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Mathematical Paradoxes Exposed Pt 1: The Nonsensical Grand Hilbert Hotel Hoax

Q: How many seconds in Eternity?

A: As many as You need.

Please excuse the inevitable typo, spell check is an evil imp. We will correct them as we find them. Thank You - AAMorris Staff.

We will try to pan the gold from the sand to see for ourselves if this supposed "paradox" has any value to it at all.

A Hotel That Cannot Exist

Parroted University nonsense sold as reason is what passes for intellectual achievement more often than not. Empty appeals to external and illogical, unreasonable authority is the order of the day in our modern global cultural Dark Age. For all the great insights and advances, both socially and technologically, this, the dawn of the 21st Century, is what a Dark Age of thinking looks like. Here we examine the University system promoted science of Mathematics. We shall see how some of it is simply incorrectly communicated.

Many of the University promoted  so-called Mathematical Paradoxes are nothing of the kind. These mental brainteasers usually rely on not using any real world demonstrable evidence. In fact many of these so-called paradoxes fail to use any real world examples at all, instead relying on focusing on numbers themselves and pretending that great insight can be achieved with this sort of navel gazing. This article, the first in a series, examines one of these great "Mathematical Mysteries". Here we look into the "Hilbert Gand Hotel Paradox" and we clearly see that the foundational premise of an infinitely sized hotel with an infinite number of guests is preposterous. This is something that can never be engineered and this is an obvious fact. It is also obvious that this idea cannot be demonstrated at all, at least not in any real world kind of way based on a literal interpretation of this "paradox".

The first question we ask is what exactly is the "paradox"? 

"Hilbert's paradox of the Grand Hotel, or simply Hilbert's hotel, is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and that this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture and was popularized through George Gamow's 1947 book One Two Three... Infinity."

How is this paradoxical? If we pretend we can have an infinite or endless supply of hotel guests and we also make believe we can have a magical endless supply of hotel rooms, what is the paradox exactly? There seems to be none. If we substitute a specific quantity like three for the word infinity we can then see that three is equal to three and that there is no paradox. Endless is equal to endless. Infinity is "equal" to infinity in the sense that infinity is infinity in the same way that a dog is a dog or a cart is a cart or an apple is an apple.

How is this a paradox at all?

Why would we expect an infinite set of guests to not be able to fit into an infinite set of rooms?

A Rose By Any Other Name

Isn't this like being surprised that twelve eggs fit into an egg carton constructed to hold a dozen eggs? A dozen is twelve, they are the same thing. If we imagine the impossible, that is the idea of an infinitely expandable hotel with an infinite stream of guests is unrealizable and simply cannot ever be actually engineered. Conservation of energy and other real considerations make this Hotel clearly nothing but an idea. As an idea it is no paradox. How is the concept of an endless stream of guests being able to find lodging in a Hotel with an endless stream of rooms be paradoxical?

This mythic mathematical Hotel would seem to be a metaphor for the idea of an eternal Universe in the sense of timelessness.

In other words time is an idea and creation of the human imagination. Existence just "is". And like with all types of measure or counting, we can continue to tick away measures of time.

A Timeless Universal Pulse

Everything that has a beginning has to have an end. Logically, the Universe must be timeless and have neither start nor ending. The Universe in this sense is simply akin to a continuous and eternal heartbeat or pulse or alternating current wave, from positive to negative and back again, in eternal round.

Do words not have meaning to Mathematicians?

"paradox (n.) 
1530s, "statement contrary to common belief or expectation," from Middle French paradoxe (14c.) and directly from Latin paradoxum"paradox, statement seemingly absurd yet really true," from Greek paradoxon, noun use of neuter of adjective paradoxos "contrary to expectation, incredible," from para- "contrary to" (see para- (1)) + doxa "opinion," from dokein "to appear, seem, think" (see decent). Meaning "statement that is seemingly self-contradictory yet not illogical or obviously untrue" is from 1560s."

paradox - Online Etymology Dictionary

Logic and common sense are not really taught to these Mathematicians or they would see that the foundational ideas they put forth are flawed. If they were trained in communication, they would see that these ideas have merits, but that they are not communicated properly and instead are sold to all of us as some kind of mysticism. We can see how this idea is useful to something like computer programming. We think it would be better to discuss these ideas in a more honest and clearly communicated manner. It seems clear that these so-called paradoxes are simply the foundation for computer programming language that we take for granted today. This can be demonstrated quite easily to anyone. There is no great mystery to how computers work.

