You could try looking up Pythagoras and music. You'll read about harmonic series and such. That's all I really know. Sorry couldn't be more helpful.

Searle (and perhaps others) correctly point out that the ontological structure of truth doesn't quite fit into an objective vs subjective dichotomy. There are certain objective features of the world which are established by prevailing human attitudes and sentiments--subjective human experiences. The value of money is a good example. You can purchase things with money because society has collectively agreed to consider it as a valid medium with which to exchange. This means the value of money is predicated on subjective human experience. However, the value of 10 dollars is not subjectively decided, but rather, the result of a complex negotiation that gives rise to the objective quality of the value; you can't just decide you'd like to buy a Ferrari with 10 dollars.

If I understand you correctly, you are saying "the set of all sets which are not subsets of themselves" is necessarily a set. — EnPassant

If we accept Frege's abstraction principle, then yes, we are committed to the position that this is a set. As you have pointed out, Russell's Paradox illustrates that such a notion is inconsistent, which is why no one accepts Frege's abstraction principle.

You seem quite motivated to learn more about this topic (which is a very good thing! :up: ). I really recommend you check out the book I mentioned in my first post. It is actually quite good and the first few chapters are quite approachable. It's Dover so its also extremely affordable. A worthwhile investment.

If the mathematical formalism is difficult at first, then you might want to try Halmos' Naive Set Theory. While I own a copy, I can't honestly say I've sat down and spent any real time with it, but I understand it is a very good place to start.

Best of luck! Hope to see more of you in the forums.

I taught college math for a couple of years right out of college. I can tell you, the students still aren't happy. :rofl:

Suppose set A = {a, b} the subsets of this set are:

{{0}, {a}, {b}, {a, b}} Now redefine set A as:

A\A. That is, A\{a, b}

Now the subsets of A are:

{{0}, {a}, {b}}

This is what I mean by X\X. — EnPassant

Ah, so this is not the correct notation. You have asked us to consider the set $2^X \setminus X$, where $2^X$ is the power set of $X$. Note that we cannot redefine X and that this set is quite distinct from $X$ itself.

In this way the paradox is avoided by defining a set that contains 'All sets...' but not X. — EnPassant

I see what you mean. I'm still sticking to my previous statement on why this set is ill-defined. In any case, I don't think one can avoid Russell's Paradox proceeding in the way you have--regardless of the logical consistency of the argument.

We have to remember that Russell's Paradox established the inconsistency of Frege's set theory, in particular, the abstraction principle which loosely stated that, given any property P, there exists a (unique) set A consisting of those and only those things that have property P.

Russell suggested we consider the collection "the set of all sets which are not subsets of themselves". Note that, by Frege's abstraction principle, this is necessarily a set. Asking that we, in effect, look the other way and consider instead another set, as you've proposed, doesn't prevent us from considering Russell's set.

The paradox establishes that the abstraction principle is unsound. Historically, this lead Zermelo to suggest the limited abstraction principle to prevent the issue--one would need to clarify which set from which the elements satisfying P are being taken. Thus we cannot speak of the set of all x having property P, but we can speak of the set of all x in A that have property P. For an explanation of how this avoids the paradox, I recommend looking at pg. 12 of Smullyan's book, linked in my first post.

Of course The Plague would be far more apt for present consideration. — Banno

:rofl:

Hmmm, I remember the book being a tad longer than two pages...

Given that this is Camus, probably theater. But this is actually what Camus meant, that one should not commit suicide because the logic of life suggests that you should be a performer, an actor. Not in a vague sense like an entity with agency or something. Literally like a reads-lines-for-a-living actor. To be fair, if I remember correctly, his general argument is that the logic of life suggests that one should have as much experience as possible. He concludes that, as an actor, one is able to derive the most experience possible as a human.

Really someone should double-check my statements, it's been years since I've read the book. But that's how I remember it.

and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material." — jgill

If I understand it correctly, I can't jump on board with this third point. It's always good to know about the applications and historical context of a subject, but often, the most compelling aspect of a mathematical subject is its elegance and simplicity. If the topic is mathematically elegant, that is usually more than enough to spark interest and curiosity and keep a student interested. Conversely, even the most applicable subjects can be arduous to study due to the clumsy and complicated math.

My idea is that it can be framed in terms of set theory alone without the invention of classes. — EnPassant

Hmm, I'm not sure I follow. ZFC does not deploy the use of classes, it is set-based. One can avoid the difficulties of Russel's Paradox without the invention of the class object. Indeed, I believe the use of classes was for alternative purposes, namely, to simplify some aspects of ZFC. From what I understand, NBG is generally regarded as more elegant than ZFC--again for more info see the book I linked in my last post. If you don't like classes, then just avoid the use of a class-set theoretic system.

Let Set X = "All sets that do not contain themselves as subsets"\X

I don't see anything wrong with this definition... — EnPassant

The problem with this definition is that the set of all sets that do not contain themselves as subsets is shown, by Russell's Paradox, to be logically contradictory. Your definition requests that we posit an object which is logically contradictory, and then remove $X$ from it. This is akin to requesting the reader to take the smallest prime number with exactly three divisors, subtract it from itself, and then insist that the answer is 0.

It's been a long time since I read The Myth Of Sisyphus--back in high school I think it was. I don't recall all the detail. However, I do remember Camus trying to pass an argument suggesting that the logic of life recommends being an actor. The tendentiousness of this line of reasoning always felt egregious and the conclusion silly.

Would your definition of $X$ imply

$X \cup X \neq X$

Indeed, it seems to imply

$X \subset X \cup X$

where containment is strict. You should consider checking out NBG class-set theory which is an alternative formulation of set theory. The theory uses the notion of classes to avoid Russell's paradox. I'm not sure I can say I'd prefer it over the standard ZFC but it's interesting to having an alternative point-of-view. There is a good dover book by Smullyan and Fitting on it as well. It can be used for self-study.

EDIT: fixed the second equation.