I am ignoring no such thing; i am igoring you desire to misrepresent set theory. if sets A and B are different, then the symmetric difference must be non-empty (either there is an element of A that is not in B or vice versa), what is the symmetric difference of the empty set and itself? feel free to declare there are distinct empty sets; do not pretend you are doing set theory as others do it though. that is misleading. you have offered no reason to suppose that we gain anything from this position. declare your conventions and stick to them.
Why would you want that? Also, what would the intersection of the set of dogs and the set of men, be? The "empty set of men", "empty set of dogs'" or "empty set of men and dogs"? How about the intersection of the "empty set of men" and the "set of men and dogs"? The intersection of the "empty set of men and dogs" and the "set of men"? I await your response.
Let M(t) denote the set of men which currently exist. Let D(t) denote the set of dogs which currently exist. Now, you ask about the intersection of these two sets. The definition of A intersect B, is as follows: A intersect B = {x| x is an element of A AND x is an element of B} There is nothing in existence which is both a dog, and a man. Therefore, the number of elements in the set A intersect B, must be zero, from the definition above. Using the poor tired out symbol for empty set, we can write... A Ç B= Æ But the thing is, if there are different empty sets, then we need something to distinguish the set of man-dogs, as follows: A Ç B= Æ = set of man-dogs One could subscript the empty set symbol with (man-dog). This would indicate simply that there is no such thing as a man-dog. In other words, that the set of man-dogs has no elements in it. The subscript (man-dog) would serve to differentiate this empty set from say, the set of ghosts, which is also empty.
I suspected as much. What is the intersection of the different kinds of empty sets? Because, surely Ø_M is a subset of M. And by your definition so is Ø_MD. What is the intersection of Ø_M and Ø_MD? Russel and Whitehead tried do differentiate different types of sets in the way you do in the concept of a "class". It doesn't work.
I don't understand your notation there. You say that 0_M is a subset of M. I don't understand that, so I cannot comment on it. Are you trying to use the theorem that "THE empty set is a subset of every set." ? PS: I am familiar with Russell's theory of ramified types, as well as principia mathematica. I couldn't understand the former, and never read the latter. But whatever 'set theory' they were using, sure the heck isn't the one I'm using.
It just says that the empty set of men (that is, the empty set associated with the sets of men) is a subset of a (random) set of men. No, just that the empty set of men (denoted as Ø_M) is a subset of a set of men M. Don't you agree? They have the same type of elements (men) and agree on them all (because there are none in Ø_M)... Clearly, for your concept of empty set(s) to be different from the set theoretic version, it needs to include the stipulation of the types corresponding, otherwise it is trivial that Ø_A = Ø_B for all types A and B, leading you to having exactly one empty set. You don't agree so apparently the type is important. I'm trying to understand A) What you wish to accomplish by such an (IMO) unnecessary complication, and B) What precisely your definition of intersection of sets (with different types) is because I honestly think that you're on a wild goose chase. But at least this is interesting, as opposed to the str stuff...
"The whole point of this thread, is that the two sets cannot ever be equivalent." Yes, we UNDERSTAND that that is the whole point of this thread. That is why people have repeatedly told you it is not true. You wrote, as a premise, that "<font face="symbol"> "</font>t[not(M(t) = D(t))] what that says is that "for all t M(t) is not the same as D(t)" do you have any reason to assert that? Of course, that contradicts M(0)= D(0) but you haven't given any reason for it. I would accept "there is some t such that M(t) is not equal to D(t)" but I don't see how you can assert "M(t) is never equal to D(t)".
how can i prove what is self evident? i mean i could type a proof, but you wouldn't believe it since you are rejecting the axioms of set theory (set theory in which it is a trivial exercise to prove A=B iff the symmetric difference is empty). i cannot prove anything to a moronic troll who has already decided the answer, and to reject the basis of proof if it doesn't fit his view of things. you have a hypothesis, one that is not in ordinary set theory and one that i see no need for (that the set of dogs is always different from the set of men - true if their not empty). set's are not bags with labels on them that we refill if they are empty.
And I stated earlier that I am not using axiomatic set theory, that I am using temporal set theory, which avoids the contradiction in axiomatic set theory. Why in the world would I care that people are telling me something isnt true, when they are using axiomatic set theory, which leads to contradiction? You clearly do not understand the point. In temporal set theory, objects can switch sets. There is another binary relation... "potential elementhood" in TST. As I said, i developed temporal set theory, I do not expect anyone else to know how to use it. You do NOT understand. The sad thing is, that I have given enough information in this thread, for you to deduce at least some of how it works, but you haven't even bothered to do that. You are basically asking me why can't a dog be a cat at least some of the time. As I said, you do not understand why I cannot use axiomatic set theory. I simply cannot and will not use it, because its axioms are inconsistent with one another, as I've already showed earlier in this thread. And if you would take the time to learn what I'm teaching you here, your understanding of the foundations of mathematics would improve.