Mathematics is supposed to be demonstrable science, not religion, right?

Why are demonstrable ideas sold as fantastic oddities?

The "Hilbert Gand Hotel Paradox" is the equivalent in many ways to Jesus and His Father's Many Roomed Mansion in Heaven or to the absurd "Big Bang Theory". An ever and impossibly expanding Universe with an eternal and everlasting and ever increasing source of energy is impossible, at least according to everything we can demonstrate, reason and common sense. 

 

Preface: Understanding What Math Actually Is & What It Is Not

Math is a tool used to model reality or to model ideas. Math is a language and is an invention of the human mind. Mathematics is not the tool of Nature nor the tool of Divinity. Neither Nature nor God needs Math, we do. Much is made of both the highly imaginative "thought experiment" and much is made of the power of Mathematics to reveal ultimate truths about our existence. Yet such claims are baseless and highly illogical. Common sense is put aside in favor of what can only be termed as an actual example of religious faith. Mathematics is bound by the same restrictions as any other human endeavor, we make mistakes and we human beings can lie, but  Nature, or Divinity, cannot.

Math is indeed foundational to any actual and realizable engineering project. Obviously math is integral to our collective global civilization and culture. Without it not only do we not have the computers and other electronic devices so common today, we'd also not have any of the engineering feats we tend to take for granted, like roads and bridges and cars and so on, for example.

Math In and Of Itself Does Not & Cannot Reveal Anything About Nature or Divinty

Math is a tool. When math is used to examine itself we end up with the mythical "Paradoxes", which are more sleight of hand, brain teasing nonsense than any actual 'paradox'. It's not that all of these so-called paradoxes are without merit, and it's not that these ideas do not have importance to us. They do. Set theory itself is important. But the way these ideas are communicated are very deceptive and lead one to think things that are not true.

Infinitesimally Yours: Division Is Not A Good Way To Model Someone Walking Across A Room

Another great mathematical brainteaser is the idea of endlessly dividing the distance a person needs to cross to reach an objective, like the wall. If we have a room that is ten feet across and we divide the distance in half we get five feet. If we continue to divide the result we can see how we will never reach zero. It is impossible to model a subtractive process, which walking across the room would be, with division. This is no paradox and would be more properly termed a mistake. The mistake being the use of the wrong tool for the job at hand.

Infinitesimals aren't useful for modeling a chicken crossing the road but are useful for modeling computer graphic scaling.

Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

Zeno's paradoxes - Wikipedia

Math Is a Language & This Sentence Doesn't like it one bit, no way!

Math is a language and like any language it is designed to describe something. It works best when it is not self referential and when all parameters are clearly defined. Math does not work so well when it becomes self referential. "This sentence is false." is put forth as a great paradox and yet it is simply bad literal grammar. We can use words to compose sentences that may be grammatically correct, but are logically meaningless or absurd. With language, whether words or numbers, we can lie or otherwise distort both reality and even our imaginative ideas. Math can be used as a medium for lies as well as words and sentences can.

This Sentence is True. This sentence's pants are on fire.

Sentences that have meaning and are logically constructed and put forth as real communication and not nonsense are supposed to be able to begin to answer some very important questions. Who, what, where, why, when and how are very important questions that every good narrative, whether fictional or factual, need to answer, otherwise the reader or listener or viewer is  left with unanswered questions and lacks complete understanding. If we state: "This sentence is a lie." we are putting forth a meaningless sentence. It might be grammatically correct, but it is a self referential sentence that is about nothing. There is no context put forth to explain how it is a lie or even ow it could be true. In what way is the sentence false? In what way a lie? The sentence is the subject of itself and this is a problem. We can state this: "This sentence does not like the idea that people can use sentences to communicate nonsense." Does this really make any sense in any real way? It, like the supposed mathematical brainteaser is a joke.

