Numbers as sets

Amazing how you have bent a relatively clear definition into something that is capricious and arbitrary. The way you bend established definitions you are either a beginning law student (with little talent) or the Queen of Hearts shrieking at the Knave "Sentence First, Verdict Afterward!".

In primary school, where I was introduced to the rudiments of set theory, I was given the following definition

Quote:

In set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc.

With a naive definition of a set, like a set being "a collectio of objects" cannot avoid at least one of the Russel's Paradoxes.
Recall that the set consisting of "all the sets" leads to a paradox.

I may be bloody stoopid, but at least I don't use capricious definitions.

Using the concept of a well defined collection of objects, cows in a herd would be a set of cows.

Rap

It doesn't lead to a paradox, but a set needs to be defined somehow, and 'collection', 'group', or 'having members' does not do the job.

Brandon9000 wrote:

John Jones wrote:

Brandon9000 wrote:

John Jones wrote:

Brandon9000 wrote:

John Jones wrote:

Setanta wrote:

Quite apart from coming here in every thread with an agenda do demonstrate the superiority of his perceptions over the arduous studies of professionals, Mr. Jones now thinks to set for us the terms of management of a thread and debate. Which is, in his own term, contemptible.

Play with the duck. Or something.

May one inquire what your qualifications are to opine about Mathematics?

We are not making a mathematical enquiry. What are your qualifications in cookery?

We can discuss me next if you like. Now, you are expressing opinions and drawing conclusions about Mathematics. Stop playing with words and tell me your qualifications to do this.

A mathematician does not draw conclusions from mathematics about mathematics. How could he? I think you came into the topic at the level of the rules that constitute it, and have never worked at its lower levels. If that's the case, there was never any possibility of discussion.

You are pontificating on the subject of Mathematics without going through the preliminary step of learning the topic first. Many of us spent years in high school and college painfully climbing this mountain a foot at a time, but you lack the self-discipline to do this. You are like someone who, speaking no Russian, offers his services as a Russian translator, making up whatever nonsense pops into his head. I'm wrong? I dare you to choose your best mathematical theory and submit it to a legitimate, peer reviewed mathematics journal.

Mathematicians are not interested in the foundations of their discipline. They are not taught it, nor is it relevant to their studies. If I wanted someone to speak about the foundations of mathematics, I would not expect a learned response from a mathematician.

I suppose you could define a set as a membership relation if you didn't like the 'collection of objects from some universe' definition. As to your initial question about whether members of a set can confer their properties onto the set, my instinct is to say no, but this assumes that I'm interpreting your question correctly.

The set defines the rule or the properties which members must follow or have. The set, interpreted this way has no properties itself. So in your example of the cows, you would define it as the "the set of all cow such that cow belongs to herd x". I'm not quite sure what you were getting at when you said a set of cows would be a herd. Why can't a set of cows be a herd?

Ok, maybe I'm no mathematician and certainly no philosopher, but I'll be damned if I can figure out what you're getting at.

Are you saying a set can't be a collection of objects because a set of cows is not a herd? I'm totally lost as to what you're proving there. Certainly you could have a set of cows which was not a herd, like the set of all cows that I've eaten this year -- they don't necessarily belong to the same herd.

As to your questions, I don't quite follow. Here let's take them one at a time.

1)If the members of a set do not confer their properties on the set then how can a set be said to have a subset?

What precludes a set from having a subset?

2)Also, how can the set of even numbers be infinite?

Again, how can they not be? The set of even numbers is the set of all numbers that have the property of being even. The set is infinite and not even so clearly the properties of the members are not conferred on the set in that case. Since there are an infinite number of even numbers, the set containing all even numbers is infinite.

3)Again, if a set has no members, how can the set be an empty set?

How can it not be?

One of the difficulties of this discussion is that I'm not quite sure which definition of set you are accepting. You appeared to be rejecting the "collection of objects" definition earlier so I can guess that you don't like that one. If we're going to talk about the logical implications of defining sets, perhaps we could all choose a definition on which build. I have in my head what I understand the concept of set to be and it could fit any of the definitions provided. But bouncing between them isn't very conducive to logical deduction.