Talk:Set theory

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Needs better overview of major topics, plus information on history and motivation. Tompw 12:13, 6 October 2006 (UTC) A vital article.

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Contents 1 Shatter 2 Formalism 3 Collections of abstract objects 4 list of set theory topics 5 Bullets 6 Self-contradictions in Set Theory 7 A few basic questions 8 Anwers to philosophical objections and a question about adding a new section to the main article

[edit] Shatter

The concept of "shatter" or "shattering" is vitally important in the fields of statistical learning theory, empirical processes, and probability theory in general. I have begun to improve the article entitled "shatter", but although it needs more content, it does indeed warrent an article of its own. Moreover, soon, I hope to begin an article on empirical processes, which is not in the Wikepedia yet, but really warrants an entry. I appreciate the editing and improvements made by Trovatore to my new article on shattering (which made many needed corrections to the earlier article, which was a good start, but had some difficulties about sets, subsets, and classes of sets) which I will continue to augment. -- sorry I am new at Wikipedia, but an old time mathematician. Sorry; forgot to sign in this time...will try to remember next time. Thanks again, Trovatore. Would like to discuss more about how we needed to invoke the Axiom of Choice in some of our work on shattering, and did indeed run into Russell's paradox.

I just remembered something I would absolutely love to see discussed in excrutiatingly fine detail! :-) Alternatives to Zermelo-Fraenkel set theory is a topic that has always fascinated me, but my limited mathematical skill (most of the material available on alternative theories seems to be at a graduate level -- i haven't gone beyond 1st year college mathematics) has meant I haven't found much information I could understand.

I'm especially interested in theories which allow the existence of a universal set, yet still somehow manage to get around Russell's Paradox? -- SJK

An example is New Foundations. -- Toby Bartels 07:58 12 Jun 2003 (UTC)

[edit] Formalism

A question: atleast in Europe I often hear talk about Formalism, how is this related to set theory? -- anon

This is part of the philosophy of mathematics. -- Toby Bartels 07:58 12 Jun 2003 (UTC)

[edit] Collections of abstract objects

Sets don't necessarily represent collections of abstract objects. A set could comprise the planets, or the Presidents of USA or... anything. Not just abstract!

[edit] list of set theory topics

This list does not yet exist. This is a hint. Michael Hardy 23:20, 14 Jun 2005 (UTC)

[edit] Bullets

For such a basic topic, I think bullet mode is too unfriendly. This is not a PowerPoint presentation! Charles Matthews 08:47, 4 October 2005 (UTC)

Ok, prompted by Charles, I've made a small start at making this article a bit less "unfriendly". I have integrated some of the ideas from the "PowerPoint" like See also section, but I have, for now left that section untouched. Obviously a lot more needs to be done. But I will wait to see what reaction this evokes in others first ;-) Paul August ☎ 22:16, 4 October 2005 (UTC)

[edit] Self-contradictions in Set Theory

Self-contradiction in set theory arises when the distinction between 2 definitionally distinct concepts is blurred. For examples:

Russell’s paradox blurs the definitions of “set” and “element”

Cantor’s paradox blurs the definitions of “set” and its “complement-set”

Burali-Forti’s paradox blurs the distinction between “set” and “ordinal number”

Cantor’s theorem’s bone-of-contention set blurs the distinction between “set” (the “power set”) and “element” (the “subset” as element of the power set and the “element” of the subset)

“Completed infinite set” blurs the distinction between “completed” and “infinite” (“incompletable” or “no last element” or “there is always a successor element to every element”)

Cantor’s diagonal argument “proving” the “uncountability” of the real numbers blurs the distinction between “set” (which cannot be completed ) and “sequence” (which may be completed); between “interval” or “variable” and real number “point” or “constant”; as well as between the countable enumeration forms x1, x2, x3, … and y1, y2, y3, …, z1, z2, z3, …

Related (as “vacuous truths”) to first-order logic’s material implication self-contradiction is the inherent self-contradiction in set theory --- the empty set is an element of any set’s power set (or a subset of any set); hence, the empty set is also an element of the power set of any given set’s complement-set (or a subset of any set’s complement-set). This obscures the distinction between an element being “included” and “excluded” in a set (or “mutual exclusiveness”) or a “set” and its “complement-set”.

Just like the statement calculus and predicate calculus of first-order mathematical logic, the self-contradictions are barred ab initio by agreeing that Aristotle’s 3 “laws of thought” (which are definitionally equivalent) as well as contraposition (which is definitioanlly equivalent to material implication) are to be “first principles” --- that is, over and above all other axioms of any first order theory --- in particular, the first principle of non-contradiction which prohibits the application of a self-contradiction (a logical formula and its negation) at the same time in the same respect.

This means that both claims “the empty set is a subset of set S” and “the empty set is a subset of the complement-set of S” could not both be used in one argument.

Consider Cantor’s “set of all sets” paradox. If the “set of all sets” exists, then it has no complement-set --- but this means that its complement-set is the empty set which, being a set, is also included in the “set of all sets”. Hence, the self-contradiction. Therefore, Cantor’s “set of all sets” is barred ab initio by Aristotle’s first principle of non-contradiction.

Please read my related Wikipedia discussion notes on “Logical conditional”, “Cantor’s diagonal argument”, “Cantor’s theorem”, “Cantor’s first uncountability proof”, “Ackermann’s function”, “Boolean satisfiability problem”, “Entscheidungsproblem”, “Definable number”, and “Computable number”. (BenCawaling@Yahoo.com [14 December 2005])

And speaking of Cantor, I see that he's not mentioned in the article. Should someone maybe add a sentence about him and a link to his page--Georg Cantor? I'm out of my depth here, so I'll leave it up to you guys. Thanks.--Staple 22:06, 26 April 2006 (UTC)

[edit] A few basic questions

Can anyone answer a few basic questions I have about set theory? I am very inexperienced in mathematics in general but in dealing with philosophy, and particularly indeterminacy in philosophy, I have begun to introduce myself to it. Philosophy, like mathematics, deals with the properties of sets, their elements, their boundaries, their unions, et cetera, and attempts to precisely and rigorously define these concepts via logical proof.

I recently began reading the freely-available book Basic Concepts of Mathematics by Elias Zakon. I was impressed at first by Zakon's statement at the beginning of the book that he found rigorous proof lacking in mathematics as it is generally taught to those inexperienced in the field; this has always been a problem for me when I've tried to become familiar with the mathematical concepts others use to describe things like sets. I was thus very surprised by what appears to me to be an extraordinary lack of rigor and, indeed, lack of proof in general of a statement of "fact" in the very first chapter:

Zakon asserts that

"[I]f M is a collection of certain sets A, B, C, ..., then these sets are elements of M, i.e., we have A is an element of M, B is an element of M, C is an element of M, ...; but the single elements of A need not be members of M, and the same applies to single elements of B, C, .... Briefly, from p is an element of A and A is an element of M, it does not follow that p is an element of M. This may be illustrated by the following examples.

Let a “nation” be defined as a certain set of individuals, and let the United Nations(U.N.)be regarded as a certain set of nations. Then single persons are elements of the nations, and the nations are members of U.N., but individuals are not members of U.N. Similarly, the Big Ten consists of ten universities, each university contains thousands of students, but no student is one of the Big Ten."

Can anyone provide me with three sets of actual numbers A, B, and C such that A is an element of B, and B is an element of C, but A is not an element of C? I currently think that this is impossible, and I really need an example of this phenomenon if one exists. This is hindering my progress in learning set theory.

Zakon speaks of "nations" as though they are discrete entities with defined boundaries. This would be necessary if we are to talk about them as sets, in my current understanding. But nations rise and fall and change during the lifetimes of men and women: think of the American Revolution or the fall of the Roman Empire. What makes a "nation" a "nation"? People differ in their answers to this question.

Furthermore, the Big Ten, being universities, consist of teachers, buildings, et cetera, as well as students. The reason that a student at one of the Big Ten is not one of the Big Ten is that a university necessarily contains other things than students; there exists a set whose elements are all elements of the set of all universities but not elements of the set of all students.

These are not "certain" sets, as Zakon says they are, in that their properties and elements are not clearly defined. And if the "fact" of the existence of this phenomenon is so obvious and so necessary to the rest of set theory, then why doesn't he illustrate it with quantifiable sets? Isn't this what mathematics is all about?

