Talk:Zermelo–Fraenkel set theory

message: User:46.109.150.165 has changed the formal statement of the union axiom from
:\forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A]
to
:\forall \mathcal{F}\, \exists A\, \forall x\, [x \in A \iff \exists Y\, (x \in Y \land Y \in \mathcal{F})].
Now, it appears to me that the new version is the one I probably would have thought of as the axiom of union.  It says \cup\mathcal{F} exists, whereas the old version says there is a set including \cup\mathcal{F}.  These are equivalent, given separation, so it doesn't make very much actual difference.
But the question is, which version is in the source we're using?  I can't remember whether we're using Kunen or Jech these days.  Is there someone else who's willing to sort this out, so I don't have to? --Trovatore (talk) 07:21, 23 January 2016 (UTC)
replies: text: For what it's worth, the article itself claims to use the axiom set from Kunen: "There are many equivalent formulations of the ZFC axioms... The following particular axiom set is from Kunen (1980)." — Tobias Bergemann (talk) 08:31, 23 January 2016 (UTC)
replies: text: Thanks, Tobias. --Trovatore (talk) 19:56, 23 January 2016 (UTC)

text: I am not an expert in this field, and am thus uncomfortable editing the page itself, but it does seem to me that regardless of the formulation used, the formal statement and English description should match (as much as it's possible for an informal statement to match a formal one).  The entry for the Axiom of Union starts with the English sentence: "The union over the elements of a set exists.", then gives an example, and proceeds to say, "Formally, for any set of sets  \mathcal{F} there is a set A containing every element that is a member of some member of  \mathcal{F}".  The former sentence corresponds to the formal statement with the biconditional, whereas the latter corresponds to the one with the conditional.  I realize that the two statements are equivalent given the other axioms, but it seems to me that the equivalence between formal statements and informal descriptions of individual axioms should be immediate, rather than dependent on other axioms.  In other words, I disagree with the claim that "it doesn't make very much actual difference".  While it makes little difference with respect to mathematical validity, it has the potential to make significant difference with respect to clarity and accessibility.  Even a trivial deduction like this one can be non-trivial to someone who's trying to understand a significant amount of new information, particularly one not used to making such formal inferences as a matter of course.  Therefore I would suggest that whichever formal statement is used, the English statement corresponding to the other should be either removed or edited to agree with it.  I don't know how important it is to stick to the formulation given by a particular mathematician, and I'll leave consideration of that question to the experts, but from my non-expert point of view, it seems preferable to go with the biconditional version (and edit the non-corresponding sentence to match it: something like "Formally, for any set of sets \mathcal{F} there is a set A containing exactly those elements that are members of some member of  \mathcal{F} ") simply because the entry is titled "The Axiom of Union", and that statement more closely captures the usual notion of set union.  Also, that's the version given in the main "Axiom of Union" article. Alternatively, the title of the entry could be changed, or, at the very least, a note could be added that clarifies why the two (informal) statements are equivalent. 207.165.235.61 (talk) 18:48, 21 July 2016 (UTC)

text: After looking at the rest of this talk page (I had previously skimmed it for references to the Axiom of Union specifically) I see that the same basic issue once existed with the Axiom of Power set.  The treatment of that axiom in the current article seems much clearer to me, and I think the same approach would work well here.  Since no one has disputed that version in the past few years, and the situation is basically identical (i.e. the formal statement from the source only asserts the existence of a superset of the more natural set, which can then be constructed by separation) I now feel confident enough to make the edit myself.207.165.235.61 (talk) 19:51, 21 July 2016 (UTC)
replies: text: At some point, User:CBM had checked that the axioms were identical to the published source.  I don't know whether that's really necessary, but it does short-circuit a lot of arguments.  Different formulations, though equivalent when taken all together, sometimes give you surprises when you start removing or weakening axioms, so there is some advantage to having a source that we can cite for the exact set of axioms. --Trovatore (talk) 20:11, 21 July 2016 (UTC)
replies: text: That makes sense.  I left the formal statement as is and just mentioned that while A needn't be \cup\mathcal{F}, the existence of \cup\mathcal{F} follows from separation.  My wording could conceivably be better, if you want to have a look, but I think it clarifies things without giving up either the source formulation or the more natural notion that \cup\mathcal{F} exists.207.165.235.61 (talk) 20:36, 21 July 2016 (UTC)

