I have understood that a number can exist in multiple sets at once, being more like a reference the same number in all places. so why can't a container have a reference to itself?
The usual axiomatization of the set theory include the axiom of regularity. From this axiom it follows that no set can contain itself; more generally, there doesn't exist an infinitely long chain of sets A, B, C, D, ... such that B is an element of A, C is an element of B, D is an element of C, and so on. But we don't get any contradiction if we omit this axiom and instead assume that there does exist a set containing itself. (These are the various "anti-foundation" or "anti-regularity" axioms.) But regardless of whether or not we accept the axiom of regularity, no set (under the rest of the usual ZF axioms) can contain all sets; in other words, a class of all sets is not a set. Given any set X it is possible to construct set Y = {x: xϵX and x∉x} - the set of all sets in X which are not its own element. Such a set Y can't be an element of X - if it were, then we get an impossibility: Y can't be an element of Y, and Y can't NOT be an element of Y - both options contradict the definition of the set Y itself. This is the famous barber paradox. Suppose that a male barber sets up shop in town, and puts up a sign: "I shave all men in town who don't shave themselves, but don't shave those who shave themselves". But one day, a customer asks the barber: "Do you shave yourself?" And it can be seen that the barber can't answer the question without contradicting his sign: if he shaves himself, then he shaves a man who shaves himself; and if he doesn't, then he doesn't shave a man who doesn't shave himself. (The only way out is that some of our assumptions wasn't true: either the barber is not telling the truth, or he doesn't live in the town, or is a woman.)
Let me summarize this. In order to build a set of all sets we would have at least give up on creation of subsets (might have to give up on other axioms). This price is too high for a set of sets. We are much better of if we conclude that a set of all sets just doesn't exist. I am in. We can live without a set of sets, but with set partitioning axioms.
Yep. He used Russell's paradox to prove that there is no universal set. This is the point of the paradox and main reason why there is no universal set. But, the powerset method is just a cool non-standard offshoot that helps you learn a little bit more about power sets.
Not necessarily. In standard ZFC there is the axiom of regularity, from which it follows that no set can be an element of itself. But let's omit it, and let's suppose to the contrary that there exists a set (also called Quine atom) containing only itself: a = {a}. Let's create a subset of this set using the formula x∉x, i.e. a set of all sets in a that do not contain themselves. Obviously, this subset of a is empty. And obviously, a does not contain an empty set; it only contains itself. No contradiction here. We have only proven that no set A can contain the set B = {x: xϵA and x∉x}.
What is actually in A? Since few sets are elements of themselves x∉x is most sets and V ∩ {x : x∉x } should be most sets. If you set x = {∅} it satisfies both conditions in {x: xϵV and x∉x} as would all sets that don't contain themselves, if I understand this correctly. DoesA contain anything?
I am not sure what you are asking. The theorem is: given any set V, there is a set which is not an element of V. And that set is the following: set of all sets for which the following is true: x∈V and x∉x. (This set exists by the axiom of separation; or, as the video calls it, axiom of subsets.) From axiom of regularity it follows that no set is an element of itself, so the set in question is simply V; but the theorem is true even in absence of axiom of regularity. (Therefore, there isn't a set of all sets.)
Couldnt I just define the universal set = {x | x is an object}, i.e. it only contains existing (= contradiction-free) objects. IMO you can rescue naive set theory with such an approach, can u?
What exactly does the proposition "x is an object" mean? Do there exist some elements which aren't an "object"? I'd say that your proposal says nothing; the universal set could as well be written as U={x: x=x}. But at the same time, you are heading in the right direction. In conventional set theory (ZF or ZFC, even if we omit the axiom of regularity) the universal set can be proven not to exist; it's the consequence of the axiom of separation. But it turns out that there's nothing a priori contradictory about the universal set; there are set theories, such as the 'New Foundation', which do allow for the set of all sets. (But obviously, such a theory will not satisfy the axiom of separation; for example, the universal set U has a definable subclass: U'={x: x∈U and x∉x}, but that cannot exist as a set.) Let's prove the non-existence of the set of all sets from bottom up, using the axiom of separation. Let A be any set. Let B={x: x∈A and x∉x}. Observe that by construction, B∉B; otherwise, B would contain a set which contains itself. B∉A; otherwise, we would have to conclude that B∈B (B contains all elements of A which don't contain themselves as an element; we have already proven that B∉B; so it follows that if B∈A, then by construction of the set B it is the case that B∈B, contradicting the previous result that B∉B). Finally, A∉B: because B is a subset of A, it follows that if A∈B, then A∈A; but that contradicts the definition of B because B only contains sets which don't contain themselves. So it follows from the axiom of separation: given any set A, there exists a set B (namely, the set {x: x∈A and x∉x}) such that B∉A, and therefore A is not a set of all sets.
