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

[wtf]

Reading through the thread, it's a little confusing whether you are referring to properties of elements or of a set. I can't imagine any specific object in the universe, physical or abstract, having exactly one property.

 

In any event, V is a set containing exactly one member of each equivalence class. The rationals are an equivalence class (the difference of any two rationals is rational) so V contains exactly one rational. But it's impossible to say which rational V contains.

 

The reason I posted this example is that it's a counterexample to your suggestion that the two ways to create a set are to specify a property its members have in common, or to enumerate its members. In the case of V, we have no idea which real numbers it contains nor what their properties are; and there's no property or set of properties that characterizes the elements of V in any way. The existence of V is guaranteed, but we can't name or characterize any of its members.

[studiot]

Reading through the thread, it's a little confusing whether you are referring to properties of elements or of a set. I can't imagine any specific object in the universe, physical or abstract, having exactly one property.

 

In any event, V is a set containing exactly one member of each equivalence class. The rationals are an equivalence class (the difference of any two rationals is rational) so V contains exactly one rational. But it's impossible to say which rational V contains.

 

The reason I posted this example is that it's a counterexample to your suggestion that the two ways to create a set are to specify a property its members have in common, or to enumerate its members. In the case of V, we have no idea which real numbers it contains nor what their properties are; and there's no property or set of properties that characterizes the elements of V in any way. The existence of V is guaranteed, but we can't name or characterize any of its members.

 

 

Firstly elements may have other properties than they possess as elements of a particular set.

Such properties are irrelevent to their member ship of any particular set.

 

I can't understand how you can accuse me of confusing the properties of elements with the properties of sets, considering the title of this thread.

 

Unfortunately your counterexample is not such.

 

You start with the set of real numbers.

 

So every member of the set has the property of being a real number. property number 1

 

You then select some memebers that have the property of being rational property number 2

 

I make that two properties.

 

So the set of rationals inherits the property of also being real.

 

I thought I gave an example of a set where the only common property of the elements was being on the membership list, but that does not prevent them having other irrelevent properties.

[wtf]

No accusations intended, I assure you. Just trying to understand the question. I'm the one who's confused and looking for clarity. I'm giving this example as it demonstrates how the Axiom of Choice can be used to produce a set whose members have no characteristic property in common, nor are they enumerated or even known. It's an interesting (to me) broadening of the discussion to give an idea of the strange kinds of sets people use in ZFC.

 

You wrote, "Firstly elements may have other properties than they possess as elements of a particular set.

Such properties are irrelevent to their member ship of any particular set."

Ok. So V is a set of reals. But if some number x real, that fact gives you no information about whether it's in V. The fact that x is real is an irrelevant property as you described it.

The members of V are impossible to identify or recognize. All the elements of V are real, but many reals are not in V. And the ones that are, can not be characterized by any property. If I show you a real number, there's no way for you to tell me whether it's in V or not by any conceivable property. That's what I find interesting about this example.

To clarify what some of the equivalence classes of V are, one class is the rationals [math]\mathbb{Q}[/math]. Another is the rationals plus pi, that is every number of the form q + pi where q is rational. Another class is the rationals plus sqrt(2).

It's clear (if you think about it) that every class is of the form [math]\mathbb{Q} + x[/math] for some real number x; but to get the unique set of classes you can't use ALL the real numbers for x, since some real numbers give the same class as others. For example [math]\mathbb{Q} + x[/math] and [math]\mathbb{Q} + y[/math] are exactly the same equivalence class just in case x and y differ by a rational.

Now V is defined as containing exactly one element of each equivalence class. Of course you are right that all the elements of V are real numbers, but isn't that a general property and not a property these numbers have specifically by virtue of being elements of V?

I hope that nothing I'm writing is coming across as argumentative or accusatory in any way at all. I find the sets given by Choice to be quite different in character than the usual sets that are given by explicit means, and interesting to contemplate. A set like V pushes against much of the intuition about sets expressed in this thread. But clearly I'm not explaining it well enough, otherwise you'd agree

Edited December 31, 2015 by wtf

[studiot]

Good morning wtf,

 

I did not use (or intend to use) ‘accuse’ in a defamatory way or feel insulted by you suggesting I might be confusing the properties assigned to sets and those assigned to elements of sets. Nor do I wish it to detract from an excellent discussion.

