A question about the meaning of a “set” in ZFC.

[Dale10101]

A question about the meaning of a “set” in ZFC.

A question about the meaning of a “set” in ZFC.

In elementary algebra a set is defined as a “collection of objects”. The meaning conveyed by example and practice is that an “object” is atomic in nature (an urelement). To say that a E A (notation for a is an element of A) means you are pointing to a single urelement in A.

In ZFC there are, by definition, no urelements, so when you say a E A are you really saying that “a” is any subset of A, “proper” or not? Or, is it that any defined set has a top level set of objects and “a” refers to only one of those?

Hmmm, maybe that is it. In elementary algebra you might write A = {b,c,d} while in ZFC you must write A = { {b}, {c}, {d} }. If I write a E A in the first case I am referring to “b” or “c” or “d”, in the second case a E A would refer to “{b}” or “{c}” or “{d}” but would not refer to, say “{ {b}, {c} }.

I must ask for confirmation or correction because I have learned that there is only a hair’s difference between “obviously right” and “obviously wrong”.

 

[Dale10101]

A question about the meaning of a “set” in ZFC.

Add on question about the last point.

given A = { {b}, {c}, {d} }

could a E A refer to the element "b" as well {b}, the set containing b?

Thanks

 

[pka]

A question about the meaning of a “set” in ZFC.
In elementary algebra a set is defined as a “collection of objects”. The meaning conveyed by example and practice is that an “object” is atomic in nature (an urelement). To say that a E A (notation for a is an element of A) means you are pointing to a single urelement in A.

There is almost no way for you to expect a simple answer to you question. There are as many equivalent formulations of the ZFC axioms as there are authors who have written on this topic. Now that is to say that you should find a satisfying text source and follow it.

Here are two that I like. Charles Pinters's Set Theory​. Pinter did his PhD in Paris in the 1960's with some of the best set theorist of the twentieth century. He develops a theory of classes and then defines a set ​as a certain restricted class. This is the way that I learned the material.

Now I have used this text to teach that material: An Outline of Set Theory by James M Henle. As you can see from his genealogy, he is more current in terminology.

 

[Dale10101]

Right

I see what you are saying, my online reading has left me with the impression that the conventions used in formulating math problems differ from one article to the next and can engender confusion. I think I will get An Outline of Set Theory by James M Henle. from Amazon with my next purchase, pretty cheap, about 10 bucks in paper back.

I don't know how deeply I need to go into set theory. Mostly I loved geometry in high school because it started from the simplest definitions and constructed everything upward as a consequence. After that everything from algebra to trig to Calculus, D.E's O.D.E etc for engineering seemed to be a bundle of recipes based on some half explained foundation. I imagined that mathematics as an edifice that could overall be constructed from fundamental axioms and it can I think but apparently not in the absolute sense that I originally thought, Henle makes that clear in his introduction. (Law and Order could probably write a script featuring murderously motivated mathematicians in contentious competition.)

Anyway, for the most part what I am looking for is how mathematics is constructed from the foundational concepts of sets and relations. For my purpose I am succeeding but keep getting drawn into curious corners and hence my questions about the minutia of definitions. Also, the endeavor is a good review of the basic math recipes, I am especially enjoying comparing and classifying different types of problems and formulas. No doubt I will continue asking somewhat odd questions. I do appreciate your help and the help of others.

 

[HallsofIvy]

All "axiomatic systems" start with "undefined terms". That is, really, the strength of mathematics. We apply mathematics to different situations by interpreting the undefined terms for that particular application.

In ZFC, "set" is the basic undefined term. As for A= {{a}, {b}, {c}}, the elements of A are the sets {a}, {b}, and {c}, NOT "a", "b", or "c" themselves.

 

[Dale10101]

Mabuti

All "axiomatic systems" start with "undefined terms". That is, really, the strength of mathematics. We apply mathematics to different situations by interpreting the undefined terms for that particular application.

In ZFC, "set" is the basic undefined term. As for A= {{a}, {b}, {c}}, the elements of A are the sets {a}, {b}, and {c}, NOT "a", "b", or "c" themselves.

Good. I thought that was the case. Thanks.