward light cone of a given point?

[studiot]

You mean something like the forward light cone of a given point?

 

Not sure I haven't worked it all out, but I was thinking about the relativity of simultaneity and how it is different for for different observers.

 

So points will be included in some observers' set but not in other observers' sets.

[wtf]

 

 

I really started by thinking about the two ways of defining a set.

 

1) List all the members

 

2) Specify a common property possessed by all members.

 

 

I hope nobody minds my resurrecting this thread. I wanted to mention that if we work in ZFC, the standard axiom system for modern math, there are sets that satisfy neither of those properties.

 

Consider the set of real numbers; and if x and y are real numbers, define the relation x ~ y if it happens to be the case that x - y is a rational number. For example, (1/3 + pi) ~ pi because (1/3 + pi) - pi = 1/3 is rational.

 

This is readily seen to be an equivalence relation. For all real numbers x, we have x ~ x; also for all x and y we have x ~ y implies y ~ x; and for x, y, and z we have x ~ y and y ~ z implies x ~ z.

 

This equivalence relation partitions the real numbers into a set of mutually disjoint equivalence classes. By the Axiom of Choice (that's the C in ZFC) there is a set, call it V, that consists of exactly one representative from each of the equivalence classes.

 

We know from the rules of set theory that V exists. We can not name any of its specific members; nor is there any property, other than membership in V, that's shared by the elements of V. We can't look at a real number and say whether that number is in V. If you ask, "Is 1/3 in V?" I answer that I don't know. I do know that V contains exactly one rational (why?) but I have no idea which one it is.

 

V is a set whose members are not known, whose members can not be recognized as belonging or not belonging to V, and share no property in common.

 

I should mention that this kind of construction is exactly why some people don't like the Axiom of Choice. However if you reject Choice, you get other unpleasant anomolies. So mathematicians just accept it and get used to using it. What that means metaphysically is a personal choice.

Edited December 28, 2015 by wtf

[studiot]

Good morning wtf, and thank you for your contribution.

 

The question in this thread is about the minimum number of properties possessed by an element, not the set properties.

In the case of your examples each element has multiple properties.

 

So I am interested in sets where the elements have only one property; I have displayed such a set.

Since such a set can be displayed, there cannot be an equivalence relation on this particular set.

 

As regards to your constructed partition of the reals,

 

As I understand it, you have partitioned the reals into two sets: the rationals (you have called set V) and the rest (I am calling W)

 

Consider any [math]a,b \in V[/math]

 

Then [math]\left( {a + b} \right) \in V[/math]

Thus a and b have at least two properties.

 

Edited December 29, 2015 by studiot