Indeed had I mixed them up you would be well justified in ‘accusing’ me of doing so, but I don’t think I did.

 

In my mind this thread probes some of the dark corners of Mathematics, so it is not surprising that some spiders crawl out.

 

J

 

The introduction of KFC or its variants limits the scope of the discussion since

 

KFC does not apply to all sets or even all mathematical sets. The foundation axiom limits KFC sets to those for which an epsilon minimum can be defined. The set of symmetries of a regular hexagon, for instance, has no minimum, though it obeys many of the other axioms.

 

I am sorry I originally misinterpreted your post 24 to mean that your set V comprised the set of rationals. I see now you were meaning a proper partition into equivalence classes. (More on this and my hexagon later)

 

I am not aware of any KFC axiom that requires all sets to be partitionable. How do you partition the zero set without invoking a circular argument?

 

 

The members of V are impossible to identify or recognize. All the elements of V are real, but many reals are not in V. And the ones that are, can not be characterized by any property. If I show you a real number, there's no way for you to tell me whether it's in V or not by any conceivable property. That's what I find interesting about this example.

 

 

If, however you start with a set such as the reals and divide it to obtain a counterexample set then surely you must start with an isolated element and show first that it is a real, before you can allocate it to your set V or the set W of the remaining reals not in V.

 

In my example I did not start with a set and cut it down I obtained elements and assigned them to my set that I was constructing. So the elements of my example set entered the set without carrying any preassigned properties with them.

 

The requirement that an element of a set be ‘distinct’ does not mean that we should be able to identify it ‘in the wild’.

In fact it does not even apply to a single isolated element, but to (at least) two elements so that they may be compared and distinguished as the same or different.

 

Nor is our ability to identify an element a bar.

Many differential equations are known to have solutions, even though we cannot solve them.

So we know there exists a set of solutions, even though we do not know a single one.

 

 

Bringing in a second element brings me back to my hexagon symmetries.

These are often stated as forming an equivalence class.

This is true so long as we are only considering an isolated symmetry.

But bring in a second symmetry and things change.

It makes no difference which vertex of a benzene ring you attach the first functional group to.

But any chemist will tell you about the difference between the ortho, meta and para positions for introducing a second group.

Chirality has raised her head.

And yet again this issue does not arise with the set of symmetries of an equilateral triangle.

 

So thank you again for thought provoking probing of my discussion.

 

As the old scribe said,

 

I look forward to your reply.

 

Edited December 31, 2015 by studiot

[wtf]

The introduction of KFC ...

 

Now I'm hungry!

 

"KFC does not apply to all sets or even all mathematical sets. The foundation axiom limits KFC sets to those for which an epsilon minimum can be defined. The set of symmetries of a regular hexagon, for instance, has no minimum, though it obeys many of the other axioms."

 

Of course you mean ZFC here. In ZFC we have the kernel of a linear transformation. In KFC we have Colonel Sanders.

 

How do you do quoted text on this forum? I can't figure out this quote window.

 

In your example I believe you are confusing well-foundedness (a set can't contain itself as a member) with having a smallest element in some order. It's true that there is no order on a set of permutations, but they're still well-founded sets. A similar example would be the integers, which have no smallest element: -1, -2, -3, ... But the integers are still a well-founded set. The set of integers does not contain itself as a member.

 

When you say that ZFC does not apply to all mathematical sets, that's only true if you are specifically working in some alternative set theory. All the sets we're talking about, including sets of permutations, are modeled in ZFC and are well-founded.

 

I'm afraid I don't know enough chemistry to respond to your comments about benzene rings. But I did look up the mathematical handling of chirality, and evidently this phenomenon does have a mathematical description. https://en.wikipedia.org/wiki/Chirality_(mathematics)

Edited January 2, 2016 by wtf

[studiot]

OMG

 

What a prime slip up.

 

My brain was clearly short of food.

 

[Strange]

How do you do quoted text on this forum? I can't figure out this quote window.

