Question About Hilbert's Hotel

[thinkandmull]

Countable infinities don’t exist except in relation to themselves. Otherwise you run into the problems I’ve described on this thread:

1. if time is infinite, then there is no “now” because it is already included in the past
2. 1 2 3 4 5 do not correspond to 1 3 5
3. you can’t pull all the odd numbers back and make it line up with all the positive numbers. That geometrically absurd
4. If all the odd numbers equal all the positive numbers, than there is no such thing as a greater infinity, because you line a few units together and say “they both go to infinity thus they are infinite”.
5. Then there is no greater infinities because infinity would them mean “including EVERYTHING”

 

[hecd2]

That irrational numbers are numbers is debatable too

Dunning-Kruger par excellence.

 

[rossum]

I am not interested in new definitions

Neither the definition of the real numbers nor the definition of the rational numbers are new. Your confusing the two, and your defensiveness about your error, are indications that you do not know enough about this area of mathematics. Go away and learn more before you return. That way you will avoid making elementary errors.

I note that you have no refutation of Cantor’s proof that the rationals are countably infinite.

rossum

 

[rossum]

Dunning-Kruger par excellence.

I’m afraid so.

rossum

 

[thinkandmull]

post 221 debunks Cantor’s “proof”

 

[thinkandmull]

Countable infinities are only countable is relation to themselves

 

[rossum]

Countable infinities are only countable is relation to themselves

All countable infinities are countable in relation to all other countable infinities. All of them can be put into a one-to-one relation with each other. In particular, all of the countable infinities can be put into a one-to-one relationship with the natural numbers: 1, 2, 3, 4, …

That is how we know that they are countable infinities: they can be placed in a one-to-one relationship with the natural numbers.

rossum

 

[thinkandmull]

There is no one to one correspondence between 1 2 3 4 5 and 1 3 5. The latter, even to infinity, is forever a part of the former whole.

 

[rossum]

There is no one to one correspondence between 1 2 3 4 5 and 1 3 5.

Correct.

The latter, even to infinity, is forever a part of the former whole.

But it has the same number of elements. Every natural number has a corresponding odd number; every odd number has a corresponding natural number. Your understanding of mathematics is lacking here.

Infinity has a lot of strange properties that do not conform to common sense. Common sense does not deal well with infinity because we do not meet infinity in ordinary life.

rossum

 

[Usagi]

There is no one to one correspondence between 1 2 3 4 5 and 1 3 5. The latter, even to infinity, is forever a part of the former whole.

Yes there is such a correspondence. We have shown it to you several times.

Permit me to use the even numbers rather than the odds, as the relation is simpler.

What is an even number? It is a number that, when divided by two, leaves no remainder. That means that every even number is twice one of the natural numbers. Two is twice one, four is twice two, and so forth. By definition, doubling any given natural number yields a unique even number. Likewise, halving any even number yields a unique natural number. Never will you find a case where that does not happen, and never will a particular number be double or half of multiple distinct numbers. The relation is one-to-one and unique. The natural numbers and the even numbers can be put in pairs of n and 2n for as far as you care to count. You will never run into a natural number that you cannot double, nor an even number that you cannot halve.

Thus, even though the set of natural numbers contains every member of the set of even numbers plus more, we can show that the two sets line up one to one and therefore have identical cardinality.

The same demonstration works for the odd numbers, but you double and then subtract one (to pair up the natural numbers with the odds) or just subtract one (to pair up the evens with the odds). Yes, it’s extremely counter-intuitive to think that one infinite set combined with another infinite set yields a third infinite set that is provably the same size as the two subsets, but math is weird sometimes.

Usagi

 

[Michael19682]

No. Pi is *defined *as the ratio of circumfrence to the diameter of a perfect Euclidean circle. To estimate the value of pi, it is not measured but can be calculated to arbitrary precision using various series expansions. However, I don’t really know what point your trying to make.

I think this is the only piece remaining to revive on this thread
and I’m curious to know you answer:
What fraction or ratio of natural numbers most perfectly defines Pi?