In philosophy and logic, the classical liar paradox or liar's paradox is the statement of a liar who states that they are lying: for instance, declaring that "I am lying" or "everything I say is false". If they are indeed lying, they are telling the truth, which means they are lying. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar themself. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
If "this sentence is false" is true, then the sentence is false, but if the sentence states that it is false, and it is false, then it must be true, and so on.

Liar paradox - Wikipedia

Math simply means we can count.

Calculus also means counting. Math is about counting. There are different ways to represent quantities. There are different ways to measure and different types of measurements, like volume, weight, distance, speed and so on. All of this is simply counting of one kind or another. All counting is compatible with other counting, since math has to do with quantities.

We can count the number of ways we can count. We can make sets of sets of sets. We can endlessly imagine subdividing the space between the lines on a ruler. We can imagine counting forever, in a very abstract way. We cannot actually do any of this of course. We can only pretend we can imagine we can. We cannot even really imagine what we claim we can. We imagine we can imagine things. Funny how recursive the human mind can be. 

Recursion - Wikipedia

A Mythic Strip That is Just an Image and Nothing More

Things like the mythic Mobius Strip are nothing but imagined imaginings with no actual basis in reality. This is what happens when a house of cards is built on a foundation of a house of cards. All depictions of the mythic strip show it clearly has two sides and is bounded by negative space, in other words, the concept cannot be demonstrated nor communicated in any kind of way. This means we should ignore it. The concept of the mythical strip means it is impossible to actually model or demonstrate. The concept itself is nonsensical. 

Look at the doughnut shape of the Moebius Strip above and take notice how we have a clear doughnut hole as represented by the white space in its center. This negative space would then have to be accounted for in some way, would it not?

Isn't this very self contradictory?

You Cannot Twist A Strip With Only One Side: There is no such thing as a strip with one side.

The Moebius Strip requires a twist. A twist by definition can only be done to an object with two (or more) sides. The very foundational concept of this mythical strip is flawed. It ignores the basic meanings of words and demonstrable physical evidence in favor of imagining imagined ideas. In other words this Moebius Strip, with its claim of a twist with only one side and with no outside, is itself self contradictory and highly illogical. We understand that this is an attempt to model the Universe as self contained. We can see how this is a very flawed way of attempting to do so. In part two we will discuss another way to model the Universe. Logically the Universe can have no “outside”. In other words the “Set of All Sets” is a Meta Set that cannot contain itself anymore so than a carton of eggs can contain a carton of eggs. IE the carton contains the eggs. The carton cannot contain itself. The Universe is a word we use to represent an idea that is somewhat beyond human understanding and ability to express. These are Metaphysical ideas, not Mathematical ones. Math is a tool and so is Metaphysics, we must learn to use the right tools for the job at hand and not get lost in our tool boxes (pun intended).

strip (n.) 
"long, narrow, flat piece," mid-15c., "narrow piece of cloth," probably related to or from Middle Low German strippe "strap, thong," and from the same source as stripe (n.1). Sense extension to wood, land, etc. first recorded 1630s.

strip - Online Etymology Dictionary

twist (v.) 
c. 1200 (implied in past tense form twaste), "to wring," from twist (n.). Sense of "to spin two or more strands of yarn into thread" is attested from late 15c. Meaning "to move in a winding fashion" is recorded from 1630s. To twist the lion's tail was U.S. slang (1895) for "to provoke British feeling" (the lion being the symbol of Britain). To twist (someone's) arm in the figurative sense of "pressure (to do something)" is from 1945. Related: Twistedtwisting.

twist - Online Etymology Dictionary

Hello. This is a video of me making a Möbius strip, a strip of paper with only one side. I'll upload another video on how to make a heart-shaped mobius strip another time. Very fascinating stuffs. Maths and Science is awesome!