text: In *this* article, we use the exact axioms from Kunen's textbook, so that we have a matching set.  Perhaps you could explain how the axiom from Kunen's book implies the existence of a universal set? The axiom that you replaced it with implies Kunen's axiom in any case... — Carl (CBM · talk) 18:08, 18 November 2017 (UTC)
thread_title:  Recent changes to union axiom 

message: There's been an unfortunate slow-motion edit war recently over this text:
:Unlike von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. These ontological restrictions are required for ZFC to avoid Russell's paradox, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of set. 
I doubt that any of us are in disagreement over the actual state of affairs, so the real problem is to find adequate wording.  Can we agree on these points?
* ZFC considered literally, as a collection of purely syntactic strings in a formal language, does not have any symbols that correspond to proper classes.  That's just a fact.  If it needs a source, it won't be hard to find.
* On the other hand, there are quite standard uses of (definable, maybe with parameters) proper classes in reasoning that can be straightforwardly if perhaps tediously formalized in ZFC.  Mathematicians who make use of these may still be quite happy to think of their arguments as being "ZFC-arguments", whatever that means exactly.
If we agree on those, then we should be able to come to some agreement on wording.
There's another, broader point I'd like to bring up about that section.  The language in question is in a section that starts with "ZFC has been criticized both for being excessively strong and for being excessively weak", and the bit about proper classes seems to be relevant to arguing for a strengthening.  But proper classes really are not much of a strengthening of ZFC.  NBG is a conservative extension of ZFC, and Kelley–Morse is only very slightly stronger.  The strengthenings relevant to the research of the last half-century or so are more along the lines of large-cardinal axioms (though certainly not limited to them).  But large cardinals are not mentioned here at all, which I think is a fairly serious omission. --Trovatore (talk) 05:42, 28 January 2016 (UTC)
thread_title:  Proper classes 

message: For human-readers is better to see sets with upper-case labels, when possible. Example:
* \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow x = y]     is ugly
* \forall X \forall Y [\forall z (z \in X \Leftrightarrow z \in Y) \Rightarrow X = Y]     is better
The intention in an encyclopedic article is also to be didactic. Krauss (talk)
: In ZFC, elements of sets are also sets, so x, y, and z are all sets. In any case, we take the axioms directly from Kunen's book for consistency, so if everything is lowercase there it will also be lowercase here.  Have you checked what case Kunen uses? — Carl (CBM · talk) 10:39, 14 November 2017 (UTC)
replies: text: The convention is to use upperclass letters for second-order variables. — Charles Stewart (talk) 11:26, 14 November 2017 (UTC)
thread_title:  Please use upper-case for sets and lower for elements 

message: Right now, we use the axioms from Kunen (1980), with only a few typographical changes (e.g. we don't have bounded quantifiers, and we sometimes rename variables). I have just double checked this against the source. I wanted to explain the reasoning behind using the axioms from a single text in this article.  There are several reasons:
# For verifiability: if we simply allow editors to write axioms from memory, the resulting list becomes unverifiable. 
# For correctness. There are many variations of individual axioms that are harmless when the full set of axioms is taken into account, but which make a difference when only some of the other axioms are taken into account. By using a single set of axioms from one source, we ensure that there are no subtle issues with the way the particular axioms have been stated. 
# Reducing arguments. There are many ways to formulate the individual axioms, some of which are obviously correct and some of which are not. Rather than wasting time discussing/arguing these things, we can use a single set of axioms which we know to be correct, so that we can focus editor time on other things.
There is no particular reason that Kunen's book was chosen, apart from its role as a standard text and reference for set theory. Some other well respected text could also be used; the key point is that all the axioms come from the same place. — Carl (CBM · talk) 18:24, 18 November 2017 (UTC)
thread_title:  Kunen's axioms 

message: The reference in section virtual classes, 
Formalism and Beyond: On the Nature of Mathematical Discourse (Logos) Digital original Edition
by Godehard Link (Editor), is extemly broad. Since it is a collection of essays:
"The essays collected in this volume focus on the role of formalist aspects in mathematical theorizing and practice, examining issues such as infinity, finiteness, and proof procedures, as well as central historical figures in the field, including Frege, Russell, Hilbert and Wittgenstein. Using modern logico-philosophical tools and systematic conceptual and logical analyses, the volume provides a thorough, up-to-date account of the subject."
Maybe the reference could say which essay?
Jan Burse (talk) 02:41, 3 March 2019 (UTC)
thread_title:  Unspecifik Link 2014 Reference 