Ok, since you are trying to understand me, I will do so for you. I have to read your post here several times, so that I can give you the genuinely correct answer. Ok, I asked you what you meant by the statement that: "0_M is a subset of M" And I think I understand your answer, which is that "given an arbitrary set of men M, that 0_M is a subset of M." So this brings us to the set theoretic concept of subset. Let A,B denote sets. A is a subset of B if and only if [for any x, if x is an element of A then x is an element of B] So let M(t) denote the set of men at moment in time t. Long ago, there was a moment in time at which the there were no men in existence, so at such moments in time the number of elements of M(t) was zero, that is #M(t)=0. And you have used 0_M to denote an empty set of men. And you simply want to know if 0_M is a subset of M. At least thats how I undestand you. For what its worth, I think you made a good post there, so I'm going to take some time to answer you carefully. In the definition of 'subset' symbols A,B denote arbitrary sets, so using universal instantiation we have: 0_M is a subset of M if and only if [for any x, if x is an element of 0_M then x is an element of M] So let me presume that your statement that 0_M is a subset of M is true. Then using modus ponens it now must be true that: for any x, if x is an element of 0_M then x is an element of M Now, let us agree to use the notation 0_Z to indicate when a set Z has no elements in it, for an arbitrary set Z. And i have explained that it is impossible for 0_z = 0_M, unless z=M. So let 0_D denote the set of dogs, which at some moment in time happens to have no elements in it. Using universal instantiation with the definition of subset would also give us this: 0_D is a subset of M if and only if [for any x, if x is an element of 0_D then x is an element of M] Suppose there is a moment in time, at which the RHS of the 'iff' above can be false. Then at such moment in time, the following statement would be true: not[for any x, if x is an element of 0_D then x is an element of M] The previous statement is logically equivalent to the following statement: There is at least one x, such that x is an element of 0_D and x is an element of M. And if the previous statement is true at some moment, then the following statement is true at that moment... There is at least one x, such that x is an element of 0_D. Supposing the previous statement to be true for a, we have: a is an element of 0_D But that statement is necessarily false, since at whatever moment in time is in question, 0_D has no elements in it by definition of 0_D. So backtracking, look at the following statement again: 0_D is a subset of M if and only if [for any x, if x is an element of 0_D then x is an element of M] The RHS is necessarily true, for any 0_z used for instantiation in the definition of subset. So if that is the definition of subset used, then an empty set of dogs is a subset of an arbitrary set of men. But, the way to think of 'empty sets' in a temporal set theory is as follows: Suppose you have eight empty plastic paint cans. Each is labelled with the name of the set on it. One can is labelled dogs, another can is labelled cats, another can is labelled human beings, and so on. Each paint can is different from any other. So any logic which allows you to conclude that one particular paint can is two different paint cans is incorrect. Now, if something isn't in the cat paint can, then it cannot possibly be a cat. If something is in the cat paint can, then it necessarily must be a cat. So, when there is nothing inside the cat paint can, there are no cats in existence. So this is the kind of thinking which lays at the axiomatic foundation of temporal set theory, which lies over and above the definition of subset. So, consider again the statement that: 0_M is a subset of M How would you arrive at it's formulation? Would you use universal instantiation on the definition of subset I gave. In fact what I really need to know, is what definition of subset are you using. Then I will pick things up from there.
