Timeline for weakening naive comprehension to avoid the paradoxes
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 29, 2012 at 23:36 | comment | added | Joel David Hamkins | I would say that it is more natural; whatever pleasant thoughts we might have had about the naive conception were largely abandoned once we realized it was contradictory. Why should we cling to a mistaken conception we know is wrong? | |
| Aug 29, 2012 at 23:31 | comment | added | Goldstern | You do not need the iterative concept to arrive at ZFC minus foundation. Zermelo formulated a version of ZC (ZFC minus replacement/foundation, admittedly with an unclear concept of "property", as "first order" was not available at the time) long before von Neumann defined the iterative hierarchy. | |
| Aug 29, 2012 at 22:57 | comment | added | user26062 | I don't mean it is perverse to adopt ZFC in the face of the paradoxes, just that the iterative conception is a <i>different</i> conception of set, one that is natural, but not <i>more</i> natural than the naive conception. That is how Boolos describes the iterative conception. The only thing wrong with the naive conception is that it's inconsistent. If we really only had the iterative conception, it wouldn't even be necessary to mention Russell's paradox. (I don't want to ague philosophy though. I was just curious about ways of avoiding the paradox by weakening comprehension.) | |
| Aug 29, 2012 at 22:03 | comment | added | Joel David Hamkins | I don't see that as perverse at all; it is exactly how I conceive of the intellectual development of the subject. When I teach set theory (as I am doing this semester), I start with the naive comprehension axiom and talk about its appeal; but then show it is actually contradictory because of the Russell theorem; then I explain that our set-theoretic intuitions didn't support it anyway (since we have an iterative conception of set); and then realize that what is left is separation and replacement. Thus, one arrives naturally at ZF from the (misguided) naive comprehension. | |
| Aug 29, 2012 at 21:46 | comment | added | user26062 | It is true that the ZFC axioms other than foundation and choice are instances of comprehension in a more straightforward way than the way in which every sentence in the language of set theory is, and I should have noticed that and pointed it out. But it is perverse to think of ZF-foundation as a weakening of naive set theory when in fact it is, as you say, an expression of the iterative conception of set, which is different conception of set. You cannot naturally arrive at ZF-foundation by removing problematic instances of comprehension. | |
| Aug 29, 2012 at 21:29 | comment | added | Ed Dean | Just to be clear, I wasn't suggesting that ZFC is in any way lacking for motivation, or that it doesn't originate from a weakening of naive comprehension, and I agree wholeheartedly that the lines you quote from the OP are mistaken as written. I meant no more and no less in my comment than that part of ZFC (Choice and Foundation, please excuse the mention of Infinity) isn't axiomatized as it seems the OP intends, and so I thought ZF-Foundation (rather than ZFC) would speak more directly to what concerns the OP. | |
| Aug 29, 2012 at 20:24 | comment | added | Joel David Hamkins | The power set, union and pairing axioms seem to be instances of naive comprehension, as well as the infinity axiom, since it asserts the existence of { n ∣ n is a finite ordinal}. So this means that ZF-Foundation is obtainable as a weakening of naive comprehension. The standard discussion of foundation shows that every model of ZF-Foundation has a model of ZF as its well-founded part and so we arrive at ZF in this way (and even at ZFC by going to the inner models). | |
| Aug 29, 2012 at 20:11 | comment | added | Ed Dean | That was my initial reaction upon reading the question as well, but I think the point that kimtown is only after theories all of whose axioms (besides extensionality) are instances of comprehension. So ZFC in its entirety doesn't make the cut, and one would have to do without foundation, infinity and choice. (Though I suppose infinity could be recovered as part of the theory while respecting kimtown's criterion by adding as axioms the finitely many instances of stratified comprehension which Specker used in his proof that NF refutes choice.) | |
| Aug 29, 2012 at 19:56 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Improved exposition
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| Aug 29, 2012 at 19:34 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |