Galileo's Paradox
in [Math]
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From: MoeBlee on 29 Nov 2006 15:00 Bob Kolker wrote: > I also got this from the wiki. > > Formal definition > > Formally, assuming the axiom of choice, the cardinality of a set X is > the least ordinal α such that there is a bijection between X and α. Right, that's the one I mentioned. (Actually, usually both the axiom of choice and the axiom schema of replacement are used for this; though, we could compromise by adopting the numeration theorem as an axiom, which entails the axiom of choice, but does not require the axiom schema of replacement since the numeration theorem is itself taken as an axiom.) Scott's method, meanwhile, uses the axiom schema of replacement and the axiom of regularity and does not require the axiom of choice. > This > definition is known as the von Neumann cardinal assignment. If the axiom > of choice is not assumed we need to do something different. The oldest > definition of the cardinality of a set X (implicit in Cantor and > explicit in Frege and Principia Mathematica) is as the set of all sets > which are equinumerous with X: this does not work in ZFC or other > related systems of axiomatic set theory because this collection is too > large to be a set, Right, as I mentioned. > but it does work in type theory and in New > Foundations and related systems. Okay, but you and I and most people don't use NF or PM for working purposes. I do not dispute that one can devise a consistent theory that has the equivalence class method of defining cardinality. My only point is that this method does not work, as Wikipedia mentions, in usual set theory. The examples of PM and NF are of type and stratification theories. > However, if we restrict from this class > to those equinumerous with X that have the least rank, then it will work > (this is a trick due to Dana Scott: it works because the collection of > objects with any given rank is a set). Right. It uses the axiom of regularity. MoeBlee
From: Virgil on 29 Nov 2006 16:06 In article <4t64brF1127iuU2(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> wrote: > Lester Zick wrote: > > > > Just out of curiosity, Bob, why is cardinality in set theory not a > > measure? I mean if you ask "how much gas" and get the answer "two > > gallons" you've certainly measured the gas. Or if you ask "how much > > space" and get the answer "two inches" you've certainly measured the > > space. It seems to me that you can obviously superimpose cardinality > > on questions like "how much" without having to count or match things. > > Consider two measure sets, disjoint. The measure of the union is the sum > of the measures of each. > > Now consider two sets of the same cardinality, disjoint. The cardinality > of the union equals the cardinality of either. This only holds for sets whose cardinality is either 0 or not finite. > > In short cardinality does not add like measure. > > Bob Kolker
From: Virgil on 29 Nov 2006 16:10 In article <1164828291.381109.186960(a)n67g2000cwd.googlegroups.com>, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > Bob Kolker wrote: > > MoeBlee wrote: > > > > > Bob Kolker wrote: > > > > > >>The cardinal number of a set is the equivalence class of sets > > >>with the same cardinality as the the given set. > > > > > > > > > In what theory is this? > > > > Standard set theory. > > Wrong. > > The usual definition is that the cardinality of a set (or the cardinal > number of a set) is the least ordinal equinumerous with the set. That > is not the class of sets with the same cardinality as the given set. In > Z set theories, the class of sets having the same cardinality as the > given set is not even itself a set, let alone a cardinal number. > > MoeBlee There are set theories in which equivalence classes need not be proper classes or cause other problems, and the equivalence class definition is commonly used in them. In ZF or NBG, the least ordinal definition works nicely.
From: MoeBlee on 29 Nov 2006 16:16 Virgil wrote: > In article <1164828291.381109.186960(a)n67g2000cwd.googlegroups.com>, > "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > > Bob Kolker wrote: > > > MoeBlee wrote: > > > > > > > Bob Kolker wrote: > > > > > > > >>The cardinal number of a set is the equivalence class of sets > > > >>with the same cardinality as the the given set. > > > > > > > > > > > > In what theory is this? > > > > > > Standard set theory. > > > > Wrong. > > > > The usual definition is that the cardinality of a set (or the cardinal > > number of a set) is the least ordinal equinumerous with the set. That > > is not the class of sets with the same cardinality as the given set. In > > Z set theories, the class of sets having the same cardinality as the > > given set is not even itself a set, let alone a cardinal number. > > > > MoeBlee > > There are set theories in which equivalence classes need not be proper > classes or cause other problems, and the equivalence class definition is > commonly used in them. Other than type theories such as PM or a stratification theory such as NF, what is an example of a set theory in which the class of all sets equinumerous with a given nonempty set is a set? (I don't deny that such a thing is possible; I'm just curious as to an example.) > In ZF or NBG, the least ordinal definition works nicely. Right, and they don't use the class of all sets equinumerous with a given nonempty set to serve as the cardinality of the given set. MoeBlee
From: Bob Kolker on 29 Nov 2006 16:53
Virgil wrote: > > > This only holds for sets whose cardinality is either 0 or not finite. Precisely. However you can take the measure of the union of two disjoint infinite (in cardinality) sets each with finite measure the the measure of the union is the sum of the measures of the constituent set. In short, measure is additive in the algebraic sense and cardiality (for infinite sets) is not. That is one of the reasons that how much and how many are different questions. Bob Kolker |