Mythic Ideas With No Basis In Reality

image source: http://www.kleinbottle.com/sliced_klein_bottles.htm

We can only draw or otherwise visually represent this mythic strip. When anyone actually tries to model one they end up with a twisted piece of paper or ribbon. Similarly, the mythic Klein Bottle is another version of this and also ends up being just another regular object when modeled. As you can see for yourselves in the above photo, a Klein Bottle is just another three dimensional object, with an inside and outside and a thickness. 

More about this in the next part of this series.

Möbius strip - Wikipedia

Klein bottle - Wikipedia

The Count counts 4 bats! For more fun games and videos for your preschooler in a safe, child-friendly environment, visit us at http://www.sesamestreet.org Sesame Street is a production of Sesame Workshop, a nonprofit educational organization which also produces Pinky Dinky Doo, The Electric Company, and other programs for children around the world.

Rudolf Stiener- HOW SCIENCE DESTROYS ITSELF- An epistymology problem yet to be corrected

Infinity simply means we can count endlessly.

"infinity (n.) 
late 14c., from Old French infinité "infinity; very large number or quantity" (13c.), from Latin infinitatem (nominative infinitas"boundlessness, endlessness," from infinitus boundless, unlimited" (see infinite). Latin infinitas was used as a loan-translation of Greek apeiria "infinity," from apeiros "endless.""

http://www.etymonline.com/index.php?term=infinity

"calculus (n.) 
1660s, from Latin calculus "reckoning, account," originally "pebble used as a reckoning counter," diminutive of calx(genitive calcis) "limestone" (see chalk (n.)). Modern mathematical sense is a shortening of differential calculus. Also used from 1732 to mean kidney stones, etc., then generally for "concretion occurring accidentally in the animal body," such as dental plaque. Related: Calculous (adj.)."

http://www.etymonline.com/index.php?term=calculus&allowed_in_frame=0

Confusing Quantity With Symbols

Is it really paradoxical to confuse the word with what those sounds or symbols represent? These words you are reading now are comprised of a collection of symbols we term the alphabet. Unlike numbers, the alphabet is limited to a certain set of characters that represent various verbal utterances we can vocally create. The written word is derived from verbal language. Can you eat the word apple?

Apples & Oranges: Counting The Ways We Can Count

Numbers are symbols. Numerals are not the ideas or objects they represent It is illogical to confuse the symbol with the substance. One refers to the other but is not the actual thing it refers to. The numeral "2" is not anything but a symbol,

1 + 1 = 2. We can represent two objects or two ideas with the symbol "2". The numeral is a representation of the concept of two objects. Let's say we have two apples. We represent the apples with the symbol "2".

Can You Eat The Number 2?

The number two - "2", is an idea. The two apples are real world three dimensional objects you can actually eat. One is imagined one is real. There is no reason to promote confusion over this easily demonstrated fact, other than to promote some kind of cultish group thinking.

1 + 1 is 2

We can also use the following symbol to represent the two apples: "1 + 1". This is why we can write, "1 + 1 = 2".

We can represent two apples as 2.0. If we wanted we can endlessly add zeros to the end of this symbol and still have the same quantity of two. We can represent the concept of two as "2.000000000000...." We can also add zeros in front of it and it will till represent the quantity of two. For example: 02 or 002 or even 02.000000!

We can represent the quantity of two as 2/1, if we want to use fractional form.

Fractions are simply another form of division. So 2 divided by 1 = 2/1

2/1 = 2.000000... and 2, 2 x 1, 1 + 1, 100 - 98, etc. These are just all ways to express the concept of two. We can represent this quantity with an assortment of symbols and collections of symbols IE an equation. We can express the quantity of two as a single digit with no decimals, or with decimals, we can express it in fractional form or in a a highly algebraic complex equation form. We can express it as a multiplication problem or as a picture of two apples. Numbers, like words and graphical depictions are meant to symbolize either ideas or real world objects. Can you eat a picture? Can you eat an idea?

