Talk:Finite set

Subsets of finite sets

The concepts of finiteness should be compatible with the notion of cardinality in two ways: (1) any set which can be mapped bijectively onto a ?-finite should also be ?-finite; and (2) any subset of a ?-finite set should also be ?-finite. There does not seem to be any problem with the first condition. But it is not clear to me that the second condition is met by: V-finite, VI-finite, or VII-finite. Can you show that it is?

There is a generalization of Ia-finite which might be of interest. Imagine forming the disjoint union of three Ia-finite sets. Then one can show that any partition of that union into four parts would result in at least one of the parts being I-finite. This could be generalized as follows

• S is Ib-finite iff there exists a I-finite set A such that for any function f from S to A there would be an element a∈A such that { x∈S | f (x) = a } is I-finite.

This should lie between Ia-finite and II-finite. JRSpriggs (talk) 10:19, 17 August 2014 (UTC)

It sounds like you've been doing some original research on this. I think that original research is a good idea, but I understand that in wikipedia, we are supposed to fairly represent the consensus of the literature. As I said above, what I added comes from "the standard literature in peer-reviewed books and journals, like by Tarski, Mostowski, Lindenbaum, Howard, Rubin, Jech, Moore and many others." If you have some improvements on their publications, you probably should get them peer-reviewed in a journal first before adding them to wikipedia. The 8 different finiteness concept definitions are as I have seen them in all of those esteemed, peer-reviewed authors. I should mention also that I do much original research of my own, and I have many ideas that are opposite to what appears in wikipedia, and which I am certain are opposite to your own beliefs. However, I do not add them to wikipedia because of the original research policy. (My ideas will appear in my own book, where they belong.) If you disagree with some of the finiteness definitions, I think you need to provide peer-reviewed references, which I have done with the 8 finiteness concepts.
--Alan U. Kennington (talk) 10:29, 17 August 2014 (UTC)

So you are writing a book. Will it be your first book? Who will be the publisher? What topics will you cover? Is it related to getting a degree? JRSpriggs (talk) 04:00, 18 August 2014 (UTC)

I shouldn't exploit wikipedia talk-pages to advertise my own book, although it's free at the moment in draft form (currently 1322 pages), and probably always will be free in electronic form. I'll just say something which could possibly be of some interest to readers of this talk-page though. My first book has four parts: I. Foundations. II. Algebra. III. Analysis/topology. IV. Differentiable manifolds (i.e. geometry). The biggest part is on foundations (logic, set theory, numbers). (I'm thinking of self-publishing this part as a separate book.) My motivation for this part arose from the discovery that very large numbers of core concepts in differential geometry rest very heavily on controversial foundational issues, like the axiom of choice. (At least I claim that there are still controversial issues in the foundations after 130 years of research by some of the best minds in mathematics.) Therefore I have read and written a lot about these issues in a book that is supposedly primarily focused on the definitions of differential geometry. (It's strongly focused on definitions as opposed to theorems.)

The reason why this potentially could have some relevance in this forum is that the foundational issues are not purely academic. Virtually all of mathematical analysis rests very heavily upon foundational assumptions, like the assumption that Dedekind-finite sets are finite, or rather that infinite sets always contain an infinite sequence of distinct elements. I am trying to reconstruct differential geometry and the underlying analysis/topology/measure-integration theory without the axiom of choice. This may seem quixotic and foolhardy. But that's what I'm doing. (E.g. linear algebra looks very different in AC, and most functional analysis textbooks would contain only 10 pages if they did not assume AC.) I've said too much already. If you look up my name plus "differential geometry" in a search engine, something will show up. It has nothing to do with a course that I'm studying or teaching. It's a personal crusade! I only mentioned it because I wanted to emphasize that although I disagree strongly with a lot of what I read in the wikipedia logic and set theory pages, I just have to hold my tongue (or keyboard) because this is not a forum for original research. (I can't even add my personal researches regarding the Tawny frogmouth, even though I know that that page has factual errors.) Thanks for asking....
--Alan U. Kennington (talk) 04:59, 18 August 2014 (UTC)

Ah, I see you have a link to your book on your user page User:Alan U. Kennington. I should have looked there first. JRSpriggs (talk) 06:13, 18 August 2014 (UTC)

I was going to ask whether my comments on finite sets would be considered reliable, per WP:SPS, as I am a recognized expert on the axiom of choice and its negation, and I do have a couple of peer-reviewed papers on Dedekind finite (IV-finite) "cardinals". Alan does not appear to be. In any case, the following characterization of Ia-, Ib-, and II- finite may be of interest.