 

http://www.scienceforums.net/topic/82164-the-quote-function-a-tutorial-in-several-parts/

[Xerxes]

How do you do quoted text on this forum? I can't figure out this quote window.

Yeah, it took me a while to figure it.

 

Bottom right click "quote". You will likely get a pretty useless box that will not allow you to split the text you want to quote

 

In the quote box look for the top left icon, click, and you then have plain ASCI text.

 

Following the portion of the text you wish to respond to enter [ /quote]. Enter your reply. Delete the rest, or continue as above, but this time remember to start the relevant portion of the remaining text with

[ quote]

[wtf]

Yeah, it took me a while to figure it.

Ah the top left icon. Thank you much.

[Xerxes]

You are welcome. Just remember - everybody - this facility is open to abuse. As in

Mathematics is all about pink unicorns, nothing else

DO NOT DO IT Edited January 2, 2016 by Xerxes

[Phi for All]

You are welcome. Just remember - everybody - this facility is open to abuse. As in

DO NOT DO IT

 

!

Moderator Note

WARNING! YOU HAVE INCORRECTLY QUOTED A FELLOW MEMBER, PURPOSELY CHANGING WHAT THEY SAID. THIS VIOLATION WILL BE MET WITH SWIFT ACTION. COOPERATE FULLY WITH THE TEAM WE'VE DISPATCHED AND YOUR PUNISHMENT WILL BE MERCIFUL. MAYBE.

[Xerxes]

I'll put the kettle on for them. Do they take sugar?

[Sensei]

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.

How about set which contains just infinity.. ?

[math]\{\infty\}[/math]

 

What can you say about property of such set?

[wtf]

How about set which contains just infinity.. ?

[math]\{\infty\}[/math]

 

What can you say about property of such set?

First you'd have to define the symbol [math]\infty[/math], then we'd know what its properties are. Typically [math]\infty[/math] is used to refer to the conceptual "endpoints" of the real number line in the extended real number system.

 

In that context, the symbol [math]\infty[/math] has definite properties such as, "[math]x < \infty[/math] for any real number [math]x[/math]".

 

The symbol [math]\infty[/math] does not refer to an infinite set these days. It's only used in the very limited context of the extended reals. There are other symbols in use for infinite sets, such as [math]\omega[/math], the first transfinite ordinal; and [math]\aleph_0[/math], the first transfinite cardinal. In each case, a symbol is introduced and defined; and from the definition, we can deduce a laundry list of associated properties.

 

If you put some symbol between curly braces you have to define what the symbol stands for. And once you do that, you're specifying its properties.

 

The easiest way to create a set without specifying a defining property or enumerating its members is to use the Axiom of Choice in the manner I showed in my earlier example.

 

By the way did you know that mathematics is all about pink unicorns, nothing else?

Edited January 8, 2016 by wtf

[studiot]

 

wtf

The easiest way to create a set without specifying a defining property or enumerating its members is to use the Axiom of Choice in the manner I showed in my earlier example.

 

 

 

I don't think you did as I subsequently pointed out.

 

Was my post not clear, I can elaborate if you like.

 

I will concede that enumeration (as a counting process) is different but not necessary.

 

Edit.

 

I just noticed some stuff in your post#30 that was added since I last looked at it.

It does not change my opinion; I will produce a properly reasoned response.

 

Edited January 8, 2016 by studiot

[Xerxes]

I don't think you did

Well I do, though I rather think his diversion into equivalence relations was not well-motivated - I'll explain why I think this in a later post (if I get time)

 

Anyway. Suppose that [math]x[/math] an object in the universe of all possible objects.

 

Say that [math]P[/math] be a propostion, and say that [math]P(x)[/math] means the proposition is true for any such [math]x[/math]. One writes

 

[math]X=\{x:P(x)\}[/math] for any [math]x[/math] denotes the set of objects for which the proposition is true. Obviously this has non-trivial content for a very large number of sets

 

But in the case of a set formed by an arbitrary selection of just one element from a (possibly infinite) number of other sets, what can we say about our proposition [math]P[/math]?