 

[rossum]

I think this is the only piece remaining to revive on this thread
and I’m curious to know you answer:
What fraction or ratio of natural numbers most perfectly defines Pi?

Most perfectly? None. For any rational approximation there can always be found a better rational approximation that is closer to pi.

rossum

 

[Michael19682]

Most perfectly? None. For any rational approximation there can always be found a better rational approximation that is closer to pi.

rossum

Thank you! And that was the point I was trying to make about the Hotel’s inability by its own rules to “assume” or take in whole the perfection of Pi. hecd2 rightly noted that Pi is transcendent as you corroborate. Therefore, there is no one to one correspondence of true Pi, which cannot be ascertained with perfection. In the context of the hotel analogy, this proves randomness and in turn a transcendent God whose actions are not always predictable except to himself.

 

[rossum]

Thank you! And that was the point I was trying to make about the Hotel’s inability by its own rules to “assume” or take in whole the perfection of Pi. hecd2 rightly noted that Pi is transcendent as you corroborate. Therefore, there is no one to one correspondence of true Pi, which cannot be ascertained with perfection. In the context of the hotel analogy, this proves randomness and in turn a transcendent God whose actions are not always predictable except to himself.

Hilbert’s Hotel deals with countably infinite sets, and only with countably infinite sets. It is no great insight to see that there are sets that it cannot deal with: the uncountably infinite sets.

rossum

 

[Michael19682]

Hilbert’s Hotel deals with countably infinite sets, and only with countably infinite sets. It is no great insight to see that there are sets that it cannot deal with: the uncountably infinite sets.

rossum

We had** modified the Hotel’s business workings** to show that the irrational numbers are accommodated for by the Hotel if we allow the proprietor an operational function call to substitute in place of a plodding count by a single number or two. E.g. we** allow him** to say 22/7, and use that to predict the next number of guests in the series of guests. We also modified it with suites, to make some uncountable sets countable by mapping their separate cardinalities into sets.

1.0000, - 1.9999, cardinality into {n}
2.0000, - 2.9999, cardinality into {n(1)}
3.0000, - 3.9999, cardinality into {n(2)}

Thus, all sets are countable or have cardinality if we have a third dimension as infinite as the first two; or a dimension (n+1).By modifying the Hotel, we can accommodate any number of guests to any cardinality n(x), by giving the Hotel suites to accommodate the arrivals in groups. ALL except a transcendent set like Pi, which involve the random acts of a creator’s f(n(x)), which is not knowable at x in time.

You may reject this on the grounds that it is not the original Hotel example. But I thought it interesting in light of the fact that the Mr. Lane takes the Hotel to imply the universe’s beginning in time. I just don’t see his proof.

 

[Tomdstone]

We had** modified the Hotel’s business workings** to show that the irrational numbers are accommodated for by the Hotel if we allow the proprietor an operational function call to substitute in place of a plodding count by a single number or two. E.g. we** allow him** to say 22/7, and use that to predict the next number of guests in the series of guests. We also modified it with suites, to make some uncountable sets countable by mapping their separate cardinalities into sets.

1.0000, - 1.9999, cardinality into {n}
2.0000, - 2.9999, cardinality into {n(1)}
3.0000, - 3.9999, cardinality into {n(2)}

Thus, all sets are countable or have cardinality if we have a third dimension as infinite as the first two; or a dimension (n+1).By modifying the Hotel, we can accommodate any number of guests to any cardinality n(x), by giving the Hotel suites to accommodate the arrivals in groups. ALL except a transcendent set like Pi, which involve the random acts of a creator’s f(n(x)), which is not knowable at x in time.

You may reject this on the grounds that it is not the original Hotel example. But I thought it interesting in light of the fact that the Mr. Lane takes the Hotel to imply the universe’s beginning in time. I just don’t see his proof.

It is not true that all sets are countable.

 

[Michael19682]

It is not true that all sets are countable.

"some" uncountable infinite sets can be made countable, by adding dimensions to the map set.

