A gap in Cantor's diagonal argument ?
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From: Reinhard Fischer on 5 Aug 2010 22:24 For a better understanding of "countable/uncountable": Which of the following statements is true/false? 1. Terminating decimals are always countable, no matter how many decimal places they consist of. 2. Decimals are uncountable, if they consist of infinitely many decimal places. 3. Numbers are countable, if they consist of a finite number of decimal places. 4. Numbers are uncountable, if they consist of an infinite number of decimal places. 5. Natural numbers are countable, if they consist of a finite number of digits. 6. Natural numbers are countable, even if they consist of an infinite number of digits.
From: Virgil on 6 Aug 2010 02:47 In article <1283897896.65519.1281075907785.JavaMail.root(a)gallium.mathforum.org>, Reinhard Fischer <reinhard_fischer(a)arcor.de> wrote: > For a better understanding of "countable/uncountable": > > Which of the following statements is true/false? > > 1. > Terminating decimals are always countable, no matter how many decimal places > they consist of. > 2. > Decimals are uncountable, if they consist of infinitely many decimal places. > 3. > Numbers are countable, if they consist of a finite number of decimal places. > 4. > Numbers are uncountable, if they consist of an infinite number of decimal > places. > 5. > Natural numbers are countable, if they consist of a finite number of digits. > 6. > Natural numbers are countable, even if they consist of an infinite number of > digits. Do you mean to ask about the countability of SETS of all such numbers? As if you do not, they are each neither true nor false, but mere nonsense.
From: Reinhard Fischer on 6 Aug 2010 01:15 I ask about the countability of SETS of all such numbers. Thanks for correcting me.
From: A N Niel on 6 Aug 2010 06:20 In article <1283897896.65519.1281075907785.JavaMail.root(a)gallium.mathforum.org>, Reinhard Fischer <reinhard_fischer(a)arcor.de> wrote: > For a better understanding of "countable/uncountable": > > Which of the following statements is true/false? > > 1. > Terminating decimals are always countable, no matter how many decimal places > they consist of. > 2. > Decimals are uncountable, if they consist of infinitely many decimal places. > 3. > Numbers are countable, if they consist of a finite number of decimal places. > 4. > Numbers are uncountable, if they consist of an infinite number of decimal > places. > 5. > Natural numbers are countable, if they consist of a finite number of digits. > 6. > Natural numbers are countable, even if they consist of an infinite number of digits. for 6 ... there is no natural number with infinitely many digits, so that set is empty, thus countable. the others (stated for "the set of all...") are true
From: jbriggs444 on 6 Aug 2010 08:38
On Aug 6, 2:24 am, Reinhard Fischer <reinhard_fisc...(a)arcor.de> wrote: > For a better understanding of "countable/uncountable": > > Which of the following statements is true/false? Hard to say -- they're so poorly phrased. > > 1. > Terminating decimals are always countable, no matter how many decimal places they consist of. A terminating decimal is a terminating decimal, not a set. So countability does not apply. The set of all real numbers with a terminating decimal expansion is countable. As are all subsets of that set. > 2. > Decimals are uncountable, if they consist of infinitely many decimal places. A decimal is a numeral denoting a real number, not a set. So countability does not apply. Some sets (of real numbers) which contain numbers with unterminating expansions are countable and some are not. The set of all real numbers with unterminating expansions is uncountable. > 3. > Numbers are countable, if they consist of a finite number of decimal places. A number is a number, not a set. So countability does not apply. There are constructions under which we identify individual real numbers with sets. e.g. As equivalence classes of Cauchy sequences of rational numbers or as Dedekind cuts of rational numbers. However, those constructions are not canonical. Countability in this sense is not an essential property of any particular real number. As above, the set of numbers whose decimal expansions have [non-zero] digits at a finite number of decimal places is countable. > 4. > Numbers are uncountable, if they consist of an infinite number of decimal places. Again, an individual number is neither countable nor uncountable, irrespective of any argument some dolt may make about its supposed canonical representation as a set. Representations are a dime a dozen. > 5. > Natural numbers are countable, if they consist of a finite number of digits. Natural numbers are numbers, not sets. They are neither countable nor uncountable. All of the standard constructions for the natural numbers as sets share the property that the set corresponding to any given natural number is countable. So I'll defer to the "dolts" on this one. All natural numbers are countable. [Though I'd be willing to argue the opposing viewpoint with equal vigor] All natural numbers have finite decimal expansions. All sets of natural numbers are countable. The set of all natural numbers is countable. > 6. > Natural numbers are countable, even if they consist of an infinite number of digits. Every natural number whose decimal expansion consists of an infinite number of digits is indeed countable. And uncountable. And appears in the tattoo on the rump of every invisible pink unicorn. And is written in letters of fire on your own forehead. |