Question About Hilbert's Hotel

[thinkandmull]

I don’t believe Cantor, for all your praises, really had a proof for this because nobody has demonstrated what it is to the exclusion of my arguments.

-2 does not have a square root. My use of “real numbers” instead of imaginary numbers I got from my friend who is a math and computers major. But none of this quibbling over words have refuted any of my arguments I’ve posted on this thread. I have a cousin and a former math teacher who I am going to email about this since the posters on here don’t really seem willing to engage with me on this

 

[thinkandmull]

Here is what I emailed them:

I’ve been in a discussion recently with people on a forum about the idea that all odd numbers equal all even plus odd numbers. Math is interesting, but it starts to get tedious when going from one function or proposition to another again and again. Anyway, I think I know enough to be able to debate this issue. I don’t think there are “countable infinities”. I think that only makes sense of an infinities’s relation to itself. Otherwise I think we run into problems. For example, if time is infinite, then there is no “now” because it is already included in the past and you can’t add one to infinity. I think the basic philosophical proof is that the odd numbers are, even to infinity, forever a PART of a whole (the positive whole numbers). 1 2 3 4 5 do not correspond to 1 3 5. If you imagine the numbers going from you infinitely in one direction, you can’t pull all the odd numbers back and make it line up with all the positive numbers. That is geometrically absurd at least. If all the odd numbers equal all the positive whole numbers, than there is no such thing of one infinity being greater than another infinity, because you line a few units together and say “they then both go to infinity thus they are equally infinite”. If by infinite Cantor meant “including EVERYTHING absolutely”, than there certainly could not be greater and lesser infinities. If units of an infinite set is what makes one infinity greater than another, adding to one set makes it larger than the formally equal set. What do you think? Was Cantor wrong?

 

[rossum]

-2 does not have a square root.

Yes it does. That root is not a natural number, not a rational number, not a real number, but it does exist. It is a complex number. The complex numbers appear to be yet another area of mathematics where you lack sufficient knowledge.

The square root of -2 is: sqrt(2) i - 1.41421356 i, where i = sqrt(-1).

Does this work? Yes. The square of sqrt(2) is 2. The square of i is -1. 2 x -1 = -2. QED.

This is an imaginary number; the imaginary component contains i, sqrt(2)i in this case. Some more mathematics for you to learn, I’m afraid.

rossum

 

[thinkandmull]

BTW, something like a flat earth implies a huge conspiracy. My position on this implies a wide spread false assumption in one aspect of the huge field of math. Big difference

 

[thinkandmull]

Yes it does. That root is not a natural number, not a rational number, not a real number, but it does exist. It is a complex number. The complex numbers appear to be yet another area of mathematics where you lack sufficient knowledge.

The square root of -2 is: sqrt(2) i - 1.41421356 i, where i = sqrt(-1).

Does this work? Yes. The square of sqrt(2) is 2. The square of i is -1. 2 x -1 = -2. QED.

This is an imaginary number; the imaginary component contains i, sqrt(2)i in this case. Some more mathematics for you to learn, I’m afraid.

rossum

That reasoning is highly debatable, but you accept things like that just because the establishment says it seems

 

[thinkandmull]

Square root means something times itself equaling that which the square root applies to. A negative times a negative is a positive, a positive times a positive is a positive. You would need to equivocate on what you mean when you say square root of negative 2

 

[rossum]

Square root means something times itself equaling that which the square root applies to. A negative times a negative is a positive, a positive times a positive is a positive. You would need to equivocate on what you mean when you say square root of negative 2

As I said, complex and imaginary numbers are obviously another area of mathematics you need to learn more about.

For imaginary numbers, i is defined as the solution to the equation i^2 = -1. That is all. The rest follows logically from that definition. As you correctly point out, i does not lie on the real number line. Complex numbers are usually represented as a point in the complex number plane. That is a plane, not a line. Again something for you to look up and learn.

rossum

 

[yppop]

Square root means something times itself equaling that which the square root applies to. A negative times a negative is a positive, a positive times a positive is a positive. You would need to equivocate on what you mean when you say square root of negative 2

TM,
If you consider what Rossum has written equivocating then there is an awful lot of “equivocating?” going on in the world that is apparently beyond you. For example, the square root of -1 =*** i*** is used extensively in solving problems in mathematics, physics and engineering. Identities such as Euler’s formula

e^(ix) = cos x + i sin x

are very useful in solving differential equations that are used to model electronic circuits without which we would not be communicating.