@@MikeRosoftJH U = {x|„x is an object“} includes all possible objects, but excludes all deceiving objects like Russell‘s set that have contradictory consequences and therefore cannot exist (as objects); so we get a set with all things that do not lead to a contradiction. That is the real universal set of naive set theory since {x|x=x} turns out to be false there and therefore non-existent. Of course in ZFC it is another story, actually I think my U would not match the conditions of being a set in ZF or ZFC, right?
@@ostihpem What does the universal set U={x: x=x} mean? It's the set for which the following is true: ∀x: x∈U if and only if x=x; because x=x for all objects of set theory, this can be simplified to: ∀x: x∈U. And obviously, ∀x ("for all x") only includes objects which exist; so it doesn't include objects which don't exist, like the impossible set {x: x∉x}. "That is the real universal set of naive set theory since {x|x=x} turns out to be false there and therefore non-existent." What does this even mean? {x: x=x} is not a proposition which could be true or false. It's a definition of a particular set, which in a given set theory may or may not exist (for example, it doesn't exist in ZF or ZFC, but it does exist in 'New Foundations' set theory). Naive set theory is inconsistent for a different reason: it assumed that for every proposition F(x) there exists a set of all sets for which F(x) is true. But that's impossible; one example of an impossible proposition for which the set of all sets which satisfy it cannot exist is x∉x. However, it is possible to do something different. You can work in a theory of classes, where every object is a class; a class may be either a set, or a proper class. We define: x is a set, if it is an element of some class. (A class which is not a set is a proper class.) And then we take as an axiom that for every proposition F(x) there exists a class of all sets for which F(x) is true. So there indeed exists a class of all sets; and it's a proper class and so it's not an element of itself. There also exists a class of all sets for which x∉x is true; assuming axiom of regularity, it's just the class of all sets (because from the axiom of regularity it follows that no set is an element of itself).
+mdphdguy1 Hello. Thank you for responding. I'm fascinated by the philosophical issues surrounding "The Universal Set" & "The Empty Set". Most people don't think that I'm serious when I tell them that I have an interest in 'nothing'. The video is reminiscent of what mathematical physicists refer to as the void (the vacuum of totality). It's a serious topic in physics forums. Dr. Lawrence Krauss is a man is search of 'nothing'. Śūnyatā is a more accurate word in reference to the nature of the empty set. It is the emptiness of 'All'. Dr. John D. Barrow "NOTHING: The Science of Emptiness" watch?v=BCUmeE8sIVo
+Thomas Rutledge It's funny that you bring this topic up because I've been following Krauss's talks and books for some time. I don't study cosmology so I cannot critique any of his work on technical grounds but I must say that I find that most of the pop science stuff on the idea of universes from "nothing" unconvincing and obscurantist. I think a lot of that topic suffers from abuse of language where someone discussing physics may have a technical definition of "nothing" and then someone studying formal systems (like in this video) might think of "nothing" as the empty set. We might mistakenly think that our ideas are in contact whereas they are not, which leads to confusion and mysticism.
+mdphdguy1 Yes, it does lead to mysticism. Especially, when one looks at modern cosmology in the context of Cantorian set theory. When a person encounters Russell's Paradox & then ponders the universal set & the empty set... it's not long before one is into Zen Buddhism & reading the Tao Te Ching. I've been able to find one critic of modern mathematics (set theory). Dr. Norman Wildberger, "Set Theory: Should You Believe? Modern mathematics as religion"
+Thomas Rutledge It's not immediately clear that any of the exotic stuff in set theory that Cantor is associated with has any immediate application in physics (at least not in 2016). Perhaps you could provide an example? I know the examples that are peddled around have to do with summing divergent series, but that doesn't need any Cantor stuff to do it, I think. One lesson I think we can learn from this exotic stuff is that there is no upper bound to the set theoretic universe, which means there will always be more math to do. It's not going to be like in physics where they think there's going to be a day where the final unifying theory will be published. I quite like Prof. Wildberger's work and I follow his stuff on his UA-cam channel. He has good points to make about real numbers and Cantor stuff and how they are way too exotic and unfeasible in the real world. Although the set theory I cover in my videos and his tempered views are in contradiction, I think it's good to learn as much as possible.
What do you think of Alternative Set Theories, e.g. NF or NFU.
I have understood that a number can exist in multiple sets at once, being more like a reference the same number in all places. so why can't a container have a reference to itself?