 

Only that [math]P(x)[/math] means that [math]x\in X[/math] if and only if [math]x \in X[/math]. Hardly a surprise, you will agree, so I say that the proposition [math]P[/math] is trivial in this case

[studiot]

Hello Xerxes, how do you define 'true'?

 

I would also be interested in understanding what the relevence of this is to the thread please?

[wtf]

I don't think you did as I subsequently pointed out.

 

Was my post not clear, I can elaborate if you like.

IMO the parts that were clear (to me at least) were wrong. The parts about benzene rings and non-wellfounded sets were off point. I'll await further elaboration. But no explicit characterization of a choice function on the equivalence classes of [math]\mathbb{R} / \mathbb{Q}[/math] is possible. That's a mathematical fact.

 

I just noticed some stuff in your post#30 that was added since I last looked at it.

It does not change my opinion; I will produce a properly reasoned response.

I shall await your response. But there is no explicit characterization of the elements of [math]V[/math].

 

By the way I did not make that example up. It's a standard idea that shows up in many different guises. These examples are not relevant to the discussion of [math]V[/math], I only mention them for interest.

 

* If you restrict [math]V[/math] to the unit interval and try to apply the standard axioms of probability, you get an example of a nonmeasurable set, a set of reals that can not possibly have a notion of size or probability assigned to it. Tthe traditional use of the letter [math]V[/math] is in honor of Giuseppe Vitali, who cooked up this example in 1905.

 

* If you consider the reals as a vector space over the rationals and then apply the theorem that every vector space has a basis, you get a set of reals [math]H[/math] such that every real number has a unique expression as a finite linear combination [math]r = q_1 h_1 + q_2 h_2 + \dots + q_n h_n[/math] where each [math]q_i[/math] is rational and each [math]h_i \in H[/math]. Once again, the set [math]H[/math] can be shown to exist but is impossible to visualize or characterize via some property. The [math]H[/math] is in honor of Georg_Hamel, and the set [math]H[/math] is known as a Hamel basis. The theorem that every vector space has a basis is logically equivalent to the Axiom of Choice (AC). If you deny AC you then have a vector space that has no basis. Point being that you get weirdness whether you accept or deny AC so we might as well accept it. That's the modern viewpoint.

 

* The same technique, defining some equivalence relation and then invoking the Axiom of Choice to create a set consisting of one element from each equivalence class, is a key step in the famous Banach-Tarski paradox, in which we decompose a sphere in Euclidean 3-space into five pieces, move the pieces around using rigid translations and rotations, and put the pieces together to create two spheres, each the same size as the original. (See step 3 of the proof outline in the linked Wiki page].

 

I'm just mentioning these examples to show that I didn't pull my idea out of thin air (or my nether regions), but rather out of modern math. Whether modern math itself comes out of thin air is another question altogether. Math really is all about pink unicorns.

 

 

Well I do, though I rather think his diversion into equivalence relations was not well-motivated - I'll explain why I think this in a later post (if I get time)

You're right that I didn't explain equivalence relations, if that's what you mean. I assumed knowledge of equivalence relations and linked the relevant Wiki page. Every equivalence relation partitions a set into a collection of pairwise disjoint equivalence classes. That's a theorem from discrete math class that most people see at some point. https://en.wikipedia.org/wiki/Equivalence_relation. Or did you mean not well motivated in some other way? I agree that there's a conceptual leap to forming sets using the Axiom of Choice. [math]V[/math] is the simplest example I know, but it's deceptively tricky to get one's mind around.

 

 

Anyway. Suppose that [math]x[/math] an object in the universe of all possible objects.

I see where you're going and you'll soon be in trouble.

 

Say that [math]P[/math] be a propostion, and say that [math]P(x)[/math] means the proposition is true for any such [math]x[/math].

 

 

One writes

 

[math]X=\{x:P(x)\}[/math] for any [math]x[/math] denotes the set of objects for which the proposition is true. Obviously this has non-trivial content for a very large number of sets

Your construction is outlawed by order of Bertrand Russell himself, who in 1901 said: "No you can't do that!!"

