Talk:Hartogs number

Invalid (or at least unclear) "Proof"[edit]

The proof given appears to be invalid or at least so unclear as to be unconvincing. JRSpriggs 06:14, 26 June 2007 (UTC)[reply]

I'm working on it. You could have tagged it with {{expert}}.... — Arthur Rubin | (talk) 18:53, 1 December 2007 (UTC)[reply]

Can we assume that, in ZF, if X is a set, then X × X is a set. The proof I'm familiar with in is NBG, where that's one of the finite set of axioms encompassing comprehension. — Arthur Rubin | (talk) 19:03, 1 December 2007 (UTC)[reply]

See Kripke–Platek set theory#Proof that Cartesian products exist for a proof that Cartesian products exist. Or use the axiom of powerset#Consequences and the axiom schema of specification. JRSpriggs 22:11, 1 December 2007 (UTC)[reply]

And thanks for reworking the proof, it is much clearer now. JRSpriggs 22:27, 1 December 2007 (UTC)[reply]

Greater than vs. not less than or equal[edit]

Forgive my ignorance, but this sentence confuses me:

If X cannot be wellordered, then we can no longer say that this α is the least wellordered cardinal greater than the cardinality of X, but it remains the least wellordered cardinal not less than or equal to the cardinality of X.

What is the difference between "greater than" and "not less than or equal to"? Solemnavalanche (talk) 19:52, 23 November 2008 (UTC)[reply]

See the article on cardinality. If A is a set of cardinality κ and B is a set of cardinality μ, then

\kappa \nleqslant \mu \,

means that there is no injection from A to B while

\kappa >\mu \,

means that there is an injection from B to A in addition to the absence of an injection from A to B. Does that clarify it? JRSpriggs (talk) 21:29, 23 November 2008 (UTC)[reply]

Thanks for the prompt response. It does clarify it somewhat, but it raises a further question. Does this mean that, absent the axiom of choice, there could exist pairs of sets for which no injection exists in either direction? That is, we could have

\kappa \nleqslant \mu \land \mu \nleqslant \kappa \,

? That clashes with my intuition, so I want to be sure I'm understanding you (and these articles) correctly. I see that the Law of Trichotomy article appears to say the same thing; but I'm having a difficult time imagining what such sets A and B would look like. Is there an article that addresses that question? Solemnavalanche (talk) 04:04, 24 November 2008 (UTC)[reply]

Well, it's good that it conflicts with your intuition, because the axiom of choice is intuitively correct.

But anyway, a good example is this: Let one of your sets be R, the set of all real numbers, and the other one be the Hartogs number of R. Then by definition there's no injection from the second to the first, and if there were an injection from the first to the second, that would imply that the reals could be wellordered.

Actually this generalizes, and shows that trichotomy implies the axiom of choice. --Trovatore (talk) 04:41, 24 November 2008 (UTC)[reply]

I think I'm starting to understand a bit more. So there exists an injection from R to its Hartogs number iff there exists a choice function on R. And so this must be related to the fact that ZF + GCH -> AC, is that right? Solemnavalanche (talk) 06:04, 24 November 2008 (UTC)[reply]

Clarity issues[edit]

The definition " $\alpha =\{\beta \in {\textrm {Ord}}\mid \exists i:\beta \hookrightarrow X\}$ " introduces several pieces of unexplained and unlinked notation.

Is $Ord$ the category of preordered sets or the ordinal numbers? The latter makes little sense because the lead is talking exclusively about cardinal numbers.
Is the hooked arrow an embedding or an inclusion map?
And what's with this variable $i$ ? $i$ is usually an index, right? It's never referred to anywhere else, so why can't it just be deleted? " $\alpha =\{\beta \in {\textrm {Ord}}\mid \beta \hookrightarrow X\}$ ". Oh, right, there's a colon. It's not a variable, it's a function named $i$ . Why not call it $f$ ?
Not to mention set-builder notation in the first place, although that's hopefully well-known enough.

It's not stated anywhere that $X\subseteq {\textrm {Ord}}$ , so I'm assuming that's false. That implies that $\hookrightarrow$ is not the natural injection $\iota$ , which opens the question of decidability: can it be determined if such a function exists? 71.41.210.146 (talk) 05:41, 21 January 2017 (UTC)[reply]

"Ord" refers to the class of ordinal numbers. If the axiom of choice holds, then the cardinal numbers can be regarded as a subclass of the ordinal numbers, namely the initial ordinals.
The hooked arrow means that "i" refers to an injective function.
Yes, "X" refers to an arbitrary set. Indeed, there would be little point in defining this (Hartog's number), if X were a set of ordinals. JRSpriggs (talk) 18:22, 21 January 2017 (UTC)[reply]

@JRSpriggs: Thanks! Article clarified, I hope. Now to just untangle the fact that the lead talks about cardinals while the proof is about ordinals. Perhaps I should just import (paraphrase) the explanation at http://planetmath.org/HartogsNumber, which is nicely clear. 71.41.210.146 (talk) 20:26, 21 January 2017 (UTC)[reply]

Thank you for clarifying the article. I am glad that I could help you. JRSpriggs (talk) 18:52, 22 January 2017 (UTC)[reply]

Hartogs['][s] function[edit]

Re the late exchange between User:JRSpriggs and User:Cherkash: I decided to go looking on Google Scholar to see how the terminology is actually used. I agree with Cherkash that the general style on WP is to keep the final s. On the other hand, if the term Hartogs' function without the s is the common usage, then we should use it. Or, if attested, we might be able to cut the Gordian knot by writing Hartogs function, using Hartogs as a noun adjunct rather than a possessive.

As it turns out, though, I was not able to verify (with any of the search terms) that this terminology is used for this at all. It seems to be used instead for some quite different notion. To quote this paper, "[a]n upper semi-continuous function h in a Stein space X is called a Hartogs function in X if the Hartogs domain $\{(z,x)\in C\times X;|z|<e^{-h(x)}\}$ is Runge–Stein in C×X." I don't know what all of that means, exactly, but it does not appear to be about ordinal numbers.

So maybe we should just remove the contested text altogether? --Trovatore (talk) 06:27, 21 December 2017 (UTC)[reply]

Going by the grammar argument alone, I agree with the dichotomy presented: we are effectively left to decide whether to use the noun adjunct form (aka noun's attributive form), or a possessive (with " 's " added). Either one is a grammatically acceptable way to use here, with the choice towards one or the other usually guided by tradition and commonality of use. In the first case, the spelling is "Hartogs function", in the second it's "Hartogs's function". As for the actual subject, if there's any doubt about the statement presented, then please by all means go ahead and either tag it with the "citation needed" tag (pending reliable sources) – or just go ahead and remove it altogether. cherkash (talk) 06:50, 21 December 2017 (UTC)[reply]

Well, if it appears standardly in the literature as Hartogs' function, then that's what we should say; I don't care what the general WP standard is in that case. However I don't see any evidence so far that that's actually the case. --Trovatore (talk) 06:54, 21 December 2017 (UTC)[reply]