what contradiction? you wrote something about sets of men and dogs but nothing mathematical and found no contradiction in any set theory of which i am aware. by all means find me an axiom of set theory that states that the "sets of men and dogs in existence at time t can never be equal even if they are empty". until then you are being intellectually fraudulent. not to say idiotic. im fact given that you still maintain set theory as we know it is wrong because of a reason that has been explained has noguht to do with set theory yuo are in fact lying. if that phrase on quotation marks is taken as an axiom in addition to those of ZFC then of course they would be contradictory; fortunately it isn't so there is no contradiction. all we can say is that in your world there are different empty sets. fine, but you derived no contradiction about, say, ZFC. and in your world it follows that almost all the rules of set theory are now false. indeed what are the rules? no, wait don't tell me; i couldn't care less what a crackpot thinks. of course in order to demonstrate a contradiction in some set theory, say ZFC, you must only reason from the axioms (you know what they are?). the axioms do not state that the class of dogs, or men, is a set. you are confusing a model with the axioms. you may find it problematic that set theories do not differentiate between the set of all pink elephants and the set of real roots of x^2+1, but that is not a contradiction of set theory. it is merely a contradiciton from attempting to apply it to a situation for which it is noty suitable. the empty set is the unique set satisfyign x in S is false for all x, thus you need to have some differnet way of specifying youir sets, as funkstart indicates below
Johnny5, Ignoring the temporal aspect for now, you just seem to be suggesting that the notion of set carries with it a specification of types. Let's denote the type (say, man, dog, cat etc.) by the subscript. A natural way to define the subset relation (and the one I've used) on sets with the same type M would be A_M subset B_M iff Ax in A_M . x in B_M (where Ax stands for "for all x") Do you agree? In this definition, the empty set whose elements are of type M, denoted as Ø_M, is certainly a subset of a random set of elements of type M, say A_M. The problem now, of course, is that we have no way of comparing sets with different types. That is what I am asking you to define; it is your logic, after all. I would probably define it as A_M subset B_N iff M is a subtype of N, Ax in A_M . x in B_N or something similar, on some suitable definition of subtype. But I'm asking you.
We are past the point where I showed the contradiction, so you need to go back and read with the intent to understand. Until you are aware of the contradiction, you shouldn't have any further negative comments. I will spell it out for you, again. Let M(a) denote the set of men at arbitrary moment in time a. Let D(a) denote the set of dogs at arbitrary moment in time a. Suppose that moment in time a, denotes the first moment in time. Therefore, the number of dogs in existence at moment in time a is zero, and the number of men in existence at moment in time a is zero. Using axiomatic set theory the following statement is true at moment in time a: M(a) = Æ AND D(a) = Æ Now, using the properties of equality (transitive, symmetric, reflexive) it now follows that: M(a) = D(a) Now, using existential generalization, it follows that: $t[M(t) = D(t)] Which translates as, there is at least one moment in time t, such that the set of men is equivalent to the set of dogs. But, the set of men is different from the set of dogs, at any arbitrary moment in time. So there is the contradiction. Again, it already showed up in the thread, I guess you just didn't read it properly is all. I am not going to respond to the rest of your post, because you say in the first sentence that you aren't aware of the contradiction. You aren't aware of the contradiction, because you don't understand the temporal logic which I am using. If you had of been aware of the contradiction, your post would never have been made. PS: The uniqueness of the empty set follows from the axiom of extensionality of axiomatic set theory. That uniqueness is what permitted usage of the transitive property of equality, in my proof that axiomatic set theory leads to a contradiction. Again, my proof already showed up in the thread, you simply failed to understand the logic.
This is a refreshing post to read, after reading the one made by 'enlightenment'. You will have your answer, let me read carefully first. My first response to you, is NO i am not implying that the notion of set carries with it the concept of types. But it could. Temporal set theory can handle 'type' sets, as well as non-type sets. The biggest difference i can see, between axiomatic set theory ZF or ZFC, is that in the temporal set theory I'm (happily) using, a reasoning agent can manipulate temporal sets, sets whose elements can vary in time. For example, consider the temporal set of logically omniscient reasoning agents. The set of reasoning agents can be broken down into two mutually exclusive and collectively exhaustive sets. The set of reasoning agents who are logically omniscient, and the set of reasoning agents who are not logically omniscient. The process of becoming logically omniscient takes place in real time, and applies to a real reasoning agent. For awhile reasoning agent W is not yet logically omniscient. Then, there comes a moment in time at which Omega becomes logically omniscient. So that as far as set theory is concerned, Omega ceases to be an element of one set, and becomes an element of another set, and this real event happens in real time. Axiomatic set theory was built for numbers, (which cannot cease to be elements of the set they are in), and therefore it should not be surprising that temporal set theory is more general than axiomatic set theory. But I repeat, I am not suggesting that the notion of set carries with it the notion of 'type'. That would be overly restrictive, for no reason at all. You state that for sets with the same type, that you use the following definition of subset... A_M subset B_M iff Ax in A_M . x in B_M Where the subscript indicates that the sets are the same type. for example, A is a set of men, and B is a set of men. I am translating your notation above) as follows: A_M is a subset of B_M if and only if For any x: if x is an element of A_M then x is an element of B_M (If that is equivalent to your meaning, then I agree) Equivalently, i could translate as: A_M is a subset of B_M if and only if For any x an element of A_M[x is an element of B_M] (The translation above fiddles with modus ponens a bit, but its ok) As for the remainder of your post, I've already explained that the notion of set does not carry with it the notion of types. In specific cases, it will involve the notion of types, but certainly not all. We can always say that elements of a set must have something in common, since by merely being elements of a common set, they have that in common even if nothing else. But again, there is no reason to necessitate that sets must carry with them the notion of a 'type', so I would not suggest using the definition which follows. As far as "the empty set" of axiomatic set theory goes, in the set theory which i am using, there is no such thing as "THE" empty set. A given set can be empty, but that could change (in a case where the set is temporal in nature). So in other words, i would say that the set of ghosts is an empty set. I would not say that the set of ghosts is THE empty set. One other thing I should mention, is the use of propositional functions in defining sets. Suppose that you have some propositional function p(x), and you wish to define set A as follows: A = {y| p(y) is true} If by substituting y for x in p(x), the resulting expression is nonsensical, then y is not substitutatble for x, in p(x). The point is, that the canditates for substitution in the propositional function, must form a proposition upon instantiation. The resulting proposition must be either true or false. Nonsense is neither true or false, hence the caveat.