 

If we could define a set that way, we could just let [math]P[/math] be the proposition [math]x \notin x[/math]. Then the resulting set both is and isn't a member of itself, a contradiction. This is the famous Russell's paradox, which blew up naive set theory in the early 1900's. [sorry no Wiki link but it can easily be looked up. Forum software doesn't like the link].

 

The resolution is to disallow exactly the type of unrestricted set formation you attempted. Rather, you have to start with some already existing set, and then apply a predicate (or proposition) to it. So if [math]Y[/math] is some set that already exists, you can take any proposition [math]P[/math] and define [math]X = \{x \in Y : P(x)\}[/math]. Without the restriction to an already existing set, you get a contradiction.

 

But in the case of a set formed by an arbitrary selection of just one element from a (possibly infinite) number of other sets, what can we say about our proposition [math]P[/math]?

In general, we can say nothing at all.

 

Only that [math]P(x)[/math] means that [math]x\in X[/math] if and only if [math]x \in X[/math]. Hardly a surprise, you will agree, so I say that the proposition [math]P[/math] is trivial in this case

That would not suffice as the defining property of the elements of some set in Studiot's sense. It's true that every set is the set of its elements. But you can't use that fact to identify which elements are in the set and which aren't.

 

Indeed, since there are only countably many propositions (a proposition being a finite-length string over some countable alphabet) and there are uncountably many sets (of natural numbers, say), it follows that most sets cannot possibly be uniquely characterized by a property. Most sets are essentially random. In terms of computability theory, which is more familiar to people these days than set theory, most sets can't have their members cranked out by a Turing machine. That's because the set of TMs is countably infinite.

 

[META: The forum software doesn't like the Wiki link to Russell's paradox, which contains a special character sequence for the apostrophe. When I try to correct the link in the editor, the software mangles my post completely. I got tired of re-writing the post over and over and finally just omitted the link. Is this a known bug?]

Edited January 8, 2016 by wtf

[StringJunky]

Link to Wiki: Russell's Paradox

 

Copy URL > Type a link word or words, like above and highlight it > Click 'Link' icon in main editor > Paste > Save

Edited January 8, 2016 by StringJunky

[wtf]

<a data-ipb="nomediaparse" data-cke-saved-href="https://en.wikipedia.org/wiki/Russell"href="https://en.wikipedia.org/wiki/Russell" s_paradox"="">Link to Wiki: Russell's Paradox

 

Copy URL > Type a link word or words, like above and highlight it > Click 'Link' icon in main editor > Paste > Save

If you click the link in your post you'll see that it's not the right link. This is a forum bug. Either that or it only happens for me. Chrome on MacOS FWIW.

Edited January 8, 2016 by wtf

[StringJunky]

If you click the link in your post you'll see that it's not the right link. This is a forum bug. Either that or it only happens for me. Chrome on MacOS FWIW.

I don't know what happened there, When I double-checked the link after writing that post it gave the 'Paradox' page. I'm using Firefox on Win. 10

Edited January 9, 2016 by StringJunky

[imatfaal]

I think it is a new bug with IP software - lots of software has trouble with special characters and spaces in web addresses and they try to parse it and fail.

It used to be OK to manually paste it in or type the tags - but even that fails now

https://en.wikipedia.org/wiki/Russell's_paradox

 

It is fine just pasting a link with an apostrophe and space - just not enclosed in url tags

[Xerxes]

wtf I am baffled by your response. I hope the following doesn't come across as aggressive.

 

Your construction {that is [math]X=\{x:P(x)\}[/math] for some proposition [math]P[/math] and arbitrary [math]x[/math]}is outlawed by order of Bertrand Russell himself, who in 1901 said: "No you can't do that!!"

In this context, he did not. In fact I used quite standard set-builder notation.

 

If we could define a set that way, we could just let [math]P[/math] be the proposition [math]x \notin x[/math].

Since for any object or set the assertion that [math]x \in x[/math] makes no sense, then likewise does its negation, I cannot see your point. You can of course have [math]P(x)[/math] as the proposition that [math]x \ne x[/math] which will define the empty set

 

The resolution is to disallow exactly the type of unrestricted set formation you attempted. Rather, you have to start with some already existing set,

On the contrary, YOUR construction may allow the existence of some "universal set", which does indeed lead lead to the Russell paradox, which I state correctly here......

