A gap in Cantor's diagonal argument ?
in [Math]
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From: netzweltler on 7 Aug 2010 14:18 > > > > Infinity is called a concept, > > By whom? > > >not a number, so infinity is not part of the set of > real numbers. > > There are any number of meanings of the word > "infinity"; while it is > true that it is not an element of the set of "real > numbers", there are > many things which are not elements of the set of > "real numbers" and > yet are numbers (e.g., nonreal complex numbers; > surreal numbers; > extended real numbers, etc). > > >What if 0.333... is also just a concept? > > Please define "concept", then define "just a > concept". Without > definitions, you aren't doing math. > > > Any terminating decimal can only approach infinity > > What do you mean by "approach infinity"? > > > as it can only approach 0.333... or pi. > > Please define "approach 0.333..." and "approach pi". > > > My idea is, that we might erroneously place > terminating and non-terminating decimals in the same > set. > > Depends entirely on what you want to do. Note that > according to the > axiom of unions and some other axioms of set theory, > given any two > sets A and B, there always is a set whose elements > are exactly those > things that are elements of A or elements of B. > > >Is it possible, that this means mixing up > terminating decimals and concepts in the same set? > > There is certainly a lot of mixing up going here, but > I think it is > mostly happening in your head. Try to give *precise* > definitions of > the concepts before philosophizing more; you will > find that it > clarifies things immensely, while the approach you > are taking > currently can only obscure them. > > -- > Arturo Magidin > Try to give *precise* definitions of the concepts > before philosophizing more; I will work on that. It might take some time. You said: > There are any number of meanings of the word > "infinity"; Is it also valid to say: There are any number of meanings of the word "number"? Reinhard
From: Arturo Magidin on 7 Aug 2010 18:30 On Aug 7, 5:18 pm, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > > > Infinity is called a concept, > > > By whom? > > > >not a number, so infinity is not part of the set of > > real numbers. > > > There are any number of meanings of the word > > "infinity"; while it is > > true that it is not an element of the set of "real > > numbers", there are > > many things which are not elements of the set of > > "real numbers" and > > yet are numbers (e.g., nonreal complex numbers; > > surreal numbers; > > extended real numbers, etc). > > > >What if 0.333... is also just a concept? > > > Please define "concept", then define "just a > > concept". Without > > definitions, you aren't doing math. > > > > Any terminating decimal can only approach infinity > > > What do you mean by "approach infinity"? > > > > as it can only approach 0.333... or pi. > > > Please define "approach 0.333..." and "approach pi". > > > > My idea is, that we might erroneously place > > terminating and non-terminating decimals in the same > > set. > > > Depends entirely on what you want to do. Note that > > according to the > > axiom of unions and some other axioms of set theory, > > given any two > > sets A and B, there always is a set whose elements > > are exactly those > > things that are elements of A or elements of B. > > > >Is it possible, that this means mixing up > > terminating decimals and concepts in the same set? > > > There is certainly a lot of mixing up going here, but > > I think it is > > mostly happening in your head. Try to give *precise* > > definitions of > > the concepts before philosophizing more; you will > > find that it > > clarifies things immensely, while the approach you > > are taking > > currently can only obscure them. > > > -- > > Arturo Magidin > > Try to give *precise* definitions of the concepts > > before philosophizing more; > > I will work on that. It might take some time. > > You said: > > > There are any number of meanings of the word > > "infinity"; > > Is it also valid to say: > There are any number of meanings of the word "number"? Indeed; "natural number", "rational number", "real number", "complex number", "cardinal number", "surreal number", "nonstandard real number" are just over half a dozen that come to mind in a few nanoseconds. -- Arturo Magidin
