Timeline for Is the analysis as taught in universities in fact the analysis of definable numbers?
Current License: CC BY-SA 4.0
44 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31 at 11:15 | comment | added | The Amplitwist |
The link to jdh.hamkins.org in a previous comment seems to be broken, but a snapshot is saved on the Wayback Machine.
|
|
| Jul 14, 2022 at 5:18 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the dead link
|
| May 13, 2020 at 11:31 | comment | added | user2103480 | @Michael I suppose uniqueness is obtained by taking the least such ordinal, since we know that one such ordinal exists | |
| Mar 31, 2020 at 21:43 | comment | added | Michael | The argument re the ordinals doesn't seem correct? Having a α that isn't definable would only constitute a definition of α if α can be proved to be unique. But that doesn't seem to be demonstrated? | |
| Apr 30, 2018 at 19:18 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Removed assertion that Wikipedia is wrong, since they have corrected.
|
| Nov 24, 2017 at 1:51 | comment | added | Joel David Hamkins | The point is that this statement is expressed in the meta-theory, about the model in the realm where one has the truth predicate, not inside the model. So one cannot argue in ZFC, say, that there are only countably many definable objects. But any model of ZFC has only countably many definable objects, enumerated in a countable sequence in the meta-theoretic context where one has the truth predicate. The difference between these two perspectives is what my post is about. | |
| Nov 24, 2017 at 1:27 | comment | added | ziggurism | It sounds tautological when you say it like that | |
| Nov 24, 2017 at 0:32 | comment | added | Joel David Hamkins | @ziggurism Yes, as a theorem of model theory, a model of set theory can have only countably many definable objects. | |
| Nov 24, 2017 at 0:15 | comment | added | ziggurism | So just to clarify, as a metamathematical statement about an uncountable model of ZFC where the reals are uncountable, the "tea cup theorem" holds, right? | |
| May 17, 2015 at 10:50 | comment | added | Christopher King | The wikipedia article has at least taken notice that you found its mess up. | |
| May 6, 2015 at 13:17 | comment | added | Johannes Hahn | Wow! Thank you for this answer, it was very informative. I'm still a bit baffled that such an obvious thing can be wrong... Has this already been mentioned in the Common-misconceptions-list? | |
| Jan 10, 2015 at 14:18 | comment | added | Joel David Hamkins | I view SOL itself as an outside-the-universe perspective, since these definitions quantify over second order objects, which are not present, and so we may be basically in agreement. If you don't mean "true" second order logic, but rather the second-order versions of set theory such as GBC, then the pointwise definable model phenomenon recurs, and that is the main theorem of our paper: every countable model of GBC has a class forcing extension that is pointwise definable, one in which every set and class is definable without parameters. | |
| Jan 10, 2015 at 13:50 | comment | added | Charles Stewart | I think [t]he concept of definability...only makes sense from an outside-the-universe perspective [because of] Tarski's theorem on the non-definability of truth is not adequately argued for in this answer, since many people (such as myself) believe that, in fact, definitions in SOL can be perfectly adequate, it is only that SOL is not a proper logic as it lacks a semi-decidable proof theory. The claim that every definition must come equipped with a complete set of proof conditions is one that I dispute. Henkin showed that SOL, like PA, is a sound-but-incomplete multi-sorted FOL. | |
| May 8, 2014 at 9:27 | comment | added | goblin GONE | @JoelDavidHamkins, thanks for your encouragement. I left a (longish) comment, so hopefully you find it interesting. (or better yet, convincing!) | |
| May 7, 2014 at 14:05 | comment | added | Joel David Hamkins | @user18921 I'd suggest that you go ahead and make a blog post about it. It is also possible to make comments on my blog post about the paper at jdh.hamkins.org/pointwisedefinablemodelsofsettheory. | |
| May 7, 2014 at 12:22 | comment | added | goblin GONE | Joel I disagree with the conclusions of the linked article (Pointwise Definable Models of Set Theory) in regards to the math tea-party argument. In my opinion, the tea-party argument does withstand scrutiny, so long as we're using the "correct" definitions. Are you interested in discussing this possibility? | |
| Oct 14, 2013 at 12:05 | comment | added | Hans-Peter Stricker | Would a hint to Skolem's paradox have been helpful in this context? | |
| Aug 2, 2011 at 17:51 | comment | added | Anixx | @Joel David Hamkins, Okay lets take standard analysis. It follows that the number of reals is uncountable (inside this model) while the number of definable numbers is countable. How can we be confident the the analysis theorems that employ definablility actually true for all reals? | |