 

"There can exist no set that contains its complement as a proper subset".

 

Or maybe you thought my use of the term "universe of objects" implied I was referring to a universal set. I was not, and there was nothing to suggest that I was.

 

 

It's true that every set is the set of its elements.

Great Heavens you don't say! I hope you are not offering this as a definition?

[studiot]

 

xerxes

Great Heavens you don't say!

 

Yes it just goes to show that my original hunch was correct.

 

There are some spiders lurking in the dark corners of set theory.

 

Oh and BTW, which set theory?

There are several versions of modern set theory.

 

Neither wtf or you have shown that it is necessary condition of membership for us to know or be able to know if an element is a set member or the properties of said member.

 

Surely an element would be a member, within the terms of reference, even if humans had never existed?

[wtf]

wtf I am baffled by your response.

I'm surprised at your bafflement but I'll try to be more clear. In truth I have nothing to add to what I've already said, but I'll try to say it better.

 

I hope the following doesn't come across as aggressive.

Not in the least. My challenge is to make myself clear.

 

In this context, he did not. In fact I used quite standard set-builder notation.

This is referring to your erroneous construction of a set as [math]\{x : P(x)\}[/math] where [math]P[/math] is some unary predicate.

 

This is not valid set-builder notation.

 

What it is, is informal set-builder notation. It's appropriate in high school, in non-rigorous math classes, in discrete math class, etc. And even in higher math, it's perfectly acceptable to use it, as long as we know in the back of our mind that we really mean [math]\{x \in Y : P(x)\}[/math] for some existing set [math]Y[/math]. In this thread, since the formal properties of sets are under discussion, it's important to be careful to understand exactly how sets may be formed according to the rules of set theory.

 

The use of "set theory without being formal about it" is called naive set theory. It's a perfectly valid thing in every context where the difference doesn't matter. But if we're talking about the properties of sets, we have to work in the realm of formal set theory. We are constrained by the axioms and their logical consequences.

 

It was believed prior to 1901 that a set could be defined entirely by a predicate. Russell's paradox falsified that belief. If you would please read the Wiki page for Russell's paradox this point would be abundantly clear to you.

 

So, what is the EXACT rule by which we can form a set based on a predicate? This is given by the Axiom schema of separation. https://en.wikipedia.org/wiki/Axiom_schema_of_specification

 

This schema is actually an infinite collection of axioms, one for each predicate. It says that if you have an EXISTING set, you can cut it down by a predicate. You can not try to cut down the universe by a predicate, that's Russell's paradox.

 

I do hope you'll take a moment to glance at the Wiki links I'm giving. All I'm doing is walking through the basics of ZFC. By the way I should mention that's Zermelo-Fraenkel set theory. https://en.wikipedia.org/wiki/ZermeloFraenkel_set_theory. The C stands for the Axiom of Choice, so ZFC is Zermelo-Fraenkel with Choice. This is the standard axiom system for modern math; although the professional set theorists study exotic variants.

 

Since for any object or set the assertion that [math]x \in x[/math] makes no sense

[math]x \in x[/math] makes perfect sense. There are in fact people who study non well-founded set theories. These are perfectly consistent theories in which sets can be elements of themselves. https://en.wikipedia.org/wiki/Non-well-founded_set_theory

 

In fact, why can't a set be an element of itself? The ONLY reason is that in ZFC, we make an explicit axiom that says that a set can't be an element of itself. Or more precisely, that there are no infinite descending membership chains. https://en.wikipedia.org/wiki/Axiom_of_regularity

 

NOTE WELL that without this axiom, sets that contain themselves are perfectly valid, and are in fact object of mathematical study.

 

We choose to accept the Axiom of Regularity simply because we only care about well-founded sets; not because [math]x \in x[/math] is logically impossible. On the contrary, from a purely logical point of view, one can choose one's sets to be well-founded or not.

 

In fact it's been noted that the Axiom of Regularity is never used in proofs. It's only purpose is to ensure that we're always dealing with well-founded sets. And if we didn't make this rule explicit, there would be no reason why sets couldn't contain themselves.

