Hilberts hotel

In pure math there is no time, only order. An infinite set is complete when it is defined even if its members are defined by the infinite iterations of some function. One may argue whether this makes mathematical infinities inapplicable to describing reality, but in the purely mathematical realm of this thought experiment those considerations do not apply.

To see how the infinite set of persons in the hotel may not include all persons, suppose that the rooms are numbered with the infinite set of positive non-zero integers [1,2,3,...] and that all the rooms are rented except room 1. So the set of rented rooms is [2,3,4,...]. Since the set of rented rooms is likewise infinite there must be an infinite set of corresponding guests. It makes no difference whether the cardinality of [1,2,3,...] is greater than that of [2,3,4,...] (it isn't), there must be the same number of guests as there are rented rooms. Just as the set of rented rooms is infinite and yet does not include 1 room, likewise, the set of persons who are guests is also infinite but may not include 1 person.


JR

north,

OK, this starts on pg 29 in my copy.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"> ... 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.

LINE 1 &gt;&gt; 3/7= 0.428571 : 428571 : 428571 : 428571 : 4...

LINE 2 &gt;&gt; = 0.(428571)

What does this segment mean?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If you do the division of 3 by 7 in longhand out to 21 or 28 decimal places you will see that the quotient comprises six digits that repeat over and over. They are: 4 2 8 5 7 1.

(So the arithmetic fraction 3/7 converts to decimal as 0.428571428571428571428571 ... )

In LINE 1 (I added some lables to help here) he uses a vertical line (which you represented here with a colon) to visually separate the groups of "repeating digits".

In LINE 2 he places the six repeating digits within parentheses. This is a math shorthand notation that means "the digits within repeat over and over to infinity".

More later ...

LB

<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 />In pure math there is no time, only order. An infinite set is complete when it is defined even if its members are defined by the infinite iterations of some function. One may argue whether this makes mathematical infinities inapplicable to describing reality, but in the purely mathematical realm of this thought experiment those considerations do not apply.

To see how the infinite set of persons in the hotel may not include all persons, suppose that the rooms are numbered with the infinite set of positive non-zero integers [1,2,3,...] and that all the rooms are rented except room 1. So the set of rented rooms is [2,3,4,...]. Since the set of rented rooms is likewise infinite there must be an infinite set of corresponding guests. It makes no difference whether the cardinality of [1,2,3,...] is greater than that of [2,3,4,...] (it isn't), there must be the same number of guests as there are rented rooms. Just as the set of rented rooms is infinite and yet does not include 1 room, likewise, the set of persons who are guests is also infinite but may not include 1 person.


JR
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If a completed infinity (if there is such an animal) does not include all possible elements then one can use the same argument Cantor used to prove the real number set uncountable to prove that the set of the naturals, represented by the rooms, is uncountably infinite with respect to the subset of the naturals, represented by the infinite persons in the room already since one can always generate a room number that the infinite number of persons had not occupied.

<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 />north,

OK, this starts on pg 29 in my copy.

... 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.

LINE 1 &gt;&gt; 3/7= 0.428571 : 428571 : 428571 : 428571 : 4...

LINE 2 &gt;&gt; = 0.(428571)

What does this segment mean?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

If you do the division of 3 by 7 in longhand out to 21 or 28 decimal places you will see that the quotient comprises six digits that repeat over and over. They are: 4 2 8 5 7 1.

(So the arithmetic fraction 3/7 converts to decimal as 0.428571428571428571428571 ... )

In LINE 1 (I added some lables to help here) he uses a vertical line (which you represented here with a colon) to visually separate the groups of "repeating digits".

In LINE 2 he places the six repeating digits within parentheses. This is a math shorthand notation that means "the digits within repeat over and over to infinity".

More later ...

LB
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Larry

thanks,it is apparent i'm ignorant of much, i do though, appreciate your time and effort! i tried to put as much down as i could to save you time, although i could go further,i relised one step at a time.


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jrich

still trying grasp what you were trying to get across,sort of there,maybe sort of!! thanks also for your time and effort!


north

123,

As I said previously, I'm undecided about Cantor's ideas about infinities beyond aleph-0, which would include his diagonal proof. But neither do I accept the idea you are proposing of the incomplete infinity. An incomplete infinity is really no infinity at all, so it seems to me your argument isn't that the hotel is full or that there is noone who is not a guest, your argument is really that the premise of the question is impossible.

Most people believe that the axioms of mathematics should reflect as closely as possible our intuitive understanding of the natural world. These people may generally be divided into at least two camps (the names escape me). One camp believes that infinities are too counter-intuitive and that this requires a rejection of the infinities. They take your view that Zeno is resolved by accepting a finite reality and that a math devoid of infinities is sufficient to describe it. The opposite camp believes that Zeno is resolved by the idea of the continuum. They have embraced the infinities and attempted to provide the axiomatic foundation for them. They have not been completely successful in that the axioms have been proven to be insufficient to prove what the cardinality of the continuum is. I think that in the MM, the cardinality of the continuum is aleph-0, but most others believe it must be at least aleph-1. If anyone can be more definitive on this, please correct me.


JR

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[north]
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?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Do you understand Cantor's rule for comparing two infinities? This is in the section "How to count infinities", and starts on page 25 in my book.

(He begins with the Hottentot and their "truncated number system" (he doesn't use this phrase), showing how a clever Hottentot could overcome this limitation to know which of two groups of objects is largest if both groups have more than three items. The basic idea is to lay an object from one group next to an object from the other group. Then repeat until you run out of objects in one group. If at that point the other group still has objects in it, the Hottentot knows that this group must be larger. He may not be able to say how much larger, or even how large each group is. But he does know that one group is larger, and he knows which group that is.

Note - the two groups could also turn out to equal in count. In this case, when he runs out of objects in the first group, he has no leftovers in the second group.

The rule is an extension of this idea.)

===

Using this rule he shows that several infinite number groups (including all integers and all ordinary arithmetic fractions) have the same number of numbers.

Let me know if we need to spend more time on this.

LB