Wikipedia:Reference desk/Archives/Mathematics/2013 November 8
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November 8
[edit]Foundations of mathematics
[edit]Does everyone agree that the problems with the foundation of mathematics has merely died down rather being resolved? I only scratched the surface with logic and set theory some time ago, but today I was reading Skolem's paradox and it's stuff like this that unnerves me. I mean when I do ordinary math I have a clear intuitive picture of the "real" cumulative hierarchy, in the sense of a realist, and think of myself as working inside it (although I'm a formalist at heart), when all the stuff I do applies to other really weird models of set theory as well (and this causes Skolem's paradox). I think the foundations of mathematics is resting on a bubble/foam, do you agree? Sorry if I sound ignorant like I said I only scratched the surface with this topic. — Preceding unsigned comment added by Money is tight (talk • contribs) 08:04, 8 November 2013 (UTC)
- Really, Skolem's paradox is just not that weird. I've never really understood why this upsets people. All it says is, you have a set that's really countable, inside a model that doesn't have a complicated enough function to witness that countability. Why is that a problem? --Trovatore (talk) 08:08, 8 November 2013 (UTC)
- It is a bit of a problem for me because it completely destroys my notion of "size". Things like "strongly inaccessible cardinals are actually countable in another model" doesn't sit well with me. Maybe you set theorists got used to the notion of cardinality being a relative notion, but ordinary people like me haven't. — Preceding unsigned comment added by Money is tight (talk • contribs) 08:17, 8 November 2013 (UTC)
- I'm not a set theorist (nor a mathematician of any kind), so take this with a grain of salt. I take Skolem's Paradox as indicating that FO isn't strong enough to capture exactly what we mean by things like "uncountable", but is strong enough to talk about them in a limited fashion, and trustworthy/usable enough to be sure we know we are making sense (see Lindström's theorem). I've always considered this to be analogous to how continuity, in point set topology, doesn't uniquely characterize what we are thinking of in the case of the reals, but that any theorem about the topological notion will carry over to the more specific case. --it just happens that we can capture what we mean in the real case. --Again, I may be just rambling stupidly, ignore if so.Phoenixia1177 (talk) 08:43, 8 November 2013 (UTC)
- One of the things you learn when you're studying analysis is that uncountable is big. I understand what you mean, but to me Skolem's paradox is troubling because it destroys one's intuition for "size". It's quite hard to believe we might be working in an universe of sets (up to the rank of some inaccessible cardinal), yet the universe is actually countable!? I do many constructions such as taking very large products/coproducts of groups/spaces with the goal of producing huge groups/spaces with desired properties, and your telling me they are all still countable??? How exactly are uncountable ordinals like so much "longer" than when they are both countable? I guess you're probably seasoned with this topic like Trovatore, or you just thought about it too much and forgot about how it defies your intuition in everyday math.
- You have to bear in mind, though, exactly what it means for a model to say "X is uncountable". "X is uncountable" means that there is no bijection X onto a countable set, even if these bijections actually exist, nothing says that they are in M- indeed, there are more than a countable collection of bijections between any two countable sets, so most of them are missing automatically. The same thing goes for how M interprets the axioms: the powerset of X, in M, according to M, will be the set of all Y in M that are a subset of X (assume M transitive so the subset stuff lines up). However, not every actual subset of X is in M, so what M calls the powerset of M falls short of the actual powerset. The main disconnect is that with regards to the model, the universal quantifier only needs to range over M, not V; so M ends up missing the sets that would witness that it isn't getting things right.
- Weird Analogy: Suppose you have a friend who is a coin collector and he tells you that his valuable coins are the ones he has less than ten of. If you, then, formalized the property "valuable coin" as "less than 10 instances in a coin collection", some small collections would end up having "valuable" coins that aren't actually valuable. The problem is in the criterion for "valuable" and the fact that these small collections are too sparse to get it right. In the same way, FO captures "uncountable" in a way that isn't quite robust enough to deal with missing sets.Phoenixia1177 (talk) 07:35, 9 November 2013 (UTC)
- Ah nice, what you said about "there are more than a countable collection of bijections between any two countable sets, so most of them are missing automatically" helped me. I think my main problem is I'm getting my realist picture (which I use when I do math) and my formalist picture (which I believe in) confused. I just need more time to think about it. Money is tight (talk) 02:58, 10 November 2013 (UTC)
- You have to bear in mind, though, exactly what it means for a model to say "X is uncountable". "X is uncountable" means that there is no bijection X onto a countable set, even if these bijections actually exist, nothing says that they are in M- indeed, there are more than a countable collection of bijections between any two countable sets, so most of them are missing automatically. The same thing goes for how M interprets the axioms: the powerset of X, in M, according to M, will be the set of all Y in M that are a subset of X (assume M transitive so the subset stuff lines up). However, not every actual subset of X is in M, so what M calls the powerset of M falls short of the actual powerset. The main disconnect is that with regards to the model, the universal quantifier only needs to range over M, not V; so M ends up missing the sets that would witness that it isn't getting things right.
