Set Theory

Philosophy Forums Science, Technology, and Math
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In preparation for my post on fractals, I came across a number of references to Set Theory. One reference mentioned that there were over 120 definitions of a set, and in fact the word set had the largest number of definitions of any word in the English language.

Additionally, there were a number of derogatory comments about Nicholas Bourbaki (and, of course, I am a fan), a group of French mathematicians that formed together and wrote under that name. Bourbaki basically tried to abstract and generalize the findings of mathematics, in a number of different areas, and set those findings, on what they thought to be, the bedrock of Set Theory. The advantage of writing Mathematics in terms of set theory is that the conclusions should hold over the broadest possible conditions.

One of the most interesting things about Set Theory is that there can be no such thing as the set of all sets. Anyway I thought that I should try to learn more about Set Theory in order to see what a set actually is.

Having way too many unfinished books, I decided that the precedent was set, so I rushed out to buy the book entitled “Basic Set Theory” by Azriel Levy.

Levy writes: “We start with the null set , and from it we obtain the set {}, from the two sets and {}we obtain the sets {,{}} and {{}}”. Note: I used for the symbol representing the null set where Levy used 0, but I think his notation is misleading.

Now my question is: Does anybody, including Levy, have a clue what he is writing about?
The Null set is not defined!!!

I am OK with the statement:
In the Universe of all fruit, the intersection of the set of apples and the set of oranges is the null set in that Universe.

But I simply can not comprehend what absolute nothing is. To me the set of no sets is analogous to the set of all sets, which as I said before does not exist. We could start by trying to throw out the things we know but, not only would it take more time than those with ADHD could stand, once you conceive of trying to throw out the Real numbers which are uncountable in number, I think there is a problem.

Set theory is a land with giants like Russell and Godel, and it is the foundation for much of Mathematics; but I think that its’ gates are guarded with pathological demons and dragons.

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All math is set theory!!! :smiley:

The null set is just the “idea of a set”. You can just think of it as zero, and the trouble you are having would be the same as the ancients had with 0. But its useful, and I find it easier to swallow the “idea of a set”, since all our thoughts are sets, then nothingness or 0. The world is weird, math is weird… who woulda thunk it, what the fuck is “i” anyways?? Dunno but its highly usefull in explaining quantum phenomena. Quite recently they have used quantum mechanics in an attempt to solve Reimanns hypothesis, (THE ANSWER IS 42!!!) and have been quite succesful, Why is there this quantum wierdness? why does zero exist, and what the fuck is “i”? Dunno but it sure as hell is interesting, and any of these answers are to be found within the relation of physics to mathematics IMHO.

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Hi: Rounder:

Thanks for the response. I have always enjoyed your posts and I particularly liked your post entitled “3 cheers for the Dali lama!!! WOOT WOOT”

I have done a little more work on the nature of the Null Set and it appears that I am in the small minority that objects to the absolute Null Set.

Technically, according to Wikipedia, the Null Set is determined by a given measure m to be a set S such that m(S) =0. However in the case I presented the term the null set is actually the empty set.

The author writes: “In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. In axiomatic set theory it is postulated to exist by the axiom of empty set and all finite sets are constructed from it. The empty set is also sometimes called the null set, but because null set means something else in measure theory, that term is generally avoided in current work.”

He goes on to write: “Does it exist or is it necessary?
While the empty set is a standard and universally accepted concept in mathematics, there are those who still entertain doubts.

Jonathan Lowe has argued that while the idea “was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.” It is not clear that such an idea makes sense. ‘All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that ‘have no members’, in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation.’’”

This stuff is actually very important, as you have noted. Not the least of which is the fact that the sequence of Null Sets can be interpreted, in a formal mathematical way, to define the integers.

I have not formulated a clear point of view at this time, but I still feel more comfortable specifying a domain/Universe to which a set is related than accepting the standard notion outright. This specification is standard practice in virtually all the mathematics with which I am familiar.

Thanks again for your response.

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I doubt most have a clue. I find that there is a trend of so called “academics” that understand or can define or explain little about an issue to hide behind vast generalizations masked in obscurity. Clearly for financial reasons.

For example. Quantum physics… string theory and 11 to 15 “dimensions”.

“Well it all works…uhmm…ahh… “MATHEMATICALY”…when we take it to 11 dimensions or more.”

Ridiculous. When asked to define what is a “dimension” every single expert comes clean that they have absolutely no clue.

I feel this is a prime example orthodox academic tactics employed by those desperate to sell their work.

Instead of saying string theory fails to work mathematically in every computer simulated model of the universe……. they choose to add in at least 11 variables/values/constants in addition to the mathematic expressions of string theory to prevent the computer simulated universe from collapsing…

They choose to label this cheating “DIMENSIONS” in order to keep the interest in their academic funding. Marketing. Thus what is the sign of utter failure… the adding in of vast sums of variables/values/constants to force the equation to work…becomes a well obscured generalization and a way to support life. In time, things may be worked out, but until then such tactics are a must to keep the academic world afloat.

