eriatarka

what?

Philip Roth is pretty cool

But many of the people who dislike the olympics enjoy sports like soccer.

First, make sure you understand what it means for a function to be surjective and injective. http://en.wikipedia.org/wiki/Injective_function http://en.wikipedia.org/wiki/Surjective Now, given any 2 sets (say X and Y), X and Y have equal cardinality if there is a surjective+injective function between the two sets. X has a greater cardinality than Y if you can find a injective function from Y to X, but there is no such function from X to Y. Intuitively, what this means is that 2 sets have the same cardinality if you can put them side by side and pair up every object in one with an element of the other. And one set is bigger than another if you 'run out' of elements in the smaller set when you try to do this. What the diagonal argument shows is that there is no injective function which can map every element of X onto the power set of X. This implies that X has a lesser cardinality than its power-set - if you start trying to pair up every element of X with an element in its power-set, then you eventually 'run out' of elements of X (even if there are infinitely many). The point of the diagonal argument is that if assume that you have a mapping which associates every element of X with an element of its power-set, then its possible to find an element of the power-set which has no element of X corresponding to it (ie youve 'ran out' of elements of X, but there are still elements of the power-set which havent been paired up). This is true no matter what you take the mapping to be, so no such mapping can exist.

I mean that it isnt interesting to watch, other than to see the competitors strive to do their best. Theres a difference between games like football or chess or starcraft which are interesting when considered purely as games, and something like running which is just competition for the sake of competition. Even something like gymnastics has spectator value in a sense which swimming doesnt.

The problem I have with the Olympics is that very few of the events are inherently interesting. Theres nothing at all interesting about watching someone run fast or pick up heavy weights - I mean yeah, it takes a lot of dedication and skill, but it also seems like a fairly pointless thing to devote your life to. Lots of things take skill, but that doesnt make them worthwhile. With real sports (football, rugby, tennis, etc) the games are actually interesting at a level beyond watching someone competing purely for the sake of competing. Most of the events in the olympic games arent actually games - theyre just isolated activities which have been ripped out of any context which would make them worthwhile. Being able to run fast is valuable if youre (eg) a Greek messenger being trusted with delivering messages to Sparta, or a football player who needs athletic ability to excel, but it has no inherent value in itself. In a sense, the Olympics is the epitome of context dropping - most of the events have been abstracted away from the situations in which theyre useful, and reified into ends-in-themselves. That being said, there are some Olympic events I enjoy watching. But I'll never understand how people can get so excited about running or swimming.

The diagonal argument is much more intuitive than a formal logic proof imo, you should look at the proof in the link I posted above if you want to see why the power set thing has to be true. Proofs directly from set theory axioms are useful iff you want to do computer proof checking but they often tend to hide the conceptual content of whats being said.

I would say that aesthetics is one of the most important (if not the most important) branch of philosophy because the aesthetics a person has will often be the main factor in his 'choice' of philosophy. I doubt many people became adherents of Objetivism based on Ayn Rands non-fictional works for example - the attraction of the philosophy lies in the strength of the vision which she put forwards in her fiction. I think that aesthetics (or 'sense of life') comes prior to philosophy for most people - someone who reads the Fountainhead and is repulsed by the character of Howard Roark isnt likely to become an Objectivist. Perhaps politics(/sociology) is more important than aesthetics (because many philosophies can be best understood in terms of their political motivations), but its one of those 2 that I'd pick as being the most important if I had to.

I think this is the wrong way to look at it. Noone is prohibiting you from doing anything - you can go away and create any set of axioms that you like. But once youve came up with some axioms, you have to accept what follows from them, and if the axioms lead to a contradiction then you have to change them. In this case, one of the problems with a universal set is that you cant define the notion of 'cardinality' in the intuitive way we would want to, because it immediately creates a contradiction as explained above (there are other problems with it too but I think this one is the easiest to explain). There cant be a universal set because the power set of the universal set will always be bigger than it, and hence cant be a subset. So the assumption of a universal set leads to a paradox. Set theory has been rife with these paradoxes ever since its creation - early set theories which allowed people to create arbitrary sets (such as Frege's) were shown to lead to contradictions. Russell's paradox ( http://en.wikipedia.org/wiki/Russell's_paradox ) is the most notorious example of this. The reason why modern ZFC set theory is more complex than this sort of naive set theory is because it was created in a way which avoids these paradoxes. Things like the assumption of universal sets seem innocuous but they can be shown to lead to contradictions. These problems were studied intensively during the first half of the 20th century and various solutions were proposed. ZFC is the solution that most modern mathematicians and logicians have adopted. edit: I want to be clear about what a universal set is here. Lets say you have a universe containing the objects 'a', 'b', and 'c'. Now, theres nothing stopping you making a set containing all these objects, say S = {a,b,c}. And the set S2 = {a} will then be a subset of S. But we can also create another set S3 = {{a}} which isnt a subset of S, since '{a}' is not an element of S (although 'a' is). The power set of S here is then {0,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}. And this obviously isnt a subset of S (and note that it has cardinality 8 whereas S only has cardinality 3). You cant have a set which has every set as its subset because the powerset of that set wont be a subset.