- One of the things you learn when you're studying analysis is that uncountable is big. I understand what you mean, but to me Skolem's paradox is troubling because it destroys one's intuition for "size". It's quite hard to believe we might be working in an universe of sets (up to the rank of some inaccessible cardinal), yet the universe is actually countable!? I do many constructions such as taking very large products/coproducts of groups/spaces with the goal of producing huge groups/spaces with desired properties, and your telling me they are all still countable??? How exactly are uncountable ordinals like so much "longer" than when they are both countable? I guess you're probably seasoned with this topic like Trovatore, or you just thought about it too much and forgot about how it defies your intuition in everyday math.
- Money, I don't say that strongly inaccessible cardinals are actually countable in another model. If they're countable in any (actual two-valued) model, then they're countable, period. Skolem's paradox doesn't say that countability simpliciter is a relative notion, not at all.
- All it says is that some models can be fooled into thinking that some sets are uncountable, when they're really not. But they're just wrong about that. There isn't any genuine relativity, just models being mistaken. --Trovatore (talk) 10:25, 8 November 2013 (UTC)
- I don't quite understand what you mean. You seem to sound like a hardcore realist in asserting if something is countable in any model then they're countable. From what I remember you can construct a countable model for ZF using as sets strings from the language of ZF, or something like that. The real numbers constructed from this countable model is uncountable inside it, but outside it's countable. To me this is still relative - although I tend to think like a realist I'm actually a hardcore formalist. I haven't really got the time to learn mathematical logic properly maybe I'll study it in detail some other time and get my head around this. Money is tight (talk) 02:49, 9 November 2013 (UTC)
- Ah, well there's your problem then. You're quite right that there's no satisfactory formalist resolution to the "foundational crisis". But that's because formalism is wrong. --Trovatore (talk) 19:08, 9 November 2013 (UTC)
- I'm confused. You don't believe there's a true model of set theory, and this doesn't seem to bother you. Yet the idea that there isn't a true notion of cardinality does bother you? Without a true model of set theory, of course there can't be a true notion of cardinality.--80.109.106.3 (talk) 19:53, 9 November 2013 (UTC)
- I think my problem is my ordinary reasoning gives me a realist view on cardinality but my formalist beliefs gets in the way. I see all three of you seem to be realists, I guess that's why you all have no problems with Skolem's paradox? Money is tight (talk) 02:58, 10 November 2013 (UTC)
- I don't consider myself a realist (at least, not beyond the hereditarily finite), but I have no problem with Skolem's paradox because I don't think there needs to be a real notion of cardinality.--80.109.106.3 (talk) 04:08, 10 November 2013 (UTC)
- That raises a question. I was reading about how forcing adds more sets to a given model to violate the continuum hypothesis (I have no idea of forcing is actually about though). I was wondering if given any model of ZF, it's possible to add more sets to form a bigger model such that the starting model is countable. This probably sounds stupid, but if this is the case I guess then I can go back to my childhood belief that infinity only has one size - , instead of having a whole proper class of cardinals shoved down my throat. Money is tight (talk) 07:36, 10 November 2013 (UTC)
- Any model of ZF is going to have nonequinumerous infinite sets, so even if this model jumping idea made perfect sense, you can at best change which pairs of sets satisfy "infinite and not equinumerous", but you'll always have them, so you can't get away with just one infinity (see Hartogs number).
