An uncountable countable set
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From: Han de Bruijn on 19 Oct 2006 03:28 MoeBlee wrote: > All cardinals are ordinals (while not all ordinals are cardinals). The > set of natural numbers is the set of finite ordinals which is the set > of finite cardinals. This is standard set theory. Ah! Therefore we can limit ourselves to the naturals and forget all the fuzz about ordinals and cardinals. Because the infinite counterparts of these beasties do not exist in _my_ universe anyway. Thank you, Moeblee, for this very clear and explicit summary. Han de Bruijn
From: Han de Bruijn on 19 Oct 2006 03:37 David Marcus wrote: > Han de Bruijn wrote: > >>David Marcus wrote: >> >>>Han de Bruijn wrote: >>> >>>>No matter how much textbooks about set theory I read, it all remains >>>>abacedabra for me. >>> >>>I just looked in two books on set theory on my shelf and both explicitly >>>state that the set in the axiom of infinity need not be the natural >>>numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set >>>Theory, An Introduction to Independence Proofs" by Kenneth Kunen. >> >>Hmm, I have that book by Halmos myself ... > > Then look at the discussion right after the axiom of infinity on page > 44. > >>But why are the finite ordinals not equivalent with the naturals (I mean >>in mainstream mathematics)? > > First you said that the axiom of infinity is the construction of the > ordinals. We said no. Then you said the axiom of infinity defines the > finite ordinals. We said no. Now you say the finite ordinals are the > natural numbers (actually you said "equivalent", but I'm not sure what > that means). Third time's the charm, i.e., you are correct. A standard > definition of the natural is as the finite ordinals. However, you seem > to think that all three of your statements are the same. I don't > understand this. The confusion stems from the fact that I cannot and shall not understand the _infinite_ counterparts of the finite cardinals and ordinals. >>>Which math textbooks have you read? Have you worked the problems in the >>>ones you've read? Taken any math courses at a university? >> >>I'm a (theoretical) physicist by education. > > Does that mean you have taken graduate courses in math (e.g., set > theory) or you haven't? Theoretical physics is mainly "advanced calculus". I've done some group theory as well. And lots of linear algebra. But only a little bit, i.e. the basics of set theory. Han de Bruijn
From: Virgil on 19 Oct 2006 03:37 In article <b653a$4537277c$82a1e228$24028(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > MoeBlee wrote: > > > Han de Bruijn wrote: > > > >>But why are the finite ordinals not equivalent with the naturals (I mean > >>in mainstream mathematics)? > > > > The set of finite ordinals IS the set of natural numbers. > > > > x is a natural number <-> x is a finite ordinal. > > > > Why don't you just read a textbook? > > It's all very confusing. Because there also "exist" infinite ordinals, > they say. The set of all finite ordinals is one of them.
From: Virgil on 19 Oct 2006 03:49 In article <deb1b$45372b26$82a1e228$27057(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David Marcus wrote: > > > Han de Bruijn wrote: > > > >>David Marcus wrote: > >> > >>>Han de Bruijn wrote: > >>> > >>>>No matter how much textbooks about set theory I read, it all remains > >>>>abacedabra for me. > >>> > >>>I just looked in two books on set theory on my shelf and both explicitly > >>>state that the set in the axiom of infinity need not be the natural > >>>numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set > >>>Theory, An Introduction to Independence Proofs" by Kenneth Kunen. > >> > >>Hmm, I have that book by Halmos myself ... > > > > Then look at the discussion right after the axiom of infinity on page > > 44. > > > >>But why are the finite ordinals not equivalent with the naturals (I mean > >>in mainstream mathematics)? > > > > First you said that the axiom of infinity is the construction of the > > ordinals. We said no. Then you said the axiom of infinity defines the > > finite ordinals. We said no. Now you say the finite ordinals are the > > natural numbers (actually you said "equivalent", but I'm not sure what > > that means). Third time's the charm, i.e., you are correct. A standard > > definition of the natural is as the finite ordinals. However, you seem > > to think that all three of your statements are the same. I don't > > understand this. > > The confusion stems from the fact that I cannot and shall not understand > the _infinite_ counterparts of the finite cardinals and ordinals. One who deliberately chooses not to understand puts himself beyond the pale. > > >>>Which math textbooks have you read? Have you worked the problems in the > >>>ones you've read? Taken any math courses at a university? > >> > >>I'm a (theoretical) physicist by education. > > > > Does that mean you have taken graduate courses in math (e.g., set > > theory) or you haven't? > > Theoretical physics is mainly "advanced calculus". I've done some group > theory as well. And lots of linear algebra. But only a little bit, i.e. > the basics of set theory. > > Han de Bruijn
From: MoeBlee on 19 Oct 2006 14:20
Han de Bruijn wrote: > MoeBlee wrote: > > > Han de Bruijn wrote: > > > >>But why are the finite ordinals not equivalent with the naturals (I mean > >>in mainstream mathematics)? > > > > The set of finite ordinals IS the set of natural numbers. > > > > x is a natural number <-> x is a finite ordinal. > > > > Why don't you just read a textbook? > > It's all very confusing. Because there also "exist" infinite ordinals, > they say. You won't read a textbook in set theory because it's confusing? So, you'd rather remain confused and completely ignorant about set theory, while spouting nonsense about it every day on the Internet? Anyway, what is confusing about the theorem that there exist finite ordinals and that there exist infinite ordinals? MoeBlee |