| Jul 26, 2011 at 4:49 | comment | added | Daniel Mehkeri | Hmmm, just read some of what was written in October about constructive set theory and describability. Actually, within constructive set theory, we can have that all sets are _sub_countable (see en.wikipedia.org/wiki/Subcountability), even from inside the theory. It makes sense to assume everything is finitely describable. Because, well, describe something that isn't! | |
| Jul 25, 2011 at 20:31 | comment | added | Joel David Hamkins | Yes, that's right. But this is not a contradiction, since the map from the definitions to the objects is not available inside the model. Thus, although the model actually has only countably many reals, as viewed from outside, inside the model it has no such bijection with $\mathbb{N}$ and thus thinks that its reals are uncountable. The main point is that the notion of a set being countable depends on the set-theoretic background, and not all models having a set in common necessarily agree on whether it is countable. | |
| Jul 25, 2011 at 20:01 | comment | added | Anixx | Well viewing from exclusively from INSIDE the model, the number of definitions is countable, but the number of objects is uncountable? | |
| Jul 25, 2011 at 12:48 | comment | added | Joel David Hamkins | The number of descriptions is countable, and there is a one-to-one function mapping definitions to the objects they define in a pointwise definable model (so all such models are countable), but this function is not one that exists inside the model. The situation is similar to that arising in Skolem's paradox (see en.wikipedia.org/wiki/Skolem_paradox), where a model is countable, but only from an outside-the-universe perspective. | |
| Jul 25, 2011 at 12:41 | comment | added | Anixx | "In these pointwise definable models, every object is uniquely specified as the unique object satisfying a certain property. The models are simply not able to assemble the definability function that maps each definition to the object it defines." But this means that the number of objects is at least smaller than the number of properties, even though there is no 1-to-1 function. As the number of properties is countable, does not it mean that the number of objects is contable? Or the number of properties itself is uncountable? | |
| Jul 25, 2011 at 12:31 | comment | added | Joel David Hamkins | Thus, the pointwise definable models of ZFC are in one-to-one correspondence with the completions of ZFC+V=HOD. | |
| Jul 25, 2011 at 12:30 | comment | added | Joel David Hamkins | Andrej, every pointwise definable model of ZFC must satisfy V=HOD, so not every model of ZFC will work in your plan. But if you take any model of ZFC+V=HOD, whether transitive or not, then there is a definable Skolem function, and this implies that the set of definable elements is an elementary substructure, which implies that this substructure is pointwise definable. Since this substructure is determined completely by the theory of the original model, I think that this amounts to the same as what you mean by the syntactic model of the Lindenbaum structure. (But you do need V=HOD.) | |
| Jul 25, 2011 at 11:03 | history | edited | gowers | CC BY-SA 3.0 |
corrected spelling of "led"
|
| Jul 25, 2011 at 10:25 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Added link to arxiv paper
|
| Oct 30, 2010 at 17:10 | comment | added | Andrés E. Caicedo | Oh, I agree. All the reasonable approaches to the still open question of the consistency of NF go through a syntactical route, which is less than ideal in a sense, but if it works, it is of course welcome. | |
| Oct 30, 2010 at 10:44 | comment | added | Andrej Bauer | @Andres: Interesting point, and I can see its validity. Nevertheless, should one prove something surprising using "artificial" constructs, it will become hard to ignore them on methodological grounds. A case in point might be non-standard models of arithmetic. I suspect many people wish they didn't exist, but they do. | |
| Oct 29, 2010 at 23:38 | comment | added | Andrés E. Caicedo | Andrej, one answer that I have heard from both Harvey Friedman and John Steel (in different contexts) is that we study set theory, not "models of set theory". As such we are interested in sets rather than "artificial" constructs like minimal models. (Of course, the study of models of set theory is in itself very interesting, and it is not always so easy to separate one from the other.) | |
| Oct 29, 2010 at 21:02 | comment | added | Andrej Bauer | Following the comment of Andreas got me thinking: as a topos theorist (which I am not) I would look at the syntactic model of ZFC (the "Lindenabaum algebra") in order to get to both the minimal model and the one in which every set is definable. But I suppose set theorists don't like that kind of model too much because they prefer transitive models that are "really made of sets". Is that so? Historically, where does this tendency come from? | |