I assert that there exist no bounded sets of numbers A, B, and C where A is an element of B and B is an element of C but A is not an element of C. If I am wrong, I'd really like to know. I need to see rigorous proof of this before I can understand anything else in set theory. The concept Zakon is trying to illustrate relates to the definition of what he calls a "family of sets". I have seen a few articles on Wikipedia that talk about things like this; although I can't think of an example at the moment, I'm sure you guys have heard of families of sets. What property could any set C have that allows it to have subsets with elements which are not elements of set C? As I understand it, an element of something is a part of it, it is within it, et cetera: how, then, can set C contain a set B which consists not only of elements of C but of elements not in set C? If the elements of set B are parts of set B, and set B is necessarily part of set C, then any element of set B must necessarily be a part of set C. Again, can anyone give me a real, concrete counterexample that doesn't rely on "intuition"? I thought that in mathematics a thing wasn't supposed to be "more" than the sum of its parts: isn't it central to the very concept of partition that all of the parts of any given thing must necessarily add up exactly to that thing?

Mathematics is supposed to be about rigorous proof, and Zakon claims to be an advocate of such rigour. But I fail to see how things like nations and universities can be considered to be discrete mathematical entities. It seems absurd to me to approach set theory this way. In philosophy, the supposed property to which I have just referred (the one set C would need in the above example to include set B but not all elements of set B) is called a thing in itself. Kant proposed its existence in his Critique of Pure Reason, and Nietzsche extensively disproved its existence. Nietzsche argued that since such a thing cannot in any way be quantified, any two observers cannot be certain that they are both observing it, and that there is thus no compelling reason to suppose its existence to begin with. Any discussion or quantification of it cannot occur, since as soon as it is defined, or bounded, the "thing in itself" would have properties and necessarily not truly be a "thing in itself".

Mathematically, one could define the "thing in itself" as the set that does not share any elements with any other set. It's not even the "empty set", in that it has defining properties which are its elements, such as the ability to be discussed in a language. This leads me to another question: how can there exist an empty set? I can see that it is useful to discuss such a set in defining what exactly an "element" is in general, but in reality there exists no set with absolutely no elements: it would not be a set at all, since a set is a group of elements. The empty set is bounded in that any element of any other set is outside of it. Why do people say that the empty set is an [element] of every other set? Sets are defined by their elements or by the formulae or functions that produce all of their elements. Occam's razor would neatly slice the empty set out of any other set, as far as I can currently tell, if we are using sets to approximate any sort of real phenomena. Even [mere description] of the "empty" set gives it some sort of definition, and thus (supposedly) includes in it certain things (such as not having any elements) and excludes other things (all possible elements of all possible sets) from it, giving it a boundary.

Am I really supposed to believe that such a set has any definite connection whatsoever to any observable phenomenon? I really hope that I'm just misunderstanding this, because if modern mathematical thought centers around concepts that are utterly unquantifiable then the world is a much weirder place than I want it to be.

I also do not understand how a set can be an element of itself. An element of a set is a subset of it, but unless sets are "granular" (I think that's at least close to the right term) there must necessarily be some element in any superset A of any set B that is [not] part of set B: otherwise they would be completely equivalent. Why consider something a subset of itself? How does this make sense? Aren't sets partitioned into their subsets? How can a set A be "partitioned" in such a way that the boundary that defines the "partition" bounds only set A itself and nothing besides? What am I missing here?

I hope that someone can find the time to explain to me exactly what a "family of sets" is, and rigorously. I haven't seen it done yet. This is hindering my progress in learning set theory, and I am having trouble finding a good definition that will allow me to move on to more advanced concepts. Mathematics shouldn't be about "intuition" but about proof. I apologise for using so much space on a discussion page meant for people who (I hope) find my argument silly and can mathematically prove it to be so.

I am also interested in the concepts of similarity and difference: they relate to set theory in that if a set A is similar to set B then it ought to share some elements with set B but not others; but in reality we can continue to find differences between any two things indefinitely. Even in a statement as simple as "1 = 1", there is a definite difference between the two "1"s, no matter how small: if there were no difference at all between them, how could we possibly be considering them as two different "1"s at all? In other words, what elements, exactly, do any two "equivalent" sets actually share? For a set to be completely equivalent to another set, both sets must contain exactly the same elements. There is no magical, mystical concept of "oneness" that allows us to make assumptions about the possibility of true equivalence; quantification is a natural part of human behavior that is necessary to our lives. It was once thought that Euclidean geometry was the only way to describe space, and there are now non-euclidean geometries. But where does this urge to falsely equate things come from? It seems to me that the concept of equality would be better termed increasingly-close approximation. No two separate things can possibly be completely equivalent or they would be a single thing. What possible difference could there be between them?