The point is that one always needs to read between the lines when human hands are involved.

The intersection? How obscure. I think defining “intersection” would be a must. To say that the intersection of a basket of apples and a basket of oranges in “nothing” is a tad… unsupported is it not?

I would think that the intersection between the two would be “fruit”. Both are fruit. But this is all based on the idea that intersection means “in common”…

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Hi something_else:

I too am skeptical about string theory; however we are probably in the odd position of being in the majority in this case. There was a wonderful article about this issue in the Discover magazine a few months back. One of the mathematicians at Columbia (I forget his name) spent some time on this matter. Ironically, Brain Greene, who has probably killed more trees than anyone promoting string theory, also teaches at Columbia.

As far as the intersection of two sets is concerned, this is actually well defined and has been widely used for many years. The intersection of two sets A and B is defined to be the set S such that x is an element of S if and only if x is an element of A, AND x is an element of B. The visual aid of the old Venn diagrams might also be useful to understanding of my statement.

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Hi Ed,

Interesting post! I’m glad to see some mathematics on here. I know quite a bit about foundational math and set theory, and would love to talk more about it and find new things to ponder, if you have any questions or intrigues.

The fact that there is no “set of all sets” is a very interesting fact. It’s actually fairly easily proven. A stronger version of this was proven by Bertrand Russell (the freakin’ man!) around the early 1900s, and is called Russell’s Paradox.

Interestingly, a branch of mathematics evolved called “Class Theory”. A class is a lot like a set, but it has no size restrictions. If you refer to “all sets”, we know that can’t be a set - but it IS a class.

Another note: it’s true that all mathematics done today is a specific category of Set Theory. But there is math done in different axiomatic systems - basically, math with different fundamental assumptions than the ones of set theory.

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Hi Twiffy:

I can not tell you what a breath of fresh air your post was to me.

There are a number of bright people on this site, but probably less than a half dozen are mathematically sophisticated.

As far as intrigues go:

A) I have been contemplating whether or not the definition of the Empty Set was self contradictory. My particular angle would be based on Class of All Sets. Since it is not a Set, could it be formally included in the Empty Set? I might be able to make mischief there.

B) Though this is not an intrigue by itself, I have been contemplating the Axiom of Comprehension in Levy’s Book “Basic Set Theory”. Can you tell me why the constrained variables are called free and the unconstrained variables are called not free? I will also mention that Russell’s Antinomy is a great counter example. And while I am on the subject, do you have a better book on Set Theory than Levy’s? I’m OK with technical literature (in fact, I would probably prefer it).

C) I have started a series of posts on Models because I believe that these human constructs reflect on our nature; and I would like to see what logical constraints might exist among these constructs. My next post will be on Model Theory itself. The remaining posts include Douglas Hoffstadtler’s comments about meaning: (from “Godel, Escher, Bach: an Eternal Golden Braid”), Comments about Equivalency Classes,
Relational Data Bases, and Artificial intelligence.

Any way that’s it for me. What about you?

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Hey Ed, sorry for the delayed reply.

A) If you have a set X, and the only restriction on X is that it contains no sets as elements, it can still contain classes, certainly. A set defined this way wouldn’t be self-contradictory, but it would be poorly defined, since there are multiple sets with this attribute. But the most common way of defining the empty set is as the unique set with no elements, sets or classes, which does the job of precisely defining it so that it is unique.

B) The book by Hrbacek and Jeck is really excellent - it’s a rigorous enough for any math class, but it also skips some formalism and explains the concepts in plain english, which is nice as an introductory book on the topic. It’s a little expensive, tho - new I think it’s about $85 on Amazon.

C) It would be terrific to see a post on Model Theory, especially as a treatment of meta-math and incompleteness theorems and the like. I’ll keep an eye out for these in the Sciences section. I’ve been posting mostly in the philosophical section, but I think the sciences will be more of an interesting forum.

I’m working on a more generalized form of Model Theory right now, which is promising in a few respects:

  1. It allows for rigorous treatment of philosophical topics, which will be a breath of fresh air for philosophy - the ability to settle topics, and to prove theorems.
  2. It has applications in how AI programming may be optimally structured
  3. It might allow for meta-proofs of incomplete statements. For example, the Continuum Hypothesis is unprovable, but it is still either true or false, even if this truth value can’t be proven from within Set Theory. But it could still be proven in meta-set theory that its truth value is “True”, although consistent techniques for this sort of proof don’t exist.

So I’m really interested in mathematical and philosophical questions that relate to these structures. What’s the real nature of propositions that are unprovable? Are there different categories of unprovable propositions? Etc.

Do you know much about Incompleteness?

-Tristan

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Oops! I totally forgot to respond to this.

Unfortunately, I have no idea why Levy does that. I haven’t read his book - in my course, we used a different text. Have you searched the web to see if it’s in the errata of the book?