Its a theorem which follows from the definition of cardinality and power sets. It applies to all cardinalities - different infinite sets can have different cardinalities and theres a hierarchy of transfinite cardinal numbers. The set of real numbers has a greater cardinality than the set of integers for example (there are 'more' real numbers than there are integers, in a sense). http://en.wikipedia.org/wiki/Cantor's_theorem

Yes of course, its an extremely basic example. But formal set theory is essentially the same, just more complex - you have an alphabet of allowed symbols, and a system of rules for producing well-formed combinations of these symbols. Semantics is the study of interpretations (''meanings') of the symbols. In the formal system I gave, we can only talk about the provability of strings (ie their derivation from the axioms) rather than their truth values, since there isnt yet a theory of semantics defined - the symbols do not yet 'mean' anything. Truth arises at the semantic level once we start creating models of the system, not the syntatic one. But the point is that the axioms and combination rules fully determines which strings are valid and provable within the system. You dont need any meanings or referents for this. Thats why its a formal system. Oh ok, I think I misinterpreted you then, I thought you were claiming the former.

What do you mean 'I accept them'? If these are strings derivable from the axioms of some formal system then they are valid and provable within that system. If you want to use them to say anything interesting about the world then youll have to start associating the symbols with 'things' (giving them meaning, as you call it), but this is a separate process from the formal definition of the system. Syntax, semantics, and interpretation ('models') are all different things, but you seem to be conflating them. Here is a (very informal) description of a basic formal system allowed symbols: {X,Y} axiom 1: 'X' is a well-formed formula axiom 2: if F is a well-formed formula, so is FY[/code] from these axioms you can show that XY, XYYYYYYYYY, etc are provable within the system (ie they can be derived from axioms) whereas XXYY and XXX cant. But none of these symbols have 'meaning' yet (we havent yet created a model of the system).

Huh? You can quantify over empty domains. Let S be the set of all natural numbers which are smaller than 0. Then we can quantify over S to form statements like ∀(x∈S) (x is divisible by 2). And this statement would be true, since there is no element of S which isnt divisible by 2. But the point is that S here is the empty set, so it isnt true that ∃x(x∈S). More generally, its true that ∀(x∈∅)(Fx) for any property F, but this doesnt imply that ∃x(Fx) or ∃x(x∈∅) Similarly, we define A ⊆ B to be true if ∀x(x∈A => x∈. So if A is the empty set, this condition holds trivially for any set B. But this doesnt imply that ∃x(x∈A), otherwise youd be saying that the empty set had members. Unless I'm misunderstanding what youre saying.

No, you can only use existential generalization here if x is a bound variable, which it isnt. edit: to clarify, AxF means that if x is an element in the universe of discourse, then Fx holds. But in the case where the universe of discourse is empty, you cant use universal instantiation to get Fx, so x here is a free variable rather than being bound.

No, it must have a meaning (which in this context implies rules of combination and transformation into other symbols). Symbols in a formal system dont need to correspond to anything outside the system - this correspondence is set up later when you create a particular interpretation of the system by assigning symbols to non-system entities. This is what I was referring to previously when I said that your interest in formal systems is their use in describing natural language, which is something I dont care about. No. Its similar to me wanting to notate a spoken language by using a roman-style alphabet, and you wanting to use chinese-style pictograms. Its a purely methodological decision on which nothing hangs (other than pragmatic concerns) and there is no 'correct' way of doing it. It has nothing to do with correct reasoning, unless you think that correct reasoning requires formalization. Natural language is too complex to be captured in a formal system and any attempt to do so is going to be filled with arbitrariness like this.

Whether this is true or not (and its not as simple as you imply, since many would disagree [including me]), its not directly relevant to set theory. Set theory is a language for describing mathematics. Whether mathematics describes reality is a separate question. I think that the relationship of set theory to mathematics is similar to the relationship that a written language has with its spoken equivalent. But many mathemaicians (and even more logicians) would probably disagree with this.

No, generally we have the theorem ∀x(Fx) -> ~∃x(~Fx) ('there is no x for which Fx fails to hold'), which doesnt imply ∃x(Fx) ('there exists an x such that Fx holds'). If what you were saying were true, then the empty set wouldnt be a subset of every set if we were using the normal definition of subset (that A is a subset of B iff every element of A is an element of . But of course, theres nothing stopping you making a logic in which ∀x(Fx) -> ∃x(Fx), it would just have different properties from standard predicate calculus.