 

[rossum]

We had** modified the Hotel’s business workings** …

Then it is no longer Hilbert’s Hotel, but something else.

We also modified it with suites, to make some uncountable sets countable by mapping their separate cardinalities into sets.

1.0000, - 1.9999, cardinality into {n}
2.0000, - 2.9999, cardinality into {n(1)}
3.0000, - 3.9999, cardinality into {n(2)}

You do realise that all your divisions here are already uncountably infinite? In effect you are assuming (incorrectly) what you want to prove. Between any two real numbers there are an uncountably infinite number of other reals (and a countably infinite number of rationals).

You may reject this on the grounds that it is not the original Hotel example.

Correct. The properties of Michael’s Hotel are not the same as the properties of Hilbert’s Hotel. They are different hotels.

rossum

 

[thinkandmull]

Yes there is such a correspondence. We have shown it to you several times.

Permit me to use the even numbers rather than the odds, as the relation is simpler.

What is an even number? It is a number that, when divided by two, leaves no remainder. That means that every even number is twice one of the natural numbers. Two is twice one, four is twice two, and so forth. By definition, doubling any given natural number yields a unique even number. Likewise, halving any even number yields a unique natural number. Never will you find a case where that does not happen, and never will a particular number be double or half of multiple distinct numbers. The relation is one-to-one and unique. The natural numbers and the even numbers can be put in pairs of n and 2n for as far as you care to count. You will never run into a natural number that you cannot double, nor an even number that you cannot halve.

Thus, even though the set of natural numbers contains every member of the set of even numbers plus more, we can show that the two sets line up one to one and therefore have identical cardinality.

The same demonstration works for the odd numbers, but you double and then subtract one (to pair up the natural numbers with the odds) or just subtract one (to pair up the evens with the odds). Yes, it’s extremely counter-intuitive to think that one infinite set combined with another infinite set yields a third infinite set that is provably the same size as the two subsets, but math is weird sometimes.

Usagi

All all of you supporters of Cantor have done is claim that all the even numbers equal all the odd plus even numbers. You haven’t provided a proof. You make a few correspondences, say they both go to infinity, then say the two infinities are therefore equal. I’ve answered your arguments already. This is not about complex math, but about your unwillingness to face up to the issue.

Line up three unreal numbers (like the square root of -2) with 1 2 3. One to one correspondence… Then say the magic words “go to infinity!” and there you have it: all numbers equal all positive natural numbers! You’ve excused me of not being mathematical, but my arguments, like this one, have not been addressed

 

[yppop]

Line up three unreal numbers (like the square root of -2) with 1 2 3. One to one correspondence… Then say the magic words “go to infinity!” and there you have it: all numbers equal all positive natural numbers! You’ve excused me of not being mathematical, but my arguments, like this one, have not been addressed

Hi Think,
Your comments on this thread prove one thing: you don’t know mathematics. The square root of 2 is a “real” number, and that makes square root of (-2) a real number and real numbers are not countable. (Given your deficiency in math knowledge, I give you the benefit of doubt and assume that by unreal number you mean “imaginary” number.)

The fact of the matter is that any countable set of numbers, meaning that they can be put in some well established order, are countable and have the same cardinality as the integers, namely aleph (0), the countable infinite. Thus we can count by even numbers (2n), odd numbers (2n-1), by tens (10n), hundreds (100n), or even the prime numbers. They all have the same cardinality as the integers.

What we are arguing about is the nature of counting and the nature of infinity not odd and even numbers. I realize this can be mind boggling. When Cantor proved that there are the same number of points in a unit square or a unit cube as there are points on a unit length of the real number line, he wrote to his friend R. Dedekind, “I see it, but I don’t believe it.” Eventually he did believe it.

However, there are still those that refuse to believe that the earth is not flat; that the earth goes around the sun; that there is no Santa Claus; that the moon is not made of green cheese; that there is such thing as immaterial, spiritual substance; and that the infinity of the odd numbers equals the infinity of the integers. Fortunately those in that crowd, but not all, are very young.

Yppop