Zero is an excepton to transitive set definition of natural numbers. because if given set is transitive, union of "the set contains empty set "(1) must be equal "empty set" (0). But union of "the set contains empty set"(1) is "the set contains emtpy set"(1) so yeah, exception.
Good video! Now I have an idea of what Peano system is. In a previous lesson, you said that there are nine ZFC axioms? Or at least nine axioms? So the three axioms covered in this video pretty much completes the list? No, I just checked "ZFC" in Wikepedia, I recognize most of them, but the three axioms of the Peano system covered in this video are not listed. I had the idea that we are still doing ZFC. Is the Peano system part of ZFC or is it something different? One reason why I tried to learn some set theory (which would have been impossible without your excellent videos!) is to understand how arithmetic can be done using sets. To do arithmetic, we need the natural numbers and so the question becomes how natural numbers can be derived using the idea of sets. Now, through your lessons, I understand that the natural numbers are an inductive set, that they are a Peano system, and that they are transitive sets. So that pretty much defines what natural numbers are? Once we have defined what the natural numbers are, using the axioms of set theory, then we can go on and do the arithmetic, which will be coming up in your subsequent lectures, I guess. I presume that everything in the Peano system fits together and is consistent throughout? Now if the axioms are inconsistent, however, we wouldn't have succeeded in defining what the natural numbers are. My problem is I don't see why we have to have a PROOF that the system is consistent. If we have a starting set of axioms and by using them we can do all the rest of the stuff without any contradictions, isn't that a proof in itself? This is an imperfect analogy, but take baseball. Given a set of simple rules, we can understand more complicated plays such as 1-2-3 double play. We play the game and everything that can happen on the field can be explained in terms of the most basic rules ("axioms"). Everything works fine and there is no inconsistency. So why would anyone ask for a PROOF that the game of baseball is consistent? So why would David Hilbert ask for "the proof of the compatibility of arithmetical axioms" in his second problem? If I may hazard a guess, is it because Hilbert was asking for a proof using finitistic means, whereas in ZFC an infinite set is taken as an axiom (namely, the Axiom of infinity)?
John Ok I think 5/9 axioms were introduced in part 2 and the remaining 4 are: infinity, foundation, replacement, and choice. It just seemed more appropriate to introduce these as we needed them. With the peano system stuff, I think pre-ZFC those were called the "Peano axioms" and this system was aimed mainly at setting up natural number arithmetic and not really sets. In the setting of ZFC, the "Peano axioms" are just a definition of what ordered triple constitutes a Peano system, just like we have three criteria for a relation being an equivalence relation. As for why someone would want a consistency proof, someone studying this subject might be super paranoid about getting into precisely the same situation as naive set theory did, where everything seemed consistent and yet the axioms are deriving contradictions. Ideally, such a person would like to prove the proposition "the truth of the following axioms implies that if a theorem is proven, its negation is demonstrably false". This is why set theory collapsed in Frege's time, because naive set theory allows one to prove a proposition and its negation, since one contradiction allows any proposition/nonsense to be proven. However, Peano arithmetic (PA) is consistent iff ZFC is consistent (I forget the theorem's name), but due to Godel's theorem, we cannot give a proof of PA's consistency within PA.
John Ok Also, to take your baseball analogy, we have to be careful because now we're talking about natural language, so all bets are off as to whether we can say with any certainty that the rules of baseball cannot be interpreted to ever give contradictions. Sure, we can make some reasonable statements about what kind of play follows the rules but we don't have the degree of certainty that we have with statements like A=B & B=C --> A=C. But, I do agree that it's not an interesting question to ask whether baseball is deductively consistent or can be tediously written out in symbolic form, just to guarantee that we won't have to ever face ambiguous moments when playing. However, one can certainly prove that some games are inconsistent. For example, let's say we're playing chess and I have the rule that "the king may move only one square in any direction" and then I have the ad hoc rule "the king may teleport himself to any square he wants". Here, the rules are clearly inconsistent.
Lectures excellent, Thank you. The Sound of the voice in each video is a bit annoying. It sounds like a bubble coming from beneath a water tank. It bothers me when I listen to the entire video.
still following along. only sad thing is you just made 10 vid's.. hope you have time for more. ! next stop for me is to do some problems..> kaharris.org/teaching/582/index.html - thanks again for doing these.
Chris Bedford Hopefully I'll be able to put out a new video every few days. We're also building up to the more interesting (some would say wacky) parts of set theory such as cardinals and ordinals. Perhaps I'll steal some of these practice problems for my videos.
mdphdguy1 Looking forward to it ! -- Here are some other problem sets.. not sure which are non copyrighted.. but this might be of interest to your other viewers. >> web.mit.edu/kayla/tcom/tcom_probs_settheory_sols.doc www.comp.nus.edu.sg/~fstephan/settheory.html math.stackexchange.com/questions/17285/basic-problems-in-elementary-set-theory saylor.org/site/wp-content/uploads/2012/11/MA111-Assessment-2-Elementary-Set-Theory-Homework-Set-Solutions-FINAL.pdf
Thanks indeed. I've watched once but I need watching again
How do we approach proving x0 implies xz
9:14 for the definition of a transitive set; for those already comfortable with peano systems. 15:00 is pretty interesting.