I think the main problem here is that we are coming at this from different backgrounds, with different concerns. I'm primarilly interested in mathematics (and I assume this applies to punk also), so I tend to see set theory as being purely a language for descrbing mathematical objects/proofs, and I judge it solely on that. However I understand that youre a linguist and I know that formal systems play a different role there. I'm not sure what your standpoint is, but I personally think the idea of applying formal logic to natural language semantics has been a horrible mistake from Frege/Russell right down to its more modern incarnations, and have very little time for any of it (I have no opinion on the Chomskian approach to use formal systems to study syntax/gramm rather than semantics though). But anyway, because I have no interest in the application of set theory to natural language semantics (since I think the whole project is misguided and couldnt possibly tell us anything useful), I tend to think that all questions regarding set theory should be decided purely on their usefulness to mathematical practice. Asking 'should the empty set be a subset of all sets?' is like asking 'would it be nice if all the verbs in a human language were regular?' - theres no absolute truth here, all that matters are the pragmatic issues. Regular verbs are a nice thing and if I were designing an artificial language from scratch then I would make it very regular, Esperanto-style. And similarly, having operations like set union obey simple properties (while minimising the number of pathological cases) is highly desirable so its a good idea to treat the empty set as being a subset of every set. This doesnt mean that theres some kind of metaphysically existing 'empty set' living in a Platonic netherworld which we are proving things about, and it isnt intended to be a statement which has direct application to reality. When we decide that the empty set is a subset of all sets, we arent learning anything new about what happens inside empty desk drawers. To put it in more Objectivist terms, what we're dealing with here are concepts of method.

In this context, a statement is any combination of symbols which is consistent with the syntax of the formal system youre working in. In general though, I'd say that a statement/proposition is a sentence which admits a truth value (which is what distinguishes statements from questions, commands, etc). I think the more important question is: what hinges on this? If you want to say that "The king of France is bald" is false, whereas I say its indeterminate, then what practical difference does this make to anything? To me, it seems like a purely methodological point and it makes no difference whatsoever which one you go with, as long as youre consistent.

But given the axioms of the system, the proofs are perfectly valid. Thats how axiom systems are often constructed - we start out with a set of properties which we want to hold, and then we choose a set of axioms which is strong enough to prove them, and 'weak' enough to not prove a lot of things which we dont want. When we define set union, it seems intuitively obvious that if C = A u B, then both A and B should be subsets of C. And if you want this property to hold universally, then you need to accept that the empty set is a subset of every set, otherwise C = A u {} would be a counter-example. Similarly, if you want it to be true that for all numbers x, x/x = 1 then this determines how you define the multiplication of negative numberes, since if you want x/x = 1 to hold universally then it must be the case that -1/-1 = 1, so that -1 * 1 = -1, and so on.

Who said it was necessary for everything to have a truth value? Within a formal system, all that matters is whether a certain statement (or its negation) is provable from the axioms of that system. And the choice of axioms isnt forced on us by reality, but is rather determined by pragmatic and aesthetic concerns. Its like asking why the product of two negative numbers has to be positive - theres no deep metaphysical reason why (-1) * (-1) has to equal 1, but we choose to define multiplication of negative numbers in this way because its desirable for the group of integers under multiplication to have certain properties, which wouldnt hold if we defined it to be anything else. Similarly, if you dont accept that the empty set is a subset of every set then certain properties of set unions fails to hold and your system starts to become ugly and needs more special cases, which can be avoided. (see punk's first post in this thread for an example of what I mean). If you want a set theory where the empty set isnt the subset of every set then youre perfectly free to make one. The only question is whether itll be as elegant and easy to work with as ZFC.

There is no 'standard formalization'. Set theory doesnt deal with the translation of English statements into formal notation. I think youre confusing the formal mathematics of set theory with the philosophical program undertaken by people like Frege/Russell/Strawson which aimed at constructing a formal calculus for dealing with natural language statements. But even within this kind of analytic philosophy, there is no 'standard' way of dealing with statements like the one mentioned, and different people would deal with them differently. For instance, the Russellian approach of assigning a truth value to every proposition is different from Strawson's approach which refuses to assign truth values to statements which deal with non-existent objects (the classic example here is the statement "the king of France is bald", which Russell would claim is false, while Strawson would say it has no truth value). But none of this is directly related to ZFC (or mathematical set theory in general), since this isnt intended to be a theory of natural language.

Because if you call that set S then the power-set of S is also a possible set, and by definition must be a subset of S. But the power-set of any given set X has a strictly greater cardinality than X and hence cant be a subset, so you have a contradiction. (This has nothing to do with the axiom of choice)

'National security' is going a bit far but you cant really underesimate how dependent the West is on oil. Securing access to oil resources for the purpose of 'national security' has been a fairly key plank in America's foreign policy over the last 30 years or so.

It's not directly related to Objectivism, but Ive always been fascinated by Nietzsche's portrayal of 'rational suicide' carried out in the name of human dignity by those who feel that life can no longer provide them with what they require: