Are distinctions in definitions of "finite" material in, eg, topology or measure theory?

Question

There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set:

(Richard Dedekind) Every one-to-one function from S into itself is onto.

(Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.

And there are more.

Do these distinctions matter when considering definitions of fundamental categories like topological spaces or measure spaces? For example,

In topology: Why do we require a topological space to be closed under finite intersection?

In measure theory: "A measure is continuous from above if [given measurable sets and closed under intersection] at least one set has finite measure," alternatively, in the definition of sigma-finite measures (same article).

Do the various definitions of finiteness lead to non-isomorphic or non-equivalent categories?

My question is not specifically about topology or measure theory, but these are basic definitions introduced at undergrad level, so I thought, better to understand the context via basic examples.

@NKS I believe there are (at least) 16 different definitions of 'finite set' and no two are equivalent. — Ittay Weiss, Jan 23 '13 at 04:59
It should be pointed out that all these definitions are equivalent under the axiom of (countable) choice. — Asaf Karagila, Jan 23 '13 at 07:47
@AsafKaragila, although I find Choice interesting, I am primarily interested in constructive mathematics. Green & Tao in their arithmetic progressions paper write that they carefully avoid Choice, like a plague of some sort. — alancalvitti, Jan 23 '13 at 21:45
alan, I wrote this comment to point to the general public why these definitions are not the same. I might post an answer of my own later. I am still contemplating this. — Asaf Karagila, Jan 23 '13 at 21:49
As far as I recall AD proves countable choice, so all Dedekind-finite sets are finite, and so all are equivalent. I should also point out that philosophically ZF+AD is a much harsher assumption than ZFC. I also don't really recall 16 definitions of finite. I recall about seven, but it's easy to generate more. — Asaf Karagila, Jan 23 '13 at 23:50

score 2 · Accepted Answer · answered Jan 23 '13 at 04:58

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The various non-equivalent notions of finiteness you allude to refer to definitions of 'finite set'. The concepts of finiteness you mention that are relevant to analysis (in general) refer to a number being finite. It is common-place in analysis to assume the standard models for Peano Arithmetic and the real numbers. Both of these are categorical models (that means that any two models are isomorphic). Then, all the elements in a model of PA are finite and all the elements in $\mathbb {R}$ are finite as well. It is common to consider the extended real numbers $\mathbb {R} \cup \{\pm \infty\}$. Then $\pm \infty$ are by convention infinite numbers.

So, there is little (if any) relation between the different definitions of 'finite set' and the notion if measure theory of a set having finite measure or in topology for taking finite intersections. Finite in the former means a real number (and not $\pm \infty$) while in the latter it means 'perform intersection a finite number of times, counted using some natural number'.

answered Jan 23 '13 at 04:58

Ittay Weiss

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I've seen this categorical defintion of Dedekind finite, no reference to sets: "An object A in a category C is Dedekind finite if all monos $j:A \to A$ are isomorphisms" http://mathoverflow.net/questions/66096/a-dedekind-pseudo-finite-set - correct? – alancalvitti Jan 23 '13 at 05:08
yes, that's a possibility. Still, this makes no difference for finiteness issues of natural numbers or real numbers. – Ittay Weiss Jan 23 '13 at 05:15
But what about for the spaces at issue, which are not numbers? Unions and intersection can be indexed by arbitrary functions right? – alancalvitti Jan 23 '13 at 05:20
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I'm not an expert on finiteness but as far as I know this has no implications to analysis. You are talking about performing a certain binary operation a finite number of times. Finite in this case means counted by some natural number, and these are categorical so there is only one meaning to that. – Ittay Weiss Jan 23 '13 at 05:25
Not necessarily to analysis, but anywhere in math that "finite" is mentioned, whether in algebra, geometry etc. Eg, Hickman's "Dedekind Finite Fields" Bull Aus M Soc 1978. Dedekind-finite rings: http://math.stackexchange.com/questions/33491/why-are-dedekind-finite-rings-called-so... – alancalvitti Jan 23 '13 at 05:31
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as far as I know in the presence of AC all these different notions of finite are equivalent. It is likely that in intuitionistic mathematics such issues as you describe are important. At least in standard analysis AC is accepted. – Ittay Weiss Jan 23 '13 at 05:42
It's all fun & games until you want to make a universal statement, such as Furstenberg's 1955 proof of the infinitude of primes via the topology on the integers generated by arithmetic progressions, where finite unions come into play. Imagine this is true with AC but false or unprovable without Choice and alternate definition of finite. That would be lame as opposed to universal. – alancalvitti Jan 23 '13 at 06:02
let us continue this discussion in chat – Ittay Weiss Jan 23 '13 at 06:05
I only saw your chat invite today. – alancalvitti Jan 23 '13 at 21:43

Are distinctions in definitions of "finite" material in, eg, topology or measure theory?

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