Zero is an excepton to transitive set definition of natural numbers. because if given set is transitive, union of "the set contains empty set "(1) must be equal "empty set" (0). But union of "the set contains empty set"(1) is "the set contains emtpy set"(1) so yeah, exception.
Good video! Now I have an idea of what Peano system is.
In a previous lesson, you said that there are nine ZFC axioms? Or at least nine axioms? So the three axioms covered in this video pretty much completes the list? No, I just checked "ZFC" in Wikepedia, I recognize most of them, but the three axioms of the Peano system covered in this video are not listed. I had the idea that we are still doing ZFC. Is the Peano system part of ZFC or is it something different?
One reason why I tried to learn some set theory (which would have been impossible without your excellent videos!) is to understand how arithmetic can be done using sets. To do arithmetic, we need the natural numbers and so the question becomes how natural numbers can be derived using the idea of sets. Now, through your lessons, I understand that the natural numbers are an inductive set, that they are a Peano system, and that they are transitive sets. So that pretty much defines what natural numbers are? Once we have defined what the natural numbers are, using the axioms of set theory, then we can go on and do the arithmetic, which will be coming up in your subsequent lectures, I guess. I presume that everything in the Peano system fits together and is consistent throughout? Now if the axioms are inconsistent, however, we wouldn't have succeeded in defining what the natural numbers are. My problem is I don't see why we have to have a PROOF that the system is consistent. If we have a starting set of axioms and by using them we can do all the rest of the stuff without any contradictions, isn't that a proof in itself? This is an imperfect analogy, but take baseball. Given a set of simple rules, we can understand more complicated plays such as 1-2-3 double play. We play the game and everything that can happen on the field can be explained in terms of the most basic rules ("axioms"). Everything works fine and there is no inconsistency. So why would anyone ask for a PROOF that the game of baseball is consistent? So why would David Hilbert ask for "the proof of the compatibility of arithmetical axioms" in his second problem? If I may hazard a guess, is it because Hilbert was asking for a proof using finitistic means, whereas in ZFC an infinite set is taken as an axiom (namely, the Axiom of infinity)?
John Ok I think 5/9 axioms were introduced in part 2 and the remaining 4 are: infinity, foundation, replacement, and choice. It just seemed more appropriate to introduce these as we needed them. With the peano system stuff, I think pre-ZFC those were called the "Peano axioms" and this system was aimed mainly at setting up natural number arithmetic and not really sets. In the setting of ZFC, the "Peano axioms" are just a definition of what ordered triple constitutes a Peano system, just like we have three criteria for a relation being an equivalence relation. As for why someone would want a consistency proof, someone studying this subject might be super paranoid about getting into precisely the same situation as naive set theory did, where everything seemed consistent and yet the axioms are deriving contradictions. Ideally, such a person would like to prove the proposition "the truth of the following axioms implies that if a theorem is proven, its negation is demonstrably false". This is why set theory collapsed in Frege's time, because naive set theory allows one to prove a proposition and its negation, since one contradiction allows any proposition/nonsense to be proven. However, Peano arithmetic (PA) is consistent iff ZFC is consistent (I forget the theorem's name), but due to Godel's theorem, we cannot give a proof of PA's consistency within PA.
John Ok Also, to take your baseball analogy, we have to be careful because now we're talking about natural language, so all bets are off as to whether we can say with any certainty that the rules of baseball cannot be interpreted to ever give contradictions. Sure, we can make some reasonable statements about what kind of play follows the rules but we don't have the degree of certainty that we have with statements like A=B & B=C --> A=C. But, I do agree that it's not an interesting question to ask whether baseball is deductively consistent or can be tediously written out in symbolic form, just to guarantee that we won't have to ever face ambiguous moments when playing. However, one can certainly prove that some games are inconsistent. For example, let's say we're playing chess and I have the rule that "the king may move only one square in any direction" and then I have the ad hoc rule "the king may teleport himself to any square he wants". Here, the rules are clearly inconsistent.
Axiom of infinity is less obvious than the axiom of choice if you ask me.
I've elsewhere heard it pronounced pe-(arr-no)?
It's an Italian name, so 'peh-AH-noh'
Lectures excellent, Thank you.
The Sound of the voice in each video is a bit annoying. It sounds like a bubble coming from beneath a water tank. It bothers me when I listen to the entire video.
still following along. only sad thing is you just made 10 vid's.. hope you have time for more. ! next stop for me is to do some problems..> kaharris.org/teaching/582/index.html - thanks again for doing these.
Chris Bedford Hopefully I'll be able to put out a new video every few days. We're also building up to the more interesting (some would say wacky) parts of set theory such as cardinals and ordinals. Perhaps I'll steal some of these practice problems for my videos.
mdphdguy1 Looking forward to it ! -- Here are some other problem sets.. not sure which are non copyrighted.. but this might be of interest to your other viewers. >> web.mit.edu/kayla/tcom/tcom_probs_settheory_sols.doc
www.comp.nus.edu.sg/~fstephan/settheory.html
math.stackexchange.com/questions/17285/basic-problems-in-elementary-set-theory
saylor.org/site/wp-content/uploads/2012/11/MA111-Assessment-2-Elementary-Set-Theory-Homework-Set-Solutions-FINAL.pdf