- That aside, the real heart of the issue is that you are getting hung up when switching from looking at things in the model M and looking at things in another model N which contains the sets in M. When you are working in N, the quantifiers range over a larger universe, this "breaks" the model properties of M -M, as a model, might satisfy "there is an uncountable set", even if countable, but that's only because you are forced to restrict all of your quantification to what is in M, once you zoom out to a bigger model N containing the sets in M, M doesn't contain an uncountable set. (add the necessary adjectives to scale this all correctly)
- For the paradox to get off the ground, you need a bigger model containing M from the start- otherwise, I'm not really sure what it means to say M is countable, M can't prove this about itself (it thinks it's a class). But, then, you're just saying that you can restrict your quantifiers and still end up with true sentences- just the sets that witness that truth end up changing, but that's not really that shocking. Or, at least, it's no more shocking than that the reals with their usual order and the rationals with their usual order both satisfy the same FO theory- I don't think this at all breaks our notion of what a real number is, it's just a quirk of restricting quantifiers and FO, the same case with sets. (sorry if this is rambly, I've been up for a lot of hours due to work (going on 37!)). :-)Phoenixia1177 (talk) 19:15, 10 November 2013 (UTC)
- That raises a question. I was reading about how forcing adds more sets to a given model to violate the continuum hypothesis (I have no idea of forcing is actually about though). I was wondering if given any model of ZF, it's possible to add more sets to form a bigger model such that the starting model is countable. This probably sounds stupid, but if this is the case I guess then I can go back to my childhood belief that infinity only has one size - , instead of having a whole proper class of cardinals shoved down my throat. Money is tight (talk) 07:36, 10 November 2013 (UTC)
- I don't consider myself a realist (at least, not beyond the hereditarily finite), but I have no problem with Skolem's paradox because I don't think there needs to be a real notion of cardinality.--80.109.106.3 (talk) 04:08, 10 November 2013 (UTC)
- I think my problem is my ordinary reasoning gives me a realist view on cardinality but my formalist beliefs gets in the way. I see all three of you seem to be realists, I guess that's why you all have no problems with Skolem's paradox? Money is tight (talk) 02:58, 10 November 2013 (UTC)
- I don't quite understand what you mean. You seem to sound like a hardcore realist in asserting if something is countable in any model then they're countable. From what I remember you can construct a countable model for ZF using as sets strings from the language of ZF, or something like that. The real numbers constructed from this countable model is uncountable inside it, but outside it's countable. To me this is still relative - although I tend to think like a realist I'm actually a hardcore formalist. I haven't really got the time to learn mathematical logic properly maybe I'll study it in detail some other time and get my head around this. Money is tight (talk) 02:49, 9 November 2013 (UTC)
- I'm not a set theorist (nor a mathematician of any kind), so take this with a grain of salt. I take Skolem's Paradox as indicating that FO isn't strong enough to capture exactly what we mean by things like "uncountable", but is strong enough to talk about them in a limited fashion, and trustworthy/usable enough to be sure we know we are making sense (see Lindström's theorem). I've always considered this to be analogous to how continuity, in point set topology, doesn't uniquely characterize what we are thinking of in the case of the reals, but that any theorem about the topological notion will carry over to the more specific case. --it just happens that we can capture what we mean in the real case. --Again, I may be just rambling stupidly, ignore if so.Phoenixia1177 (talk) 08:43, 8 November 2013 (UTC)
- It is a bit of a problem for me because it completely destroys my notion of "size". Things like "strongly inaccessible cardinals are actually countable in another model" doesn't sit well with me. Maybe you set theorists got used to the notion of cardinality being a relative notion, but ordinary people like me haven't. — Preceding unsigned comment added by Money is tight (talk • contribs) 08:17, 8 November 2013 (UTC)
All of math that mankind will ever be able to invent can be captured within an ultrafinitistic framework. To see this, consider simulating the entire Earth with all the mathematicians on a very powerful computer. The computer is powerful enough to simulate how real mathematicians behave. Then because the computer is just a finite state machine, any notion of the continuum, uncountable sets etc. that the virtual mathematicans come up with are clearly just fantasies. They can all be re-interpreted in terms of finite concepts. If one insists that these fantasies about things that cannot exist in the universe are to be taken serious, then why not any other of the zillions of other fantasies one can come up with? Count Iblis (talk) 16:34, 10 November 2013 (UTC)
- Does this have anything to do with what is being discussed? What does your "neat" proof that mathematical philosophy is bullshit have to do with helping someone develop a better intuition for set theory? This is the equivalent of replying with a link to pomo griping about things being "theory laden" when someone is confused about quantum mechanics.Phoenixia1177 (talk) 19:15, 10 November 2013 (UTC)
- Your framework can't explain the observed consistency of the mathematics. If you don't care about explanation, then why bother with fantasies like electrons and protons, to explain physical observations? --Trovatore (talk) 19:51, 10 November 2013 (UTC)