Yppop

 

[thinkandmull]

I am reading Heisenberg on quantum physics. Maybe that fits into the rather subjective system.

My math-computer major friend told me that unreal numbers were imaginary, and pi is an irrational number.

 

[rossum]

My math-computer major friend told me that unreal numbers were imaginary, and pi is an irrational number.

Your friend is correct. Now go and look up the maths-specific meanings of “imaginary” and “irrational” in a Dictionary of Mathematics. They are technical terms, and do not have the meanings assigned to those words in ordinary life.

For instance, your computer friend will use the word “bit” differently to the way it is used in ordinary life. “Bit” is another technical term in computing and information theory.

rossum

 

[hecd2]

My math-computer major friend told me that unreal numbers were imaginary, and pi is an irrational number.

pi is irrational and transcendental, whereas the square root of 2 is irrational and algebraic, and the square root of -2 is imaginary, has an irrational imaginary part and is algebraic

 

[Michael19682]

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

“Naturally”

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).

I was thinking that the set of all reals between two natural numbers could be explained as follows:

1. the set of all reals is either reals with repeating decimal digits or terminating decimal digits, e.g.
   4.567567567, or 4.57869860000,. (sorry but my computer doesn’t have a convenient horizontal bar as is traditional to use to indicate repetition.) OR IS
2. those irrational, theoretically non repeating reals with decimal digits governed by an arbitrary f(a/b) = x, where x is the real, irrational number.
   Thus, in my Hotel model, a modified Hilbert’s, there IS a one to one correspondence of the repeating (decimal) reals if:
   the repeating decimal sequences are treated as one guest arrival/party and put into suites, with the original guests occupying rooms between the suites
   Furthermore, there is created a countable infinity of the irrational, non repeating decimals when there is an allowable function call used to predict the next (decimal) digit of the real irrationals; this call is f(a/b), and the proprietor can keep track of where he is in the inflow of guests, to the end of always knowing the next decimal digit;
   Lastly, (1) if the Hotel grew by a mere Y dimensions, each irrational, which exists among an uncountable infinite number of irrationals in the set between the two naturals, could be mapped or corresponded one to one on that dimension if some other continuous function f(magic-relationship) existed to describe the procession of all irrationals.
   (2) if there is a mappable correspondence between the infinities between any two reals, then adding yet another dimension to the Hotel would capture all possible reals in the same way as the original Hotel captured all reals, one to one correspondence, countable infinities.

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

Yes. But I still want to know what you think because I believe my modifications are in response to the proposition of the Mr. Lane, who suggests the Hilbert’s Hotel proves or leads one to suspect a beginning in time to the universe. I withhold my opinion on that conclusion, but say that no Hotel, no matter how robust in accommodating infinities, can prove or give that inkling because it cannot accommodate the transcendent number Pi

 

[hecd2]

I was thinking that the set of all reals between two natural numbers could be explained as follows:

1. the set of all reals is either reals with repeating decimal digits or terminating decimal digits, e.g.
   4.567567567, or 4.57869860000,. (sorry but my computer doesn’t have a convenient horizontal bar as is traditional to use to indicate repetition.) OR IS
2. those irrational, theoretically non repeating reals with decimal digits governed by an arbitrary f(a/b) = x, where x is the real, irrational number.
   Thus, in my Hotel model, a modified Hilbert’s, there IS a one to one correspondence of the repeating (decimal) reals if:
   the repeating decimal sequences are treated as one guest arrival/party and put into suites, with the original guests occupying rooms between the suites
   Furthermore, there is created a countable infinity of the irrational, non repeating decimals when there is an allowable function call used to predict the next (decimal) digit of the real irrationals; this call is f(a/b), and the proprietor can keep track of where he is in the inflow of guests, to the end of always knowing the next decimal digit;
   Lastly, (1) if the Hotel grew by a mere Y dimensions, each irrational, which exists among an uncountable infinite number of irrationals in the set between the two naturals, could be mapped or corresponded one to one on that dimension if some other continuous function f(magic-relationship) existed to describe the procession of all irrationals.
   (2) if there is a mappable correspondence between the infinities between any two reals, then adding yet another dimension to the Hotel would capture all possible reals in the same way as the original Hotel captured all reals, one to one correspondence, countable infinities.