The usual axiomatization of the set theory include the axiom of regularity. From this axiom it follows that no set can contain itself; more generally, there doesn't exist an infinitely long chain of sets A, B, C, D, ... such that B is an element of A, C is an element of B, D is an element of C, and so on. But we don't get any contradiction if we omit this axiom and instead assume that there does exist a set containing itself. (These are the various "anti-foundation" or "anti-regularity" axioms.)
But regardless of whether or not we accept the axiom of regularity, no set (under the rest of the usual ZF axioms) can contain all sets; in other words, a class of all sets is not a set. Given any set X it is possible to construct set Y = {x: xϵX and x∉x} - the set of all sets in X which are not its own element. Such a set Y can't be an element of X - if it were, then we get an impossibility: Y can't be an element of Y, and Y can't NOT be an element of Y - both options contradict the definition of the set Y itself.
This is the famous barber paradox. Suppose that a male barber sets up shop in town, and puts up a sign: "I shave all men in town who don't shave themselves, but don't shave those who shave themselves". But one day, a customer asks the barber: "Do you shave yourself?" And it can be seen that the barber can't answer the question without contradicting his sign: if he shaves himself, then he shaves a man who shaves himself; and if he doesn't, then he doesn't shave a man who doesn't shave himself. (The only way out is that some of our assumptions wasn't true: either the barber is not telling the truth, or he doesn't live in the town, or is a woman.)
Let me summarize this. In order to build a set of all sets we would have at least give up on creation of subsets (might have to give up on other axioms). This price is too high for a set of sets. We are much better of if we conclude that a set of all sets just doesn't exist.
I am in. We can live without a set of sets, but with set partitioning axioms.
Doesn't the first proof also imply that there are no sets that can contain themselves?
Yep. He used Russell's paradox to prove that there is no universal set. This is the point of the paradox and main reason why there is no universal set. But, the powerset method is just a cool non-standard offshoot that helps you learn a little bit more about power sets.
Not necessarily. In standard ZFC there is the axiom of regularity, from which it follows that no set can be an element of itself. But let's omit it, and let's suppose to the contrary that there exists a set (also called Quine atom) containing only itself: a = {a}. Let's create a subset of this set using the formula x∉x, i.e. a set of all sets in a that do not contain themselves. Obviously, this subset of a is empty. And obviously, a does not contain an empty set; it only contains itself. No contradiction here.
We have only proven that no set A can contain the set B = {x: xϵA and x∉x}.
What is actually in A? Since few sets are elements of themselves x∉x is most sets and V ∩ {x : x∉x } should be most sets. If you set x = {∅} it satisfies both conditions in {x: xϵV and x∉x} as would all sets that don't contain themselves, if I understand this correctly. DoesA contain anything?
I am not sure what you are asking. The theorem is: given any set V, there is a set which is not an element of V. And that set is the following: set of all sets for which the following is true: x∈V and x∉x. (This set exists by the axiom of separation; or, as the video calls it, axiom of subsets.) From axiom of regularity it follows that no set is an element of itself, so the set in question is simply V; but the theorem is true even in absence of axiom of regularity. (Therefore, there isn't a set of all sets.)
Couldnt I just define the universal set = {x | x is an object}, i.e. it only contains existing (= contradiction-free) objects. IMO you can rescue naive set theory with such an approach, can u?
What exactly does the proposition "x is an object" mean? Do there exist some elements which aren't an "object"? I'd say that your proposal says nothing; the universal set could as well be written as U={x: x=x}. But at the same time, you are heading in the right direction. In conventional set theory (ZF or ZFC, even if we omit the axiom of regularity) the universal set can be proven not to exist; it's the consequence of the axiom of separation. But it turns out that there's nothing a priori contradictory about the universal set; there are set theories, such as the 'New Foundation', which do allow for the set of all sets. (But obviously, such a theory will not satisfy the axiom of separation; for example, the universal set U has a definable subclass: U'={x: x∈U and x∉x}, but that cannot exist as a set.)
Let's prove the non-existence of the set of all sets from bottom up, using the axiom of separation. Let A be any set. Let B={x: x∈A and x∉x}. Observe that by construction, B∉B; otherwise, B would contain a set which contains itself. B∉A; otherwise, we would have to conclude that B∈B (B contains all elements of A which don't contain themselves as an element; we have already proven that B∉B; so it follows that if B∈A, then by construction of the set B it is the case that B∈B, contradicting the previous result that B∉B). Finally, A∉B: because B is a subset of A, it follows that if A∈B, then A∈A; but that contradicts the definition of B because B only contains sets which don't contain themselves. So it follows from the axiom of separation: given any set A, there exists a set B (namely, the set {x: x∈A and x∉x}) such that B∉A, and therefore A is not a set of all sets.
