Thread topic: Logic, Set Theory, and the Empty Set

[PseuTonym]

The two main approaches to set theory in mathematics nowadays seem to be ZF and another approach. The other approach seems to be not exactly one system, but a collection of different systems that are all very similar to each other, and are known by names such as NB, NBG, or Kelley-Morse.

In ZF, there is no universal set. In NBG there is universal class, but it isn’t a set. However, it seems that sometimes there’s a strategy that allows one to reformulate a statement that involves a universal set. In the reformulation, no universal set is referred to. Instead, the empty set takes on the role that was played by the universal set.

Please feel free to post questions in this thread. If you have a very short and difficult question, then I would appreciate disclosure of your train of thought as you try to answer your own question. If you prefer, you can send your train of thought to me in a private message, rather than posting it for everybody to see.

 

[GEddie]

Is the universal set supposed to be the opposite of the empty set?

 

[PseuTonym]

Is the universal set supposed to be the opposite of the empty set?

The concepts “universal class” and “empty class” are more precise than the vague concept of “opposite.”

Some examples of the vagueness of the concept of “opposite”:
"Mary gave birth to Jesus "
is in a sense the opposite of …
(** anomalous sentence ahead **)
… “Jesus gave birth to Mary.”

However, one could also say that
“Mary didn’t give birth to Jesus”
is in a sense the opposite of
“Mary gave birth to Jesus.”

Every set is an element of the universal class.
No set is an element of the empty class.
One of the first things that one needs to prove in NBG is that the empty class is a set.

 

[GEddie]

Point taken.

But I couldn’t think of the right mathematical term. “inverse” is too specific.

ICXC NIKA

 

[svid2]

Pax Christi!

Huh??!! A CAF discussion of set theory? Cool!

Since I’ve only studied elementary set theory, I need some help.

A set is a collection of objects or things. But what is a class?

Thanks for your patience.

God bless.

 

[svid2]

Pax Christi!

Huh??!! A CAF discussion of set theory? Cool!

Since I’ve only studied elementary set theory, I need some help.

A set is a collection of objects or things. But what is a class?

Thanks for your patience.

God bless.

 

[PseuTonym]

I’ve studied set theory, but only elementary set theory.

Where would you draw the line between elementary and advanced set theory? Some people draw the line at the moment when the manifest goal is to produce rigorous and non-trivial independence proofs, and when we are presented with theorems whose presentation otherwise seems unmotivated.

A set is a collection of objects or things.

What is a collection?

I haven’t forgotten your other question (“What is a class?”). We’ll get there eventually, I think.

Please be patient with me, and please let me know when I’m not being patient enough with you. Thank you!

 

[PseuTonym]

What are we talking about in set theory when we use the phrase “independence proofs”?

Many mathematical problems have the following format. Some assumptions or made. Then you are presented with a statement, and you are to either prove that the given statement is true, or prove that the given statement is false. However, there are other possibilities. We might be able to prove that the assumptions aren’t powerful enough to prove that the given statement is true. We might also be able to prove that the assumptions aren’t powerful enough to prove that the given statement is false.

It may be helpful to understand why this situation arises. In a formal system of mathematics, we may have in mind some analogy with real experience (such as of physical space). However, just as we avoid an infinite regress in our deductions by beginning with some statements that we don’t attempt to prove, and use as the basis of all proofs, we avoid an infinite regress in our definitions by beginning with some concepts that we don’t attempt to define. We use those concepts as the basis for all definitions of concepts that we need to define.

The assumptions might not be restrictive enough to uniquely determine the meaning of a given primitive concept. Instead, we can consider a variety of different meanings. As Russell said, in mathematics we never know what we’re talking about. In that respect, mathematics is like accounting. A column of numbers could represent numbers of shoes, numbers of ships, or numbers of cans of sealing wax.

We never know what we’re talking about because if our reasoning is valid, then we can give a rose another name, and our conclusions about roses will still follow from our assumptions about roses. Theorems about roses, such as theorems about how sweet they smell, don’t rely upon the fact that we happen to use the word “roses” to denote roses.