No. The set of reals includes all irrational numbers whether algebraic or transcendental, is uncountable and cannot ever be put in a one to one correspondence with the natural numbers or any countable set.

But I still want to know what you think because I believe my modifications are in response to the proposition of the Mr. Lane, who suggests the Hilbert’s Hotel proves or leads one to suspect a beginning in time to the universe. I withhold my opinion on that conclusion, but say that no Hotel, no matter how robust in accommodating infinities, can prove or give that inkling because it cannot accommodate the transcendent number Pi

I think you misunderstand William Lane Craig who is using not the hotel per se, but what he regards as the absurdity of an actual Hilbert’s hotel in support of his argument.

 

[Michael19682]

No. The set of reals includes all irrational numbers whether algebraic or transcendental, is uncountable and cannot ever be put in a one to one correspondence with the natural numbers or any countable set.

 

[Michael19682]

Also, how can the Hilbert’s Hotel, which does not sample values in time, be used to hint at a beginning to time? How does a man made model’s absurdity show a beginning to a variable external to it. It’s like saying, “Because infinity can’t bound my imagination, my imagination itself must be finite.” Perhaps in the Hilbert’s Hotel, guests check in, but they don’t check out.

 

[Michael19682]

]No. The set of reals includes all irrational numbers whether algebraic or transcendental, is uncountable and cannot ever be put in a one to one correspondence with the natural numbers or any countable set.

This sounds like a law. Laws in mathematics presume certain philosophical conditions. I will not hear “cannot ever” as long as I believe all things are possible.

 

[hecd2]

hecd2;13123014:

Rule out the transcendentals for the moment.

Then you are not considering the set of real numbers which includes transcendental numbers. You are considering the set of algebraic numbers which is countably infinite. However, if you include the transcendental numbers, which you must do if you are considering the set of real numbers, then it is uncountably infinite.

Let x vary from -infinity to +infinity. The real number line.
Use a continuous function as an example. f(sin(x/y) + 5) as y → abs((90* integer(j)/90)) (* (-1) if x<0) or (*+1 if x>0) for each j; and let j be a co variant → both + infinity and - infinity continuously in arbitrary increments; all to give this functional variance f(sin(x/y) + 5) a curve and make it differentiable and raised up a bit. If f ’ (sin(x/y) + 5) exists as above, then all reals are real and countable as per the allowable function call condition of the proprietor , and can be mapped onto the Hotel.

No, because the reals include transcendental umbers and the set of the real numbers is uncountably infinite as is the set of numbers which are a continuous function of the real numbers. This is easily seen as the output of any continuous function for which the (name removed by moderator)ut is transcendental is itself transcendental.

 

[hecd2]

hecd2;13123014:

It’s a theorem which can be *proven *

in Zermelo-Fraenkel set theory with the axiom of choice. Which is what we are talking about.

 

[Michael19682]

Michael19682;13123097:

It’s a theorem which can be *proven *

in Zermelo-Fraenkel set theory with the axiom of choice. Which is what we are talking about.

I’ll look into it. In the meantime I’ll check back in to see what Rossum puts down in the way of comment, or you hecd2 for that matter. Or anyone else.

 

[yppop]

I decided to listen to Craig:

His conclusion: Since an infinite number of events can’t exist, the number of events must be finite, therefore reality must have had a beginning.

His argument: An infinite number of events can’t exist because, the existence of an infinity is absurd; it is absurd because if an infinity of rooms are full how can you add another infinity of guests to it?

However, his argument is based on a really bad assumption, namely, that all the rooms in the infinite hotel are full. This assumption is bad because:

It is the nature of the infinite that it can’t be filled. This a concept that gives so much trouble for people like ThinkandMull who haven’t thought through this concept, but once grasped helps to clarify the ideas that seem paradoxical, when they are not if one accepts the rigorously defined set theory. I myself, join with the vast majority of the world’s mathematicians, physicists, and other scientifically trained persons, and accept the logic of set theory.

And I also believe that the universe is finite and had a beginning, but not because of Craig’s specious argument.
Yppop