Naive (intuitive) definition of set

[Shellsunde]

naive (intuitive) definition of "set"

I happened upon a book by a Joseph Landin, once head of the math department at University of Chicago and subsequently Ohio State University, in which he gives this as a definition of a set and states this property:

A set is a collection of objects; the nature of the objects is immaterial. The essential characteristic of a set is this: Given an object and a set, then exactly one of the following two statements is true.
a) The given object is a member of the given set.
b) The given object is not a member of the given set.

Shortly thereafter, he writes,

It might be tempting to speak of "the set of people who will visit the city of Chicago during 2050." But, clearly, such a collection cannot qualify as a set according to our understanding of this term.

Would you please explain why his second statement is so? I cannot fathom why this is not a perfectly consistent and constructible set.

 

[Office_Shredder]

I think what he's getting at is that if I ask you, right now "Is this person inside of your set?" you cannot say yes or no because you can't tell the future.

 

[phinds]

Would you please explain why his second statement is so? I cannot fathom why this is not a perfectly consistent and constructible set.

As OfficeShredder said, how would you construct the set? Do you have a time machine? For example, tell me this --- am I in the set or not?

 

[gopher_p]

The definition given does not state that we should be able to determine whether or not an object is a member of a proposed set. We can't determine whether or not $e+\pi$ is a rational number, but I doubt the author would claim that the collection of rational numbers is not a set. There must be something else going on, but I can't tell. It's certainly not "clear".

 

[Shellsunde]

I think there are two problems. One, as you said, is that it is not yet known or even knowable who will enter Chicago in 2050. The second is, what precisely constitutes the boundary of Chicago. I think the future problem is the one the author wished to emphasize.

My problem with that is the author's overly restrictive and pedantic usage of the word "will". When one says something will happen, the speaker means the something "will", in the future, change from potential to actual. I can't even say that without using the word; that's what "will" means and it's self-evident. I took it as reasonable to overlook that I can't actually identify the elements of the set in favor of admitting the hypothetical of the author's description. Why say it will happen when he means you can't yet know if it will happen? Just to trick the reader?

Clearly the author wants to illustrate that a set must be constructible, but he displays such a rigidity in his thinking that it made me wonder if his manner of thinking and use of language is considered part of 'proper' or normative mathematical or set-theoretical thought and language. (Admittedly, this is no longer a set-theoretical question or discussion, but one concerning the philosophy of mathematics.)

Thanks for your replies and consideration.

 

[Hornbein]

I don't agree with this author. That Chicago set could be useful. He seems to be requiring that every set be constructable, which I find weird. I would say that any collection of well-defined objects without duplicates is a set.

However, teacher is God in the classroom, so whatever he/she says goes.

 

[Office_Shredder]

I don't think the set needs to be constructable, but I don't think the future set is well-defined is the whole point. If the set of people who will be in Chicago is a well defined-set, then I am either in it or not in it. If I am in it, I can choose to not go to Chicago just to invalidate the set. If I am not in it, I can choose to go to Chicago just to screw you over as well.

The set of rationals/irrationals has the same problem that we can't determine whether some numbers are in it or not, but we DO know provably that every number is either rational, or irrational, and if it's one it can never be the other. Don't get too caught up in the philosophy of whether the future is set in stone and people are predestined to go to Chicago, just understand the broader point which is that the set needs to have a well-defined membership, even if you can't determine whether every object is in it or not.

 

[phinds]

The whole problem would not exist if instead of

"the set of people who will visit the city of Chicago during 2050

He had said

"the set of people who will have visited the city of Chicago during 2050

The latter is a well-defined set even though you won't be able to construct it until the end of 2050

 

[Shellsunde]

I thought there was complete agreement that a set has to be constructable. Evidently not, OK. That a set's elements must be well-defined is an essential characteristic of a set, too. That's an important conition, too.

To Office_Shredder, are you using 'will' in the sense of 'intend', so as to say since you can change your intent, you can thus change the elements of the set? I'm sure the author meant 'will' as a statement of actual being, the people who in fact were or are to be in Chicago, however their choices may have wavered.

 

[Office_Shredder]

I think trying to deconstruct the author's intent with regards to grammar and his belief in free will is a futile effort, and it's better to just take away the point of the example even if the example itself sucks. As we can see apparently everyone has a slightly different idea of why this example works, but everyone agrees with the main point that it is intended to show that you can write down a sentence which fails to make the elements of a set well defined.

 

[D H]

I happened upon a book by a Joseph Landin, once head of the math department at University of Chicago and subsequently Ohio State University, in which he gives this as a definition of a set and states this property ...

I suspect you are writing about An Introduction to Algebraic Structures by Joseph Landin.

Some generic advice: When reading a math text, don't get hung up on what is said in the first few pages. Or ten pages, or in some (bad) cases, the first 60 pages.

The title of this thread is naive (intuitive) definition of "set". I don't have the book, but a number of sellers show the first few pages. From that, it's rather obvious that Landin is not an intuitionist. His theorem 2 uses proof by contradiction. Intuitionists reject such proofs. The law of the excluded middle doesn't exist to a strict intuitionist.

Apparently Landin is a mathematical constructivist, but he does not go quite so far down that rabbit hole as do intuitionists.