In other words, if set A is exactly equivalent to set B then they should contain exactly the same elements and neither set should contain any additional elements; but if they are to be discretely named set A and set B, then there surely must be some element not shared between them, or "they" would simply be named set A! Occam's razor would rid us of set B in this example as unneccessary if set A is supposed to approximate some part of reality. What elements could possibly be in a set that is exactly equivalent to a different set?

I realise that set theory is very basic to mathematics, and that is why I must ask these questions. Again, if there is a real, mathematical proof of the thing-in-itself-- or if, more likely, I've mistaken some other concept for the thing-in-itself-- then I apologize for wandering into the wrong "department". But I don't know how else to get this straight than by asking here. Have I stumbled on some of the "contradictions in set theory" mentioned below? Or am I just being really dumb? A little help would go a long way here.

Tastyummy 08:53, 22 August 2006 (UTC)

Hi Tastyummy, you ask: Can anyone provide me with three sets of actual numbers A, B, and C such that A is an element of B, and B is an element of C, but A is not an element of C? Consider the sets A = {1, 3}, B = {1, {1, 3}} and C = {{1, {1, 3}}, then B has two elements: the number one, the set whose only elements are the numbers one and three. Since A is "the set whose only elements are the numbers one and three", A is an element of B. Likewise C has one element the set B. Thus B is an element of C. But A is not an element of C, since C's only element B, is not equal to A. Does this make sense? Paul August ☎ 21:19, 22 August 2006 (UTC)

Thanks very much for your reply.

I think this makes sense, but let me pose another question to make sure I'm understanding this correctly:

Is the set containing no elements different from a superset containing only that set? In other words, if the only "element" of a set A is simply that set which contains no elements, then am I to consider A to have an element after all? I mean, this seems to make sense only if we consider set A { { B, C } } as fundamentally different from { B, C }, right? Is the only thing that distinguishes {no elements} [please exsuse my not knowing how to use the symbol for the empty set] from { { no elements } } and from { { { no elements } } } that we are considering them as supersets and subsets of one another?

As I said in my questions above, I am having trouble understanding the concept of the "empty set" altogether. If a set is a group of elements, and the empty "set" contains no elements, then how is it a set at all? I'm sorry if this sounds incredibly stupid. I just don't understand how a set containing nothing is different from a set containing a set containing nothing, et cetera. Surely our consideration of things as sets comes from our ability to group things, right? (Or does it? Am I just reading too much into it?)

Zakon says in the beginning of his book that

"A set is often described as a collection (“aggregate”,“class”,“totality”,“family”) of objects of any specified kind. However, such descriptions are no definitions, as they merely replace the term “set” by other undefined terms. Thus the term “set” must be accepted as a primitive notion, without definition."

If something is altogether undefined, how can we possibly discuss its mathematical properties at all? Isn't definition pretty necessary if we are to "prove" the properties of sets? And if it is an "intuitively-defined group", or something, then ought we not to start at the beginning with what makes a group a group before we move on to other aspects of set theory? Elements of sets always share at least one aspect: that they are the members of that set. But how can we call a set with no elements a set? Where is the property that makes the set containing absolutely no elements different from nothingness in general?

I mean, what is not shared between "nothing" and "containership of nothing"? "Containership" itself? What can this possibly mean without any elements being "contained"? Don't we derive our concept of continence from observing actual things which are contained within one another? How can there be "containership" without anything being "contained"? How can this be observed in reality, or is it not even supposed to be? The problem I am having is that if I use a set to approximate a real phenomenon, such as "the set of all atoms of carbon", then it would seem that I could approximate all actual atoms of carbon in set theory equally well by calling that set "the set containing all atoms of carbon and the empty set", and I could also call it "the set containing only the set whose elements are all atoms of carbon and the empty set and another set containing two empty sets", and so forth, without adding anything to or taking anything away from how closely this concept approximates all actual atoms of carbon. Is this where mathematics and science differ, because Occam's Razor seems to suggest this. How can there be such a thing as a set with no elements, if all sets are defined in terms of their elements, and how is a set A { B,C } different from a set D { { B,C } }? Are they only different in that we consider them separately?

Ben Cawaling says above that

"the empty set is an element of any set’s power set (or a subset of any set); hence, the empty set is also an element of the power set of any given set’s complement-set (or a subset of any set’s complement-set). This obscures the distinction between an element being “included” and “excluded” in a set (or “mutual exclusiveness”) or a “set” and its “complement-set”."