| Oct 29, 2010 at 16:44 | comment | added | Andreas Blass | A minor technical comment on the first bullet point in Joel's answer: To use the minimal transitive model, one needs to assume that ZFC has well-founded models, not just that it's consistent. The main claim there, that there is a pointwise definable model of ZFC, is nevertheless correct on the basis of mere consistency, essentially by the second bullet point plus the consistency of V=HOD relative to ZFC. | |
| Oct 29, 2010 at 15:29 | comment | added | Anixx | A "definable-modified ZFC" theory should be somewhere in the middle, with classically(but not definably)-countable continuum. Since constructivism makes serious impact on analysis, it would interestiong to investigate what impact on analysis makes such "definable" theory. | |
| Oct 29, 2010 at 15:22 | comment | added | Anixx | @ Andrej Bauer regarding first comment, yes. This is also analogous to constructivism: you cannot express in a constructivcist set theory that something is not constructive. Not constructive means it does not exist in this theory. Regarding second comment, no, classical theory in in fact more general than constructivist theory. That's why not anything classically valid is constructively valid. | |
| Oct 29, 2010 at 15:15 | comment | added | Andrej Bauer | This is off-topic, but: it makes no sense to claim that "constructivist continuum is countable in ZFC sense". What might be the case is that there is a model of constructive mathematics in ZFC such that the continuum is interpreted by a countable set. Indeed, we can find such a model, but we can also find a model in which this is not the case. Moreover, any model of ZFC is a model of constructive set theory. You see, constructive mathematics is more general than classical mathematics, and so in particular anything that is constructively valid is also classically valid. | |
| Oct 29, 2010 at 15:13 | comment | added | Andrej Bauer | Joel made a very fine answer, please study it carefully. Joel states that there are models of ZFC such that every element of the model is definable. This does not mean that inside the model the statement "every element is definable" is valid. The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement. | |
| Oct 29, 2010 at 15:07 | comment | added | Anixx | So I suppose, by analogy, that the definable continuum of such modified theories as put forward by Joel while is not countable by terminology of such theories, is still perfectly countable in terms of standard ZFC. | |
| Oct 29, 2010 at 15:02 | comment | added | Anixx | @ Andrej Bauer He clearly says there can be models of ZFC where undefinable numbers do not exist. So I suppose they are either modifications of ZFC or ZFC with an additional axiom. Regarding constructivism, in constructivism "not countable" has different meaning than in ZFC. So the constructivist continuum is not countable in constructivist sense but perfectly countable in ZFC sense. | |
| Oct 29, 2010 at 14:50 | comment | added | Andrej Bauer | @Anixx: No, this is not what Joel was saying. He did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the notion "is definable in ZFC". And no, this has absolutely nothing to do with constructivism (also please note that even in constructivism uncountable means "not countable", whereas you stated that it means "no practical enumeration" whatever that might mean). | |
| Oct 29, 2010 at 14:40 | comment | added | Anixx | And also how these theories deal with axion of choice. Take for example the sequence formed as described on the question: with throwing dice. In conventional analysis such sequence will necessary have limit and that limit will be undefinable. But what if the theory postulates that there are no undefinable numbers? What explanation then to choose for such experiment? - 1. There cannot be such sequence because undefinable (undeterministic) functions are not permitted - 2. Undeterministic functions permitted but the numbers do not form a sequence - 3. The sequence does not converge? | |
| Oct 29, 2010 at 14:25 | comment | added | Anixx | Well, now I see what you said, Joel. As I understand you say it can be postulated in ZFC that undefinable numbers simply do not exist. And also such models use the term 'uncountable' in the constructivist meaning (uncountable=no practical way to enumerate). So, how an analysis built on top of such theories would differ from the conventional analysis based on a theory where the set of definable numbers has lower cardinality than that of continuum? | |
| Oct 29, 2010 at 13:59 | comment | added | arsmath | So would you say "For all you know, there are only a countable number of sets"? | |
| Oct 29, 2010 at 13:28 | vote | accept | Anixx | ||
| Oct 29, 2010 at 13:15 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |