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

[Xerxes]

wtf I thank you for your forbearance and for the above interesting and informative post.

 

I also apologize for doubting that your knowledge of the subject exceeds mine. Sorry!

 

The use of "set theory without being formal about it" is called naive set theory.

Exactly so. At my college, set theory was introduced as "something you need to be aware of", then dismissed as "dry, not to say arid".

 

We were pointed to a nice little book by Paul Halmos (yellow as I recall), which I read and enjoyed but which was unequivocally titled "Naive Set Theory".

 

But just you wait - one these days I'll best you!! *wink*

[StringJunky]

Nice to see someone concede gracefully... rare as the proverbial rockinghorse...

[wtf]

wtf I thank you for your forbearance and for the above interesting and informative post.

 

I also apologize for doubting that your knowledge of the subject exceeds mine. Sorry!

Hey thanks, no prob!

 

 

We were pointed to a nice little book by Paul Halmos (yellow as I recall), which I read and enjoyed but which was unequivocally titled "Naive Set Theory".

Interesting point. NST is actually an axiomatic treatment of set theory but not excessively formal. But I definitely agree that a lot of people get quite far in math without paying much attention to the fine points of set theory.

 

But just you wait - one these days I'll best you!! *wink*

Honestly I wasn't trying to best anyone, just toss out an example of how sets can be created without lists of elements and without predicates. You responded to a question on category theory the other day so I'm sure you can best me there Anyway I'll still go with "zero" as the number of properties necessarily possessed by all the elements of a set that characterize the set, if that was the original question.

Edited January 12, 2016 by wtf

[studiot]

First I would like to thank both wtf and Xerxes for the large amount of hard work and thoughtful interesting material they have put into this.

 

Perhaps unfortunately this has introduced the issue of 'what is a set?', which was not the subject of this thread, interesting though it may be. Indeed I would support the contention that that issue deserves a thread of its own.

 

I think it was Xerxes, though I cannot locate it at the moment, that offered a link to Professor Weiss's lecture notes and thoughts on the subject of what constitutes a set. These notes use Richard’s paradox to exemplify what is not a set, in his definition.

 

In my original post I did not specify any particular definition or restriction on what might constitute a set.
Indeed the fact that we have so many words for an assembly of objects suggests that there are many (some subtle) shades of meaning. So we use

Aggregate, agglomerate, collection, class, group, assembly, category and many others.

 

Now during my working life I was an applied mathematician, which means I was interested primarily in obtaining answers by fair means or foul.

I have more time now to lift the covers and look underneath and this is what I was attempting in starting this thread.

 

To an applied mathematician, restrictions of set definition based on Zermelo does not cut it. Rather it introduces severe difficulties, for example the set of all men or all men in Denmark or whatever as heavily used in actuarial mathematics.

 

In fact an effect I have noted is for restrictions to be stated at the beginning of something, a course, a book, a development of theory etc and then to be forgotten later when attempts to apply that theory outside the domain of definition are made.

 

Zermolo’s theory is like this and placed somewhere in the middle of the pecking order of set generality. In particular it was developed for and strictly is restricted to sets with numbers as elements. This does not prevent it being useful sometimes elsewhere.

 

I cannot put this idea of generalisation v restriction more clearly than Professor Phillips. This is extracted from his Analysis, Cambridge University Press.

 

 

…The other method of pursuing the study of mathematics is the reverse of the historical order of progress. Instead of pursuing the constructive process towards increasing complexity, we proceed, by analysing, to greater abstractness and logical simplicity. Instead of considering what can be defined and deduced from our initial assumptions, we examine whether more general ideas and principles can be found in terms of which our original starting point can be defined or deduced. This second method is what characterises the study of what has come to be known as mathematical philosophy…

 

So to the original post.

Although zero elements imply total properties = elements times available properties = 0 I am still not sure about this as I said in post#8.

[wtf]

Perhaps unfortunately this has introduced the issue of 'what is a set?', which was not the subject of this thread, interesting though it may be. Indeed I would support the contention that that issue deserves a thread of its own.[/size][/font][/color]

If the subject of the thread was, "How many properties does a flapdoodle have," then the natural response would be: "Well, first tell me what you mean by a flapdoodle."

 

 

I see two cases here.

 

* If you are interested in how mathematical sets work, I'm happy to give my perspective.

 

* But if you only want to say that well, for all we know a set is a category or a group (when in fact is it NOT these things because these things have very specific, widely agreed on technical meanings) then I can't argue with you because you are not doing math. You're doing something else. You listed words like class, group, and category, which are each technical terms that do NOT mean set. So if you want to say, "Well if we deny the standard terminology of math, then I can make the following point," then I can't argue with you or disagree with you or frankly even conversate with you. Because I'm not interested in the point of view that says, "Let's pretend to talk math while denying and ignoring the actual content of math."

 

I tossed out that [math]\mathbb{R}/\mathbb{Q}[/math] example for its inherent interest in showing readers how modern mathematicians use the axioms of set theory. And for people who have maybe heard of Banach-Tarski or nonmeasurable sets, a little step in the direction of those constructions. And of course IMO it does support my point that there are sets not characterized by properties. But I can already prove that via a cardinality argument so I've presented now a formal proof plus a concrete example.

 

I really just don't get where you're coming from. If you want to say well nobody really knows what a set is so for all we know each element has exactly 47 properties, it's ok by me. That's doesn't invalidate my point

 

Maybe I'm just not understanding you. Can you clarify your intent? I feel like I'm supposed to defend something but I actually made my point several posts ago and don't understand yours.

 

Or perhaps ... would you say that perhaps you're struggling to understand more about the philosophy of set theory? You know quite a lot of thought has gone into all this by people far smarter than me. From Cantor through Zermelo and Turing and Gödel and Frege and Wittgenstein (who thought it was ALL wordplay) right up to the modern geniuses like Hamkins and Woodin, exploring and defining the very forefront of what humans think about sets.

 

So are you perhaps just trying to make some point about the philosophy of sets?

 

Although zero elements imply total properties = elements times available properties = 0 I am still not sure about this as I said in post#8.[/font][/size]

I can't parse that. Can you explain please?

 

I went back to #8, you asked if the empty set has properties. Yes the empty set does have properties, lots of them. Let's list a few.

 

* It's the set of flying purple unicorns.

 

* It's the set of married bachelors.

 

* It's the set of real numbers not equal to themselves.

 

* etc.

 

So the empty set is not an example of a set characterized by no properties. It's an example of a set characterized by MANY properties.

Edited January 19, 2016 by wtf