Thread: All Sets

Start to list all sets starting with A,B,C,…
you can’t get past A:
A, {A},{{A}}, {{{A}}}, …..
Let alone A, B:
A, B, {A,B}, {A,{A,B}}, {B,{A,B}}, {A, {A,{A,B}}}, {A, {B,{A,B}}},
{B,{A,{A,B}}}, {B,{B,{A,B}}},…..

True, the fact that you can’t list them all doesn’t mean they don’t exist. However, it appears to me this construction renders the notion of “all sets” meaningless, especially if you start with A any object or concept in the universe. It's not mathematics anymore, its just words and combinations of words- philosophy (in my opinion).

And each one of these sets has a different definiton that have to be combined.

If you are saying that we cannot have "sets of sets", yes, that's true.

But surely, not for the reason you state. It is not, for example, possible to list all of the rational numbers between 0 and 1 so you can't even "get past 1" "let alone 2". Does that show that rational numbers do not exist?

{A}

This is not in your list of sets. It is a set which contains a set that you listed, so its completely different.

I´m supposing you are saying one cannot have the set of all sets.
In this case, yes, one can´t. For example, in ZFC the axiom of regularity forbids it by not allowing in the language to exist a set that is a member of itself (solving the Russel´s paradox).

However, the "set" of all sets can be very treated mathematically. For this, on needs a stronger language that encompasses the language of ZFC and build a stronger axiomatic. For example, in Neumann-Bernays-Gödel set theory the "set" of all sets is not a set but it is a proper class and you still can formally work with classes and sets in the same formal theory (in the case, NBG).

Unfortunately or not, the boundaries of philosophy and mathematics are not well defined.

Originally Posted by FelipeAbraham

I´m supposing you are saying one cannot have the set of all sets.
In this case, yes, one can´t. For example, in ZFC the axiom of regularity forbids it by not allowing in the language to exist a set that is a member of itself (solving the Russel´s paradox).

However, the "set" of all sets can be very treated mathematically. For this, on needs a stronger language that encompasses the language of ZFC and build a stronger axiomatic. For example, in Neumann-Bernays-Gödel set theory the "set" of all sets is not a set but it is a proper class and you still can formally work with classes and sets in the same formal theory (in the case, NBG).

Unfortunately or not, the boundaries of philosophy and mathematics are not well defined.

I was thinking there might be a way in that direction. But I was thinking of categories. What is the difference between a class and a category?

-Dan

Hi Dan,

Sorry being late.

A category is a diferent kind of mathematical object and it can be more general than the usual notion of set. It is closer notion to groups than to sets, since it was formerly conceived as an algebraic structure. They are usually defined in NBG set theory, but we can weaken it to ZFC.

You can found a good explanation on Wikipedia.
A category is always a pair: the class of the objects and the class of the morphims. So only makes sense of saying the category of all sets if we add morphisms to it. For example, the satisfying functions of model theory.
There you can found there is no category of all large categories. Analogously, there cannot be the class of all classes.

Originally Posted by Hartlw

Start to list all sets starting with A,B,C,…
you can’t get past A:
A, {A},{{A}}, {{{A}}}, …..
Let alone A, B:
A, B, {A,B}, {A,{A,B}}, {B,{A,B}}, {A, {A,{A,B}}}, {A, {B,{A,B}}},
{B,{A,{A,B}}}, {B,{B,{A,B}}},…..

True, the fact that you can’t list them all doesn’t mean they don’t exist. However, it appears to me this construction renders the notion of “all sets” meaningless, especially if you start with A any object or concept in the universe. It's not mathematics anymore, its just words and combinations of words- philosophy (in my opinion).

And each one of these sets has a different definiton that have to be combined.

All you are saying is that sets cannot be listed linearly- they are not countable. Nothing new in that.