 

 

then likewise does its negation, I cannot see your point.

I hope that looking at the Wiki pages I linked will make the point clear. In fact you only need to give the Wiki page for Russell's paradox a good read. It's unfortunate that the forum software doesn't like that particular link, but please do Google Russell's paradox and give a careful read.

 

 

You can of course have [math]P(x)[/math] as the proposition that [math]x \ne x[/math] which will define the empty set

I'm grateful for small points of agreement. Perhaps this is a good sign. I should add, though, that in fact the definition of the empty set is a bit of a controversial point, since we can't define it unless we already have some other set, and where'd that one come from? Let's leave this aside for now, but do be aware that your way of defining the empty set is not without problems.

 

 

On the contrary, YOUR construction may allow the existence of some "universal set", which does indeed lead lead to the Russell paradox, which I state correctly here......

 

"There can exist no set that contains its complement as a proper subset".

Very peculiar way of putting it. But you are confusing well-foundedness with unrestricted comprehension. This is a very common false belief, that Russell's paradox says a set can't be a member of itself [or your alternative phrasing, if that's the same]. Rather, that is the principle of well-foundedness; and as I noted, you CAN have non-wellfounded sets by simply dropping the Axom of Regularity.

 

Russell's paradox shows that you CAN NOT have unrestricted comprehension. That means that you can always cut down or qualify a set by a predicate; but you can not qualify over the universe. The principle that lets you qualify sets by predicates is the Axiom schema of specification, as I mentioned.

 

Or maybe you thought my use of the term "universe of objects" implied I was referring to a universal set. I was not, and there was nothing to suggest that I was.

Well, nothing but the fact that in your very next sentence you used it exactly that way

 

Great Heavens you don't say! I hope you are not offering this as a definition?

I think this was in response to my saying back to you exactly what you were using as a definition. But this point may be lost in the cross-quoting.

 

 

Oh and BTW, which set theory?

There are several versions of modern set theory.

Surely you haven't forgotten that we were talking about ZFC. If you're talking about some other version, please say so.

 

By the way I looked up KFC set theory. There is such a thing. The Google references were VERY obscure and I never found a full definition, but the K stands for Kripke and this is some arcane inside baseball in modal logic.

 

 

Neither wtf or you have shown that it is necessary condition of membership for us to know or be able to know if an element is a set member or the properties of said member.

I think this is getting to be a point of semantics. I already noted that there are only countably many predicates but uncountably many sets, so most sets can't be chararized by predicates; and I also gave an explicit construction of a set that can not be characterized by any property of its members. But if you have some vague notion in your mind about this, it's not a point worth arguing.

 

Frankly I was hoping people would find these examples interesting, particularly since I pointed out that the example I gave is the first step in understanding the famous Banach-Tarski paradox. But if I'm not making my point at all, I take responsibility for failing to communicate with sufficient clarity.

 

 

Surely an element would be a member, within the terms of reference, even if humans had never existed?

Oh what a fascinating thing to say. Set theory didn't exist before 1874 (Cantor) and didn't achieve its modern form till 1922, with the publication of the ZFC axioms. Set theory is a historically contingent activity of humans.

 

Now of course you are right that there is a human notion of "collections" that is very ancient (though I'd dispute that it's independent of humans entirely).

 

But at least the idea of collections does predate set theory by thousands of years. Formal set theory is only a recent attempt to formalize our vague ideas about collections.

 

But in the process of formalization, certain naive notions must be abandoned. We have to transition from intuition to doing only what's allowed by the axioms.

 

I think that's the impedance mismatch we're having. If you're using sets for something, naive intuitions are generally ok. But if you're trying to think about the exact nature of sets, then you have to work in the context of formal set theory. And in fact as you note ... set theory really is quite strange.

 

Sets in mathematics are exactly what the axioms say they are; no more and no less.

 

And even then there are all sorts of nonstandard models for even the standard axioms. So yes, set theory is very strange, and the sets of mathematics are not really the collections of our intuition.

Edited January 11, 2016 by wtf