Hilberts hotel

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Larry Burford</i>
<br />jrich,

My tongue is bleeding. You up for some company?

LB
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I hope you didn't bite through it when you read my first response.


JR

north,

I think you were on the right track. I hope my glaring mistake didn't throw you off.


JR

north,

[north] "I'm stuck from pg.19 to pg.24. I just cannot understand why he's doing what he does (combining two numbers? decimal) what on earth is he doing!! I have read this over and over to no avail, I'm frustrated."


Are you refering to, for instance, the example of the grains of wheat on the chess board?

LB

jrich,

No, it was not you.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by jrich</i>
<br />Woops! I made a mistake. With infinite sets, a set and a proper subset of that set can have the same cardinality. Therefore, the equivalence of the cardinalities of the sets of guests and possible guests cannot be used to show the sets are equivalent. So Larry you are correct after all. I don't know why I ever doubted you.


JR
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If we make this wrong assumption that an infinite set is completed (like Cantor did in his diagonal proof), the logical conclusion is to say that this set includes all persons so there are no persons left. OTOh, we could argue circularly like Cantor did that since a new guest can show up at the hotel even though there are infinite number of persons there already (and this new person must represent another natural number), the cardinality of this new natural set is larger than the infinite natural set of the persons in the hotel. Using the same argument as in his diagonal proof for reals, we get the contradiction that naturals and their infinite subsets have the same cardinalities (according to his one-to-one correspondence for
finite numbers), and also that they don't (since a new guest is generated that wasn't in the original rooms).

The more accurate way of representing the situation is to say that there can't be an infinite number of persons in the hotel already since there are no completed infinities.

<i>Originally posted by Larry Burford</i>
<br />north,

[north] "I'm stuck from pg.19 to pg.24. I just cannot understand why he's doing what he does (combining two numbers? decimal) what on earth is he doing!! I have read this over and over to no avail, I'm frustrated."


Are you refering to, for instance, the example of the grains of wheat on the chess board?

LB
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ANS:no,on pg.19 he talks about the difference between infinite decimal fractions and arithmetical fractions 3/7 or 8/277 then he convertes into decimal fractions e.g. 3/7= 0.4285571 : 428571 : 428571:428571:4...=0.(428571),what does this segment mean?

he goes on to say: WE have proved above that the number of all ordinary arithmetical fractions is the same as the number of all integers; so the number of all Periodic decimal fractions must also be the same as the number of all integers. what has he done here to prove this?

further he goes on to say "but the points on a line are not necessarily represented by Periodic decimal fractions, and in most cases we shall get the infinite fractions in which the decimal figures appear without any periodicity at all. and it is easy to show that in such case no linear arrangement is possible. what does this mean?

Suppose that somebody claims to have made such an arrangement,and looks like this:
N
1 O.38602563078....
2 0.57350762050....
3 0.99356753207....
4 0.25763200456....
5 0.00005320562....
6 0.99035638567....
7 0.55522730567....
8 0.05277365642....

he goes on to say,"of course,since it is impossible actually to write the infinity of numbers with the infinite number of decimals in each,the above claim means that the author of the table has some general rule(similar to one used by us for arrangement of ordinary fractions,what rule is this?) according to which he has constructsd the table , and this rule guarantees that every single decimal fraction one can think of will appear sooner or later in the table.

"Well,it is not at all difficult to show that any claim of that kind is unsound,since we can always write an infinite decimal fraction that is NOT contained in this infinite table.How can we do it? Oh,very simply.just write the fraction with the first decimal different from that of N1 in the table,second decimal different from that in N2,why is this possible?

the number he gets is this
3 7 3 6
not not not not
0. 5 2 7 4 etc. what is he doing?( in preview this part is not coming up right)

thanks larry, north