I'm having, I think, the same problems here: how can something (the empty set) be an element of both a set and its complement, if the complement of a given set is defined as all elements not in that set? Isn't this a reductio in absurdum? We now have "The empty set is both and element of any given set and not an element of it". How can something be the exact opposite of itself?

What, exactly, isn't an element of any superset? I can't think of any real object that I can't categorise in some way or another along with other objects. Even "the entire universe" is an element of "the set of conceptions of reality", et cetera.

Thanks very much for your example, though. At least now I see what Zakon is trying to say (I think). I can try to accept that { 1 } is different from { { 1 } } at least in learning the rest of set theory; you've been a great help to me in this and I appreciate it a lot.

One more question: is there any category like "philosophy of mathematics" or something to which it might be appropriate to add my article on the indeterminacy of definition? I really think the concept of the "empty set" is strongly related to Kant's noumenon, and I have also cited Zakon's assertion above that sets are not clearly defined (and "indeterminacy in philosophy" refers to such non-definition, or, alternately, indefinability).

Tastyummy 22:59, 22 August 2006 (UTC)

One more thing: Is this discussion inappropriate for this page? If so, why? I am considering adding a section about set theory to my article on indeterminacy in philosophy, and I just want to be sure that I understand it correctly before doing this. If I add this section, it will cite mathematical theorems, et cetera, at all appropriate points, such as those referred to by Ben Cawalling above on this page. (In other words, I won't make anything up; I'll only quote people who are well-informed on the subject.) If a mathematical theory relies upon undefined concepts at its root, this should be openly discussed.

Also, in order to make this discussion officially relevant to changes in the article on set theory, I propose that an article specifically on contradictions in set theory be written and linked to in the main article.

Thanks again, Tastyummy 23:22, 22 August 2006 (UTC)\

To answer your last question first, yes such long and general discussion, not directly about writing the article are not really appropriate for this page. The appropriate venue for such questions is Wikipedia:Reference desk/Mathematics. However if you want I would be willing to try to answer some of your questions on your talk page. — Paul August ☎ 01:56, 23 August 2006 (UTC)

The wikipedia talk pages are really only for dicussing article improvements. But I will try to answer some of your queries. Note, all your misunderstandings stem mainly from your informal understanding of set theory (well, ZF, at any rate).

Regarding the empty set: A set being "a group of elements" doesn't not preclude a set being a group of zero elements. Just like zero dollars is a valid amount of money to have, zero items is a perfect valid number of things for a set to contain. At any rate, the existence of the empty set can be deduced from the other axioms of set theory.

Regarding undefined symbols: At this point, I really must insist you look at the ZF. You don't mention it, but in fact there are two undefined symbols in set theory: 'set' and 'is a member of' (which is equivalent to 'is an element of'). The mathematical properties of these symbols are precisely those properties that the axioms, and any theorems deducible from them, allow them to have. Mainly, what you confuse is what it means to be 'defined' in a theory, and what it means for a theory to have a model. A 'definition' in set theory is simply a shorthand for some expression that is ultimately reducible to the symbols or first-order logic, 'set', and 'is a member of'. That these symbols are undefined simply means that you cannot go on further reducing an expression that's purely in <first-order logic + 'set' + 'is a member of'> language into simpler symbols. What it doesn't mean is that there is no meaning to the symbols. Finding a meaningful connection between a theory and other mathematical objects is the domain of model theory.

Regarding powersets: You have confused 'is a member of' (which is equivalent to 'is an element of') with 'is a subset of'. Some examples:

• 3 is member of {1, 2, 3, 4, 5}
• {1, 4, 5} is a subset of {1, 2, 3, 4, 5}
• {1, 4, 5} is not a member of {1, 2, 3, 4, 5}
• {1, 4, 5} is a member of {1, {1, 4, 5}, 9, 10}.

By the definition of a subset, set A is a subset of set B if and only if all memebers of A are also members of B. Which means that the empty set {} is a subset of every set. And since the powerset of a set S is the set of all it's subsets, the empty set is a member of every powerset.

Regarding philosophy of mathematics: Yes, there's an article on mathematical philosophy already. I suppose the concept of 'definability' in maths is fairly interesting, but note that maths, and especially foundations (set theory, mathematical logic, model theory, some other areas) is extremely formal, and so whether ideas like 'definability' and even 'truth' are fleshed out in maths or not doesn't really matter; we can derive almost everything from the formal manipulation of symbols from axioms and rules of inference.