Addition, Subtraction, Multiplication & Division Are All Ways Of Counting or Expressing Quantities

expression (n.) - Online Etymology Dictionary

Expression (mathematics) - Wikipedia

Equal = Is

Equal means the idea on the one side of the equal sign is the same as the idea on the other. The equal sign shows us we can use either symbol to represent the concept of two. We should not confuse the concept or the reality with the symbol we use to represent that idea. This is the basic problem with many of these so-called Mathematical Paradoxes, the authors simply ignore that they arena longer discussing anything real but instead are focused on the symbols themselves.

Numbers Are Symbols: Numerals Are Limited

 Numbers are also symbols and the language those symbols belong to is properly termed Math.

Like with the alphabet, we only have a fixed set of Numerals. These symbols are limited to this set:

( 0,1,2,3,4,5,6,7,8,9). With these ten digits we can endlessly count. We can rearrange these symbols into countless forms that can be used to represent an endless number of quantities.

Infinity Is Not a Number

Calculus means counting. Infinity means boundless. Infinity plus one or minus one is nonsense. Infinity is not a number and anyone who claims otherwise is being intellectually dishonest, whether they intend to be or not. It's interesting and telling how the actual history of science and math and the original meaning of terms are clearly ignored in favor what what can only be described as creating myth or rather, propaganda. Infinity is a word ( a symbol ) that represent the concept of "endless" or "boundless", useful for modeling the endless generative process of a Universe for a Metaphysical discussion. Here we can see that there is value to these ideas, when put in the proper context, of course.

By definition the ultimate source of all reality has to have no beginning nor can it have an end. This is not science, this is the subject of Metaphysics. Here the concept of infinity or boundless has its proper place.

Metaphysics - Online Etymology Dictionary

Math is Not a Perfect Tool - The Limits of Math as Language Exposed: 0 does not equal 1

0 does not = 1. There's a mistake in the proof or perhaps it is meant as a clever puzzle.

This is an excellent example that shows the limits of mathematics.

See Video Below:
At (time) 1.15  81/4 is needlessly added to both sides.

At about 2:30, during the factoring, the 2 x 4 and the 2 x 5 parts of the equations get lost.

left side: 2 x 4 x 9/2 is 2 x 4 x 4.5 = 36
right side: 2 x 5 x 9/2 is 2 x 5 x 4.5 = 45

These values are ignored and this is why the result is wrong.

-20 = -20
this becomes
16-36 = 25 - 45
towards the end we have 4 = 5
4 is the square root of 16 and 5 is the square root of 25 of course. The problem is the - 36 (on the left) and the -45 (on the right) are now missing from the equation so there is an obvious imbalance.

We'd have to express it like this to get the correct and logical result.
16 - 36 = 25 - 45

At around 2.50:
(4 - 9/2)(4 - 9/2)=(5 - 9/2)(5 - 9/2)
4 - 9/2 = 4 - 4.5 = - 0.5
5 - 9/2 = 5 - 4.5 = +0.5
These numbers times themselves are:

-.05 x -.05 = 2.5
0.5 x 0.5 = 2.5
Obviously they are equal.

Any problems that occur simply show the limits of the human construct and tool we know as math. The problem would seem to have to do with the use of the negative numbers and the square roots. This "Proof" shows us that we must always keep in mind that mathematical equation and symbol are meant to model quantities. When we forget this we can end up using this wonderful tool in ways that make no sense. Much in the same way we can type or otherwise communicate "This sentence is a lie." We can use language and math to communicate absurdities. Negative numbers are useful for modeling certain things and not so useful with others. Canceling out parts of equations from either side of the equal sign has its uses as well, but it this too has its limits as we can see from this video. We can also seed how we can represent the quantity of - 20 in  different ways and we can see if we forget that we are representing quantities, we can end up with something that is nothing but nonsense. Math is not a perfect tool.

A Proof that 0 = 1. I do not know what to believe any more.

Confusing Quantity With Counting

Mathematics is the codification of counting. We can count stones. We can use these stones to represent work, to be used to represent a simple system of currency exchange. If a farmer grows corn and another makes and repairs shoes, one can trade some corn for a new pair of boots. As people gain more specialized areas of skill and begin to profit from it, we can see the need for a symbol to be used to trade our work for a product of one kind or another.