The Axiom of Comprehension is more frequently (and more accurately) called the Axiom of Subsets. It simply states that, given a set X and a formula A(x), there exists a subset of X defined by all members of X that satisfy A(x).

In more precise terms, you insist that a variable is free if it isn’t being quantified by a “there exists” or other such quantifiers. Because of this I’m tempted to guess that Levy just made a mistake; however, I won’t commit to that, because the context of what he said could fill in some important gaps there. If it’s worth it to you to take the time to quote the relevant text to me, I could give you a better answer.

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0 is not in the null set. I don’t know why anyone would put that. The whole point of null set is that it is a set with nothing inside of it. Null set has to be defined otherwise we cannot even talk about it. Null set is the set which has no element in it. It is not denoted {phi} or {0} because phi can be seen as an element and 0 is in real. Why not just use { }? It is perfectly fine I think to use that notation to denote the set with nothing in it. Of course, then we would have to think about nothingness which can be kind of pointless considering by definition, nothingness does not exist. But it makes mathematics clearer in that when we have other definitions involving sets, we usually talk about non-empty sets. Like the axiom (some use it as an axiom, others do not) that every non-empty set of real numbers that has an upper bound has a least upper bound. But this can be proven taking another more basic axiom.

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Hi Twiffy:

Thanks for your response.

I am still having trouble with the Empty Set, { } (I like light_eclipseca’s notation best) , but I am having trouble formulating an articulate response.

One related question that I do have is: Do you know if hierarchies are countable in nature or uncountable and why? I am thinking of this in the following terms:

H[size=75]0[/size] = [size=100]Sets[/size],
H[size=75]1[/size] = [size=100]Collection[/size] of all Sets (Classes),
H[size=75]2[/size] = [size=100]Collection[/size] of all Classes,
H[size=75]3[/size] = [size=100]Collection[/size] of all H[size=75]2[/size][size=100],[/size]
[size=59].
.
.[/size]
[size=100]H[/size][size=75]N[/size] = [size=100]Collection[/size] of all H[size=75]N-1[/size]

[size=100]Your[/size] comments in response to my questions about Levy’s book seem right on, and I have purchased the book “Introduction to Set Theory, Vol. 220” by Hrbacek and Jeck.

My knowledge concerning Incompleteness comes primarily from reading the recent edition of the book “Godel’s Proof” By Ernest Nagel and James R. Newman which was edited by Douglas Hofstadter. (I learned new things and generally was very excited by the book.)

I can not recall, whether or not, axiomatic geometry is complete ( I think it is). From the book “God Created the Integers” edited by Stephen Hawking I learned that Godel showed that Predicate Calculus was complete in his doctorial dissertation. Obviously, the big deal is that Principa Mathematica is not.

As far as Model Theory goes, this is new territory for me. However, my preliminary work on the subject turned up an online Stanford Encyclopedia of Philosophy article on First-order Model Theory by Wilfrid Hodges. This was interesting because the first work on Model Theory was apparently on formal languages, but the later more mathematical work on First-order Model Theory can serve as a paradigm for the work on languages.

In any case I am looking forward to your work.

Hi light_eclipseca:

Aside from your notation of { } which I find preferable you wrote:

“Of course, then we would have to think about nothingness which can be kind of pointless considering by definition, nothingness does not exist”

This is exactly my problem! Probably some personality flaw on my part.

Thanks again.

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The {} notation is definitely easiest for internet forums. Sometimes using the phi-esque symbol can be misleading, too.

Heirarchies are definitely countable. There are really only two; sets and classes.

Classes are actually restricted from being elements of either classes or sets. This is because, again, of Russells’ Paradox; if A = set of all sets that don’t contain themselves is contradictory, and the resolution is to make A a class, you could still form A’ = class of all classes that don’t contain themselves. If classes could contain themselves, you’d still have a contradiction.

Thus, you have sets and classes. Now, if you’re talkin’ about heirarchies of ordinals, they are very much uncountable. In fact, the collection of all ordinals itself cannot be a set because there are so many; that collection is a class.

Yeah, axiomatic geo. is complete. The real and complex number systems can be axiomatized so that they are complete, too. But Set Theory (and thus Principia Mathematica) is very much incomplete.

It’s really neat that you can give the intuitive version of the proof really easily: “this sentence is unprovable” does the job. The only reason the proof is difficult is because you have to prove that that sentence can actually be encoded in Set Theory.

-Tristan

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Hi Twiffy:

I got the book “Introduction To Set Theory” by Hrbacek and Jeck today and upon a quick perusal it looks very readable. Thanks.

If you don’t mind me asking, what is a collection of Classes called, and why would that collection, if it consisted of all Classes, not be a hierarchy of Classes? I, vaguely, remember dealing with this in a Topology class.

One thing that I found interesting in reading the Godel Proof, was Godel’s numbering system and the concept of numbering sentences. I thought it was brilliant, and I doubt that very many people have pondered this.