Regarding contraditions in set-theory: Depends what you mean by 'contradiction'. There are paradoxes, but these are usually theorems that merely bend our intuition. There is 'inconsistency', which means that you can prove a proposition 'P' while also being able to prove a proposition 'not P'. There are no such inconsistent propositions in current, ZF set theory. The Russell paradox is an inconsistency in naive set theory, but the Russell-paradox set cannot be formed in ZF.

Note I am not an expert - my set theory knowledge consists of vaguely recalled bits of undergrad maths glued together by what I can understand from mathworld and wikipedia. But if you've got any other questions, do feel free to continue this on my talk page. Again, this talk page is for discussing article improvements. Tez 02:36, 23 August 2006 (UTC)

Thanks again for your patience. I'll continue to discuss this on user talk pages as you've both requested, and I apologise for taking up so much space here. I did, in fact, realise (but didn't mention) that membership in a set is also, as you put it, an "undefined symbol": this is what I meant by asking what it is that makes us consider things as members of sets or groups at all.

I suppose that my questions about definability in general relate more to a discussion of first-order logic than to set theory. Again, sorry for the inconvenience and thanks very much for your time. I will look into ZF set theory and model theory more closely; I greatly appreciate these suggestions.

-Tastyummy (forgot to log in)

Regarding one more thing:

"We can derive almost everything from the formal manipulation of symbols from axioms and rules of inference."

We certainly can, but the axioms and rules of inference of which you speak are not self-evident; they are simply a model and approximation of reality deriving from our continued observation of real phenomena. Nothing is self-evident; hence my inquiry into the origin of even the most basic concepts.

I won't waste any more space here. Thanks again to everyone for helping me with this, and for being patient with my long-winded rants.

-Tastyummy

[edit] Anwers to philosophical objections and a question about adding a new section to the main article

I've managed to answer my own questions about the "existence" of the empty set, etc., mainly via the help of user:tezh, but also via an interesting quotation of Charles Proteus Steinmetz:

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional."

From a post I made on Tezh's talk page (our discussion consisted of my posting on his page and his posting on mine) after my having figured out something quite obvious mainly because of his patient attention:

"If everything must be considered as a set, then there must be some set with no elements."

In other words, in my opinion it is perfectly reasonable to argue against the consideration of all possible phenomena as sets, since to consider every phenomenon as sharing some common attribute with every other phenomenon is to propose the "existence" of a "self-evident" thing in itself; however, it is also perfectly sensible to conclude, from a given set of axioms, that the empty set "exists", or that { { 1 } } is different than { 1 }, since per the axioms of set theory membership in a set is not the same as being a subset of it, even though I still don't fully understand the necessity of this concept.

Tezh further informed me that my assertions about the empty set's ability to approximate any real phenomenon were more relevant to, for example, model theory than to set theory. I agree with this, although I still differ with Tezh over the possibility of the existence of Platonic ideas (or, similarly, of any "self-evident" truths or "immediate certainties" in mathematics or elsewhere), and agree with Steinmetz that mathematical truths are, like all other truths, conditional.

I am considering writing a section on "set theory and philosophy", in which I would explain philosophical objections to various aspects of set theory and expose them as being erroneous for the aforementioned (and other philosophical) reasons. Does anyone object to my doing this? I'd be careful not to draw mathematical conclusions therein; it would be a statement and refutation of the philosophical objection to the seemingly-Platonic or Kantian "implications" of set theory.

Please let me know if this is inappropriate in the article on set theory. I think it would be useful to readers who, like me, struggle with mathematics in general on philosophical grounds in a world where many eminent mathematicians, like Roger Penrose, espouse a Platonic view:

"Indeed, I would regard mathematical objectivity as really what mathematical Platonism is all about" - Penrose, in his The Road to Reality: A Complete Guide to the Laws of the Universe

Whether or not "objective truth" is possible, various philosophers have been critical of it for centuries; Nietzsche was especially influential on modern philosophy and, arguably, science in his criticisms of Platonism and Kantianism, and eminent mathematicians like Steinmetz have taken the view that mathematical truth is not "objective" as well.

I wouldn't even go into all of these vagaries if I were to write the proposed section; I'd simply address and refute the specific objections to certain conclusions in set theory which I raised earlier, and as concisely as possible. Again, please let me know whether this is appropriate.

In fact, what I'll probably do is post a draft of the section on this discussion page and then ask whether it's appropriate for the main article. But if there are any objections to my even beginning to do this, by all means, let me know.

Thanks,

Tastyummy 20:24, 13 September 2006 (UTC)