@@MikeRosoftJH U = {x|„x is an object“} includes all possible objects, but excludes all deceiving objects like Russell‘s set that have contradictory consequences and therefore cannot exist (as objects); so we get a set with all things that do not lead to a contradiction. That is the real universal set of naive set theory since {x|x=x} turns out to be false there and therefore non-existent. Of course in ZFC it is another story, actually I think my U would not match the conditions of being a set in ZF or ZFC, right?
@@ostihpem What does the universal set U={x: x=x} mean? It's the set for which the following is true: ∀x: x∈U if and only if x=x; because x=x for all objects of set theory, this can be simplified to: ∀x: x∈U. And obviously, ∀x ("for all x") only includes objects which exist; so it doesn't include objects which don't exist, like the impossible set {x: x∉x}. "That is the real universal set of naive set theory since {x|x=x} turns out to be false there and therefore non-existent." What does this even mean? {x: x=x} is not a proposition which could be true or false. It's a definition of a particular set, which in a given set theory may or may not exist (for example, it doesn't exist in ZF or ZFC, but it does exist in 'New Foundations' set theory). Naive set theory is inconsistent for a different reason: it assumed that for every proposition F(x) there exists a set of all sets for which F(x) is true. But that's impossible; one example of an impossible proposition for which the set of all sets which satisfy it cannot exist is x∉x.
However, it is possible to do something different. You can work in a theory of classes, where every object is a class; a class may be either a set, or a proper class. We define: x is a set, if it is an element of some class. (A class which is not a set is a proper class.) And then we take as an axiom that for every proposition F(x) there exists a class of all sets for which F(x) is true. So there indeed exists a class of all sets; and it's a proper class and so it's not an element of itself. There also exists a class of all sets for which x∉x is true; assuming axiom of regularity, it's just the class of all sets (because from the axiom of regularity it follows that no set is an element of itself).
When mathematicians embrace "the universal set", they rest in its absolute complement, "the empty set". watch?v=HFWKJ2FUiAQ
+Thomas Rutledge If you say so. And what does this link have to do with the video?
+mdphdguy1 Hello. Thank you for responding. I'm fascinated by the philosophical issues surrounding "The Universal Set" & "The Empty Set". Most people don't think that I'm serious when I tell them that I have an interest in 'nothing'. The video is reminiscent of what mathematical physicists refer to as the void (the vacuum of totality). It's a serious topic in physics forums. Dr. Lawrence Krauss is a man is search of 'nothing'.
Śūnyatā is a more accurate word in reference to the nature of the empty set. It is the emptiness of 'All'.
Dr. John D. Barrow "NOTHING: The Science of Emptiness" watch?v=BCUmeE8sIVo
+Thomas Rutledge It's funny that you bring this topic up because I've been following Krauss's talks and books for some time. I don't study cosmology so I cannot critique any of his work on technical grounds but I must say that I find that most of the pop science stuff on the idea of universes from "nothing" unconvincing and obscurantist. I think a lot of that topic suffers from abuse of language where someone discussing physics may have a technical definition of "nothing" and then someone studying formal systems (like in this video) might think of "nothing" as the empty set. We might mistakenly think that our ideas are in contact whereas they are not, which leads to confusion and mysticism.
+mdphdguy1 Yes, it does lead to mysticism. Especially, when one looks at modern cosmology in the context of Cantorian set theory. When a person encounters Russell's Paradox & then ponders the universal set & the empty set... it's not long before one is into Zen Buddhism & reading the Tao Te Ching.
I've been able to find one critic of modern mathematics (set theory). Dr. Norman Wildberger, "Set Theory: Should You Believe? Modern mathematics as religion"
+Thomas Rutledge It's not immediately clear that any of the exotic stuff in set theory that Cantor is associated with has any immediate application in physics (at least not in 2016). Perhaps you could provide an example? I know the examples that are peddled around have to do with summing divergent series, but that doesn't need any Cantor stuff to do it, I think.
One lesson I think we can learn from this exotic stuff is that there is no upper bound to the set theoretic universe, which means there will always be more math to do. It's not going to be like in physics where they think there's going to be a day where the final unifying theory will be published.
I quite like Prof. Wildberger's work and I follow his stuff on his UA-cam channel. He has good points to make about real numbers and Cantor stuff and how they are way too exotic and unfeasible in the real world. Although the set theory I cover in my videos and his tempered views are in contradiction, I think it's good to learn as much as possible.