How, then, do you compare sets? What are the definitions of intersection and equality of sets? Because if you agree with the above definition of subset, and the type is not important for the subset relation, then any empty set will be a subset of any other set (specifically, all other empty sets). This leads to any empty set including every other empty set, and thus making all the empty sets equal except for name. Of course, this could just as easily be shown by an equality on their elements (there are none), which is why I ask how you define equality between sets. It cannot be the straightforward definition, because this leads to all empty sets being the same. I don't understand what you mean by "if substituting y for x in p(x) makes p(x) nonsensical" btw. What exactly is nonsensical?
yes, that is exactly what you wrote before, and it still isn't a cotradiction in set theory. 1. no set theory states that any class of dogs is actually a set; set theories actually specify very little about what the elements of sets are. 2. this isn't a contadiction in set theory, it is a contradiction of applying set theory to a situation that has the extra codition that a set empty of things recalls what it is empty of. thus your model with the extra axiom here is inconsistent with, say, ZFC, but that does not make ZFC inconsistent. as long as you insist on rubbishing these facts (do you even know what the axioms of ZFC are? are you aware of the concept of a model?) i will still think you are lying - deliberately propogating a false hood. i am not denying that there is a contradiction of something if we insist on extra rules, but set theory does not have those rules, you are not reasoming about set theory and the ignorance and arrogance you are displaying in claimng you are is frankly disgusting.
Ok, I am going to try to keep this thread from descending into chaos... I don't know what you mean by "compare" sets. Definition: Let A,B denote sets. The set "A intersect B" is defined as follows: AÇB = {x|x Î A AND x Î B} Definition: Let A,B denote sets. The set theoretic statement "A is a subset of B" is defined as follows: AÍB if and only if For any x: if x Î A then x Î B. In the case of non-empty sets A,B, set equality is defined as follows: A = B if and only if (A is a subset of B AND B is a subset of A) As there are a multiplicity of empty sets, there is no reason to devise a means to conclude that different empty sets are the same. That would be absurd. If this fails to answer your question about different empty sets, then make your question a little clearer please, and only ask one question at a time, so that I can focus. A statement must be either true or false. When you quantify, or instantiate a propositional function with a free variable, it is supposed to turn into a proposition. But a meaningless sentence cannot have truth value. So for example, suppose that A denotes the first moment in time, and that x is a free variable. consider the following propositional function in one variable... A before x Let R denote a randomly selected moment in time. If we instantiate x with R, we obtain the following proposition... A before R If R denotes any moment in time OTHER THAN A, then the proposition above is true. However, if the randomly selected moment in time R denotes the first moment in time, then the proposition above is false. The key thing to realize, is that R had to be chosen from the correct set. The binary relation 'before' being used above, is from the set of moments in time, to the set of moments in time. In other words, its a binary relation on the set of moments in time. If someone comes along and instantiated x with "atlantic ocean" the resulting sentence... A before atlantic ocean Would have been nonsensical. It would not be true, it would not be false, hence it would not be a proposition. This becomes an issue when one tries to define a set with no members, using a propositional function. I believe Frege made the error that any propositional function p(x) can be used to define a set, without paying attention to the relations (and their associated sets) used to formulate the propositional function.
It is a contradiction in set theory. You cannot have only one empty set. How the hell can I be lying, when I haven't said anything false? :bugeye: You are entitled to your opinion, but you should realize that I can perpetually isolate the error in axiomatic set theory. Perhaps this will finally make what I am telling you register... A ghost is not, i repeat not, an elf. Even though neither exist. Get it now? Axiomatic set theory allows you to conclude that a Ghost is an Elf. Temporal set theory does not.
