Timeline for Don't the axioms of set theory implicitly assume numbers?
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| Dec 2, 2010 at 15:50 | comment | added | Deniz | Yes, I agree with both points -- I understand that whatever gives us a working formalism where we can prove things & resolve uncertainties is good enough. It sounds like we take our everyday world for granted, and then agree on a formal language & rules like ZFC, define functions as "ordered pairs" etc. simply to resolve uncertainties & inherent ambiguities in our intuitive notions. I agree Stefan's last sentence about "this sort of circularity". Can we comprehend anything except in light of new combinations of what we already comprehend? This gets very 'philosophical' very fast. | |
| Dec 2, 2010 at 15:38 | vote | accept | Deniz | ||
| Nov 27, 2010 at 21:46 | vote | accept | Deniz | ||
| Nov 27, 2010 at 21:46 | |||||
| Nov 26, 2010 at 19:43 | comment | added | Stefan Geschke | @Ketil: I wouldn't say that NGB is all that different from ZFC. They are pretty much formalizations of the same concepts. This is why it doesn't make a difference for usual mathematics which system we are working in. | |
| Nov 26, 2010 at 13:05 | comment | added | Ketil Tveiten | ... which is model how the natural numbers behave. So, it doesn't matter what foundations we use (and none of them are really 'more natural' than the others), so long as they let us do what we want. ZFC is just a neat way of formalising how sets (should) behave, where things work the way they are supposed to (w.r.t. 'ordinary' mathematics) and that's all we need. We might as well have been working in NBG, (so far as I know) the 'ordinary' mathematics wouldn't know the difference. | |
| Nov 26, 2010 at 13:02 | comment | added | Ketil Tveiten | @Deniz: It doesn't matter whether something in (say) ZFC that behaves like the natural numbers is the 'natural' foundation, indeed it doesn't really make sense to ask if it is. To put it this way: what are the natural numbers, really? One could say they are 'the set of finite ordinals' (i.e. quoting the ZFC definition), but how is that different from [however they are defined in NBG]? From the perspective of how they behave, not at all. Is either one the Right True Answer? Yes and no. No, because neither is any more right than any other model; Yes, because they do what they're supposed to ... | |
| Nov 26, 2010 at 12:53 | comment | added | Ketil Tveiten | @Stefan: I didn't mean to say we should somehow try to avoid this circularity, I meant to say that we shouldn't worry about it. That natural numbers show up in the prerequisites to formalisation of mathematics probably reflects the fact that the natural numbers are a very basic structure, rather than that we need knowledge about them to construct the formalisation to get it right (even if we do, kinda). | |
| Nov 26, 2010 at 10:59 | comment | added | Todd Trimble | @Stefan: I would say that practically all mathematicians work with sets and functions (with the category of sets and functions, if you like). The further reduction of functions to sets (thus founding mathematics on a single membership predicate, leading to ZFC if taken seriously) is an optional move which has pluses and minuses, but it leads to issues most mathematicians have little practical interest in. | |
| Nov 26, 2010 at 10:12 | comment | added | Stefan Geschke | Some critics of my point of view say that this is just a social phenomenon: Basically every mathematician is educated to works within this framework. While there is certainly some point to this, I cannot fully agree. | |
| Nov 26, 2010 at 10:10 | comment | added | Stefan Geschke | @Deniz: Well, it is one foundation, and it is one that turned out to be extremely successful and one that was developed after quite a bit of struggle with other attempts. I agree that ZFC might actually be stronger than what is realistically needed to carry out most of mathematics. But I believe that the most striking argument for some theory like ZFC is that practically all mathematician work within the system without necessarily being familiar with it on a formal level. This indicates that ZFC is natural after all. (To be continued.) | |
| Nov 26, 2010 at 9:58 | comment | added | Deniz | @Stefan -- exactly! We suspend our disbelief for a bit, and try to 'simulate' set theory, and then construct numbers and so on in this make-believe universe. We already assumed numbers but that was "in the real world". Two ways to think about this: If we forego any notion of "meaning", then we are not constructing numbers, just manipulating strings to get other strings. This dissolves the question. But if we do imbue our axioms with the intended "meaning", it is unclear to me why assuming ZFC and building something that corresponds to our intuitive notion of numbers is the natural foundation | |
| Nov 26, 2010 at 9:45 | comment | added | Stefan Geschke | @Ketil: I agree that the circularity is by design, but I also don't see a way it could be avoided. You have to built on something in order to built a foundation of mathematics. There is one striking example where we seem to built something from nothing, and that is the von Neumann hierarchy of sets. Starting from the empty set, we construct all sets by just iterating the power sets operation. But in order to carry out this approach, we need to be able to define things using formulas. And hence we need formulas. And hence we need strings. And then we are back to the natural numbers. | |
| Nov 26, 2010 at 9:38 | history | edited | Stefan Geschke | CC BY-SA 2.5 |
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| Nov 26, 2010 at 9:02 | comment | added | Ketil Tveiten | In some sense, we formalise the things we want to study (sets, numbers, whatever), and that formalisation necessarily is built on the (observed) properties of the things we want to study. I.e. we observe that natural numbers have certain properties, and invent a formalisation (e.g. Peano arithmetic) that fits. So in this sense, the circularity is by design. | |
| Nov 26, 2010 at 8:37 | history | answered | Stefan Geschke | CC BY-SA 2.5 |