message: The articles for Dimitry Mirimanoff, the axiom of regularity, and John Von Neumann all assert that Von Neumann proposed the axiom of regularity not Mirimanoff, but the history section of this page says Mirimanoff proposed the axiom of regularity. Furthermore, Zermelo set theory includes the axiom of choice, so adding replacement and regularity should yield ZFC not ZF.
WingsOfEpsilon (talk) 17:14, 8 May 2019 (UTC)
replies: text: Please feel free to change the History section, if you have references to reliable sources to back you up. Notice that Wikipedia is not a reliable source. JRSpriggs (talk) 02:35, 9 May 2019 (UTC)
thread_title:  History section contradicts other pages 

message: "the cumulative hierarchy of sets introduced by John von Neumann.[8]"  But [8] refers to Shoenfield's text.  Should it not refer to a paper of von Neumann?  (Unfortunately I am unable to say which paper.) 31.49.9.229 (talk) 00:27, 30 December 2019 (UTC)
replies: text: This is just speculation, but if John Von Neumann's paper is not a suitable reference for some reason (in German rather than English, expression is too archaic to be understandable to most modern students, out of print, makes no claim of originality for John, is an unpublished letter or lecture, or whatever), then a suitable reference by Shoenfield attributing the idea to John would be preferable. JRSpriggs (talk) 08:42, 30 December 2019 (UTC)
thread_title:  Wrong reference? 

message: In line with the program of reverse mathematics, I want to derive ZF from a weaker set theory together with a definition of V. I thought of using Kripke–Platek set theory, but even that may have too much. Henri Poincaré criticized the axiom of union as being impredicative. And the axiom of collection (replacement) maybe too strong. Anyway, here is my current thinking:
* Axiom of extensionality: Two sets are the same if and only if they have the same elements.
* Axiom of empty set: There exists a set with no members, called the empty set and denoted {}.
* Axiom of adjunction: Given a set and an element, there is a set whose elements are the elements of the given set together with the given element.
* Axiom of Δ0-separation: Given any set and any Δ0 formula φ(x), there is a subset of the original set containing precisely those elements x for which φ(x) holds. (This is an axiom schema.)
* Axiom of infinity
* Axiom of Hartogs number: For every set there is an ordinal which cannot be mapped injectively into it.
* V:
** For every ordinal α, there is a unique set Vα such that  \forall x ( x \in V_\alpha \leftrightarrow \exist \beta < \alpha ( x \subseteq V_\beta ) ) .
** For every set s, there is an ordinal α such that s is an element of Vα.
OK? JRSpriggs (talk) 23:16, 5 October 2022 (UTC)
replies: text: Not the place to discuss it, but what you seem to be asking for is basically impossible.  Reasoning about the cumulative hierarchy in a formal theory will never give you more consistency strength than the theory you start with.  The cumulative hierarchy as a motivation is a different matter; that's done by informal argument, not in a formal theory. --Trovatore (talk) 00:30, 6 October 2022 (UTC)
replies: text: This is not as strong as ZF (let alone ZFC), but it is stronger than it would be if one left out the definition of V part. That part allows one to prove the axioms of union, regularity, and powerset. Pairing follows from empty set and adjunction, of course. JRSpriggs (talk) 14:48, 6 October 2022 (UTC)
replies: text: Add an axiom saying that Ord is Mahlo — for every closed unbounded class of ordinals C (definable by a formula with parameters), there is a regular ordinal in C. JRSpriggs (talk) 21:57, 8 October 2022 (UTC)
replies: text: Beginning with any ordinal α form a sequence as follows: take Vα, take the Hartogs number of that set, take the next regular ordinal after that, then repeat. The limits of this sequence will be a closed unbounded class of ordinals, so there will be a regular ordinal Κ among them (perhaps we could call this a semi-strong inaccessible cardinal). Then using Δ0-separation we can prove that the axioms of separation and replacement hold in VΚ as well as the other axioms mentioned above (except Mahlo). The only difficulty is showing that the image set provided by the axiom of replacement is bounded in VΚ. If it were unbounded, then by the regularity of Κ the ranks of its elements would have to form a set with order type Κ. This could then be mapped injectively backwards through the function to subsets of the domain of the function which lie in Vα+1 and that contradicts the fact that Κ is larger than the Hartogs number of that set by the construction of Κ. So we have a model of ZF without choice. JRSpriggs (talk) 13:57, 9 October 2022 (UTC)
thread_title:  Axiom system more attuned to the cumulative hierarchy ''V''