One Third of Three Apples is One

Much is made of the repeating decimal as if it was some great insight. When we use real world examples we see that this is clearly not the case at all. If we have three apples, each apple is 0.3333333... (1/3) of the total. The group of three apples is considered to be the whole or one thing ( the 1 ). Each apple is one third of this group or "whole".

A Question of Resolution: 1/3 = 3.3333333333333333........

All the repeating decimal does is reveal the limits of human ingenuity and not much else. The use of the decimal place is best understood in terms of actual real world usage rather than abstract self referential metaphysics. Math is a great tool for building real things, whether physical or on a computer screen. Math is not a good tool for explaining Metaphysical ideas and this is the mistake many make. We can keep adding decimal places to the numbers we use. The reality is that we can only physically measure down to a certain size and then the rest of it is imagined. In reality we only need to measure to the decimal place appropriate for the job at hand and we move on. We do not spend the rest of our lives needlessly counting decimal places just because the math does not work out so we get a nice clean result.

If we measure substances for chemistry or we measure a length of lumber we are only concerning ourselves with the resolution of measure we need for the task at hand.

Division Means We Are Grouping Quantities Into Sets

We can have a dozen eggs or three sets of four eggs, for example. We can have two sets of six eggs or any other combination.

Repeating Repeated Memes

"A repeating or recurring decimal is decimal representation of a number whose decimal digits are periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333…. A more complicated example is 3227/555, whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144…. At present, there is no single universally accepted notation or phrasing for repeating decimals.
The infinitely-repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this decimal representation is called a terminating decimal rather than a repeating decimal, since the zeros can be omitted and the decimal terminates before these zeros."
"Every terminating decimal representation can be written as a decimal fraction, a fraction whose divisor is a power of 10 (e.g. 1.585 = 1585/1000); it may also be written as a ratio of the form k/2n5m (e.g. 1.585 = 317/2352). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9. This is obtained by decreasing the final non-zero digit by one and appending a repetend of 9, a fact that some find puzzling1.000... = 0.999… and 1.585000... = 1.584999… are two examples of this. (This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.[2])
Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats but extends for ever without regular repetition. Examples of such irrational numbers are the square root of 2 and pi."

Repeating decimal - Wikipedia

Converting repeating decimals to fractions

Negative Numbers Are Ideas

When we get into money and accounting, we can see the value of the negative number as a tool. If we desire to model electrical phenomena we can see the value of the negative number as a symbol and tool used to model a real world phenomena. We can see how it is used to create a model that we can examine and draw conclusions from that we can then go to actually demonstrate.

$$

If we use the monetary system as an example we can easily see that with debt anyone can indeed "have" a negative balance, or amount of money. In other words the use of a negative number is to model the fact that we have a deficit of funds and owe more than we are worth.

AC/DC

If we wish to model an electromagnetic wave with its positive, neutral and negative modes, or aspects, we can see how the negative number would repent the negative polarity and the positive obviously models its namesake. We can see how the symbol of zero is useful as a symbol for the neutral position between the two poles.

Revealing The Magic Trick

The sleight of hand deception works by the simple confusing of the symbol with the real world quantity (or idea) it is supposed to represent.

Mainstream Science & Math Are Religion

The creation and promotion of an unsolvable Mystery (or mythic paradox) is of upmost importance in the creation of a Cult or Religion.  Personas are also promoted over ideas. Thus we get the mythic and incorrect image of Einstein and all the rest of the demigods of our global cartoon culture. All good religions need idols to worship and look up to. External appeal to authority is the order of the day. We must not learn how to think for ourselves. We must simply nod our heads and consent to accepting nonsense as reason. We must learn to parrot the various University promoted memes in order to achieve the all desired and all important University degree. Any who step outside the accepted mainstream University maintained boundaries face a less profitable career, to say the least.