From: netzweltler on 7 Aug 2010 15:39 > On Aug 7, 5:18 pm, netzweltler > <reinhard_fisc...(a)arcor.de> wrote: > > > > Infinity is called a concept, > > > > > By whom? > > > > > >not a number, so infinity is not part of the set > of > > > real numbers. > > > > > There are any number of meanings of the word > > > "infinity"; while it is > > > true that it is not an element of the set of > "real > > > numbers", there are > > > many things which are not elements of the set of > > > "real numbers" and > > > yet are numbers (e.g., nonreal complex numbers; > > > surreal numbers; > > > extended real numbers, etc). > > > > > >What if 0.333... is also just a concept? > > > > > Please define "concept", then define "just a > > > concept". Without > > > definitions, you aren't doing math. > > > > > > Any terminating decimal can only approach > infinity > > > > > What do you mean by "approach infinity"? > > > > > > as it can only approach 0.333... or pi. > > > > > Please define "approach 0.333..." and "approach > pi". > > > > > > My idea is, that we might erroneously place > > > terminating and non-terminating decimals in the > same > > > set. > > > > > Depends entirely on what you want to do. Note > that > > > according to the > > > axiom of unions and some other axioms of set > theory, > > > given any two > > > sets A and B, there always is a set whose > elements > > > are exactly those > > > things that are elements of A or elements of B. > > > > > >Is it possible, that this means mixing up > > > terminating decimals and concepts in the same > set? > > > > > There is certainly a lot of mixing up going here, > but > > > I think it is > > > mostly happening in your head. Try to give > *precise* > > > definitions of > > > the concepts before philosophizing more; you will > > > find that it > > > clarifies things immensely, while the approach > you > > > are taking > > > currently can only obscure them. > > > > > -- > > > Arturo Magidin > > > Try to give *precise* definitions of the concepts > > > before philosophizing more; > > > > I will work on that. It might take some time. > > > > You said: > > > > > There are any number of meanings of the word > > > "infinity"; > > > > Is it also valid to say: > > There are any number of meanings of the word > > "number"? > > Indeed; "natural number", "rational number", "real > number", "complex > number", "cardinal number", "surreal number", > "nonstandard real > number" are just over half a dozen that come to mind > in a few > nanoseconds. What is common to all these different meanings of the word number? Reinhard
From: Arturo Magidin on 7 Aug 2010 20:05 On Aug 7, 6:39 pm, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > > > Is it also valid to say: > > > There are any number of meanings of the word > > > "number"? > > > Indeed; "natural number", "rational number", "real > > number", "complex > > number", "cardinal number", "surreal number", > > "nonstandard real > > number" are just over half a dozen that come to mind > > in a few > > nanoseconds. > > What is common to all these different meanings of the word number? They all refer to specific concepts, defined in a precise, explicit, and careful manner? They all refer to concepts called "number"? They are all made up by human beings? I have no idea what you are driving at, but your question is hopelessly vague, much like most of your musings in this thread so far. -- Arturo Magidin
From: netzweltler on 8 Aug 2010 00:16
> On Aug 7, 6:39 pm, netzweltler > <reinhard_fisc...(a)arcor.de> wrote: > > > > > Is it also valid to say: > > > > There are any number of meanings of the word > > > > "number"? > > > > > Indeed; "natural number", "rational number", > > > "real > > > number", "complex > > > number", "cardinal number", "surreal number", > > > "nonstandard real > > > number" are just over half a dozen that come to > > > mind > > > in a few > > > nanoseconds. > > > > What is common to all these different meanings of > > the word number? > > They all refer to specific concepts, defined in a > precise, explicit, > and careful manner? They all refer to concepts called > "number"? They > are all made up by human beings? > > I have no idea what you are driving at, but your > question is > hopelessly vague, much like most of your musings in > this thread so > far. A number is a mathematical object used in counting and measuring. That´s what you can read in Wikipedia. Is it valid to measure the distance between New York and Paris and to define, this is a number now? No. All you can say is, you can measure the distance between New York and Paris and your result is a terminating decimal. So, why is pi called a number? All you can say is, there is an algorithm to determine pi as precisely as you need it. And again, your result is a terminating decimal. This terminating decimal is the number, not pi (as a non-terminating decimal). By the way, why is > > What is common to all these different meanings of > > the word number? a vague question? Is it allowed to define any concept and call it a number? There must be something, they all have in common, like "they are all used in counting and measuring". Reinhard |