Galileo's Paradox
in [Math]
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From: Bob Kolker on 29 Nov 2006 14:28 MoeBlee wrote: > > > 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. I stand corrected. However I thought the number two was the class of all sets equinumerous with {a,b} a not = b. Bob Kolker
From: MoeBlee on 29 Nov 2006 14:33 Bob Kolker wrote: > However I thought the number two was the class of all sets equinumerous > with {a,b} a not = b. Frege thought so too. But, as you know, his system is inconsistent. MoeBlee
From: Bob Kolker on 29 Nov 2006 14:34 MoeBlee 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 Here is what I got from the wikipedia article on cardinality: Main article: Cardinal number Note that, up until this point, we have defined the term "cardinality" in a strictly functional role: We have not actually defined the "cardinality" of a set as a specified object itself. We now outline such an approach. The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A. There are then two main approaches to the definition of "cardinality of a set": 1. The cardinality of a set A is defined as its equivalence class under equinumerosity. 2. A particular class of representatives of the equivalence classes is specified. The most common choice is the Von Neumann cardinal assignment. This is usually taken as the definition of cardinal number in axiomatic set theory. Cardinality of set S is denoted | S | . Cardinality of its power set is denoted 2 | S | . Cardinalities of the infinite sets are denoted \aleph_0 < \aleph_1 < \aleph_2 < ... (for each ordinal α, \aleph_{\alpha+1} is the first cardinality greater than \aleph_\alpha). The cardinality of the natural numbers is denoted aleph-null ({\aleph_0}), while the cardinality of the real numbers is denoted \mathbf{c}. It can be shown that \mathbf{c} = 2^{\aleph_0} > {\aleph_0}. (see: Cantor's diagonal argument). The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, and so \mathbf{c} = \aleph_1. [edit] Do you see anything wrong with this definition? If so, what is wrong? Bob Kolker
From: Bob Kolker on 29 Nov 2006 14:38 MoeBlee wrote: > Bob Kolker wrote: > >>However I thought the number two was the class of all sets equinumerous >>with {a,b} a not = b. > > > Frege thought so too. But, as you know, his system is inconsistent. 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. 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, but it does work in type theory and in New Foundations and related systems. 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). -------------------------------------------------------------------------------- Do you see a problem with this? If so, what is it? Bob Kolker
From: MoeBlee on 29 Nov 2006 14:50
Bob Kolker quoted Wikipedia: > The relation of having the same cardinality is called equinumerosity, I would have said: The relation of equinumerosity is called 'having the same cardinality'. The article seems to have reversed the order of definitions. 'equinumerous' usually comes before 'same cardinality'. > and this is an equivalence relation on the class of all sets. The > equivalence class of a set A under this relation then consists of all > those sets which have the same cardinality as A. There are then two main > approaches to the definition of "cardinality of a set": > > 1. The cardinality of a set A is defined as its equivalence class > under equinumerosity. > 2. A particular class of representatives of the equivalence classes > is specified. The most common choice is the Von Neumann cardinal > assignment. This is usually taken as the definition of cardinal number > in axiomatic set theory. 1. is okay as long as we have a consistent theory that proves the existence of such as class as a SET (and if it's only a proper class, then we need to figure out how to make it still work as a cardinal number). 2., the von Neumann definition, is the usual way. And there are also variations on 2., such as Scott's method. MoeBlee |