A set S is Ia-finite if any partition of S into I-infinite sets has at most 1 element.

A set S is Ib-finite if any partition of S into I-infinite sets is I-finite, and bounded.

A set S is Ic-finite if any partition of S into I-infinite sets is I-finite.

A set S is II-finite if no partition of S has an I-infinite linearly-ordered subset [without a largest element]. (I believe I can show that any I-infinite linearly-ordered set has a I-infinite linearly-ordered partition without largest element, which is sufficient to remove the bracketed text.

A set S is III-finite if no partition of S [into IV-infinite sets] is IV-infinite.

I consider Ic more interesting than Ib, but it's not obvious that Ic-finite implies II-finite. — Arthur Rubin (talk) 08:39, 20 August 2014 (UTC)

To Arthur: Regarding your note, why would not $F(x)=\{y\vert y\in X\land y\approx x\}$ work, since F:X→P(X) ? JRSpriggs (talk) 09:53, 20 August 2014 (UTC)

You're right. I was confusing ZFU with NBGU (or KMU). What I meant to say is that there isn't a "class function" F (in NBGU) (or property φ in ZFU where $(\forall x)(\exists !y)\varphi (x,y,p_{1},p_{2},\ldots ,p_{n})$, and we make the convention that $\varphi (x,F(x),p_{1},p_{2},\ldots ,p_{n})$) such that $(\forall x,y)(x\approx y\leftrightarrow F(x)=F(y)).$ — Arthur Rubin (talk) 14:12, 20 August 2014 (UTC)

Notes

Other types of finiteness with respect to ω

In the section "Other types of finiteness", it gives different definitions of "finiteness" for ZF without choice.
It specifically mentions that all these defintions are pure set-theoretic definitions that don't explicitly involve ordinals.
But in ZF without choice, it is still possible to construct ω.
Thus we can actually get two new definitions of finiteness:

• S is A-finite iff there exists an injection from S to ω but no bijection from S to ω. (i.e. the |S|<|ω| definition)
• S is B-finite iff there does not exist an injection from ω to S. (i.e. the ¬|S|≥|ω| definition)

I'm just wondering where these definitions fit on the list of the "other types of finiteness". Are they equivalent to any of the definitions there? If not, where do they each fit in the hierarchy? --AndreRD (talk) 09:41, 18 June 2019 (UTC)

See Talk:Finite set#Is it really true that I-finite does not imply T-finite in ZF? above.

I-finite at Finite set#Other concepts of finiteness is the same as Tarski's definition (see #6 at Finite set#Necessary and sufficient conditions for finiteness). Your A-finite is the same as finite (see #1). So I-finite and A-finite are equivalent.

Your B-finite is the same as Dedekind finite which is IV-finite. JRSpriggs (talk) 05:11, 19 June 2019 (UTC)

Problems with the claims about Kuratowski finite

It seems to me that the there are several problems with the claims made about Kuratowski-finite. First, the notion of an object having "all properties" of such-and-such a kind is not generally expressible in ZF set theory, in light of Tarski's theorem on the non-definability of truth. That is, the definition of Kuratowski finite seems explicitly to use a truth predicate, which is not available in ZF. But secondly, since it refers to provability, the notion would seem to trivialize in a model of ZF + ¬Con(ZF), since everything is thought to be provable in such a model. In such a model, there can be no finite sets at all, not even the empty set or singletons. JoelDavid (talk) 18:27, 7 July 2024 (UTC)