Cantor Confusion
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From: Tony Orlow on 11 Oct 2006 11:58 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> [...] >> >>> There is no largest natural! There is a finite set of >>> arbitrarily large naturals. The size of the numbers is unbounded. >>> >> I can only conclude you have knocked youself out. > > Try the following gedanken-experiment to become accustomed with it: > a) How many different natural numbers can you store using a maximum of > 100 bits? > b) What is the largest natural number you can store with a maximum of > 100 bits? > > Regards, WM > > Answer to a) less than 100. > Answer to b) unknown, depends on representation. > No, if they are binary digits, or bits, then there are 2^100 unique strings possible with 100 digits.
From: Tony Orlow on 11 Oct 2006 12:00 Dik T. Winter wrote: > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > David Marcus schrieb: > ... > > > I don't follow. How do you know that the procedure that you gave > > > actually "defines/constructs" a natural number d? It seems that you keep > > > adding more and more digits to the number that you are constructing. > > > > What is the difference to the diagonal argument by Cantor? > > That a (to the right after a decimal point) infinite string of decimal > digits defines a real number, but that a (to the left) infinite string > of decimal digits does not define a natural number. It defines something. What do you call that? If the value up to and including every digit is finite, how can the string represetn anything but a finite value? Tony
From: William Hughes on 11 Oct 2006 12:05 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > [...] > > > > > There is no largest natural! There is a finite set of > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > I can only conclude you have knocked youself out. > > Try the following gedanken-experiment to become accustomed with it: > a) How many different natural numbers can you store using a maximum of > 100 bits? > b) What is the largest natural number you can store with a maximum of > 100 bits? > c) Once you have decided on a representation, what is the largest number you can store with a maximum of 100 bits? You are confusing "unknown" with "arbitrary". A natural number X may have unknown size. It cannot have arbitrary size. if you pick an even natural number you end up with an even natural number. If you pick an arbitrarially large natural number, you do not end up with an arbitrarially large natural number. - William Hughes
From: Randy Poe on 11 Oct 2006 12:15 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > > David Marcus schrieb: > > ... > > > > I don't follow. How do you know that the procedure that you gave > > > > actually "defines/constructs" a natural number d? It seems that you keep > > > > adding more and more digits to the number that you are constructing. > > > > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > It defines something. But not necessarily a number. > What do you call that? If the value up to and > including every digit is finite, how can the string represetn anything > but a finite value? Because representations of finite values end, and the string doesn't end, so it breaks the rules of "strings that represent finite values". - Randy
From: William Hughes on 11 Oct 2006 12:37
Tony Orlow wrote: > Dik T. Winter wrote: > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > > David Marcus schrieb: > > ... > > > > I don't follow. How do you know that the procedure that you gave > > > > actually "defines/constructs" a natural number d? It seems that you keep > > > > adding more and more digits to the number that you are constructing. > > > > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > It defines something. What do you call that? If the value up to and > including every digit is finite, how can the string represetn anything > but a finite value? > Because there are two types or strings. Strings that end and strings that don't end. Only strings that end represent finite values. -William Hughes |