Recursive Nonsense: Imagining Imagined Assumptions

A Hotel That Cannot Be Built

Of course the famed Hotel cannot actually be engineered. This is a great example of cartoonish reasoning sold as logic. We'd run out of space eventually, of course and we cannot have a physically infinitely sized hotel. Infinity means boundless or endless as in no beginning and no end. The Hotel Paradox is not a good analogy or model for the concept of endless or boundless. This paradox is not an example of clear communication.

Teeth Cannot Bite Themselves

An Egg Carton Must Contains Eggs - The Egg Carton Cannot Contain Itself

Counting Counting

Set theory is a very interesting area of study, but the way it is often sold to both the public and the University student is extremely misleading, Instead of counting quantities we end up counting ways we can count.

Toasting The Roach Hotel

Here we examine the Hotel Paradox:

"Hilbert's paradox of the Grand Hotel, or simply Hilbert's hotel, is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and that this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture and was popularized through George Gamow's 1947 book One Two Three... Infinity."

A Whole Lotta Nonsense

"Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply."

From the start we are dealing with something clearly labeled "hypothetical". What exactly is the hypothesis? Is it laid out in a clearly communicated matter with real world examples that we can demonstrate? Is the idea logically constructed and communicated in such a manner that we can imagine what the author is proposing and we can see how it makes sense, with no real world demonstration? The answer to both questions is no.

What is this idea based on? What real world phenomena is it meant to model? If not a real world and demonsrrable natural principle, then what imagined idea is this paradox meant to show us? 

Let's Get Metaphysical

All the so-called paradox does is clearly illustrate that if we suppose an infinite or endless stream of guests, and we imagine an infinitely or boundless set of rooms to put these guests in, we can see how one boundless idea can be contained within another. This very basic concept is clouded and concealed in a literal foggy day dream - LSD like nonsense. A better way to explain this simple concept is the idea of the uncreated and unending, timeless Universe that simply waves or beats like a heart. This cosmic heart was never born and will never die and will beat away forever. Time is meaningless to it. All it "knows" is the positive, the neutral and the negative aspects of the wave. In other words all there is, is a ceaseless pulsing that knows no end or beginning. This would better represent the concept of boundless or "infinite". 

Time is an Idea, Time is a Human Construct

The boundless or infinite is better expressed in terms of time or rather the lack of it. Boundless and infinite make less sense when describing space. The concept of an infinite and endless amount of space clearly contradicts what we can demonstrate in terms of the conservation of energy. A self contained Universe of some kind, is not only logical, but makes sense in terms of demonstrable electromagnetic principle. A self contained Universe would be immune to losing its energy as there would be no outside to it. Such a Universe would "ring" or "wave" forever as the energy would not be carried away ever. A self contained Universe is insulated from any energy loss. There simply cannot be an outside environment to lose the energy to.

Can You Guess The Number of Guests?

"Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests."

As we can read for ourselves in the quote above, the paradox ignores the very real fact that this Hotel is something that we cannot engineer or build. The entire premise is illogical and only makes use of real world things like people and hotels in a dream like fashion and this is the very stuff of science fiction and fantasy and not real science or even clearly communicated Metaphysics.

Not a Care In The World About Actually Having To Build All The Rooms

"Finitely many new guests
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests."

Above we can read for ourselves how there is no concern over constructing the rooms for this imagined hotel. No concern over all the details that would inevitably be involved with such a scheme. Instead we get the fine art of creating definitions and sets and organizing information. Here we can see the value to this so-called paradox, but it is no "paradox" nor is it a mystery.

The Non Paradoxical Nature of Set Theory's Use in Computer Science

We can see how this reasoning can be used to build computer systems that involve physical objects like computer storage and to program the software that keeps track of the increasing storage and ever increasing number of files.

Flights of Fancy Instead of Clearly Defined Reasoning

"Infinitely many new guests
It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
Infinitely many coaches with infinitely many guests each
For more details on this topic, see Pairing function.
It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (or use the axiom of countable choice). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be n, and their coach number to be c, and the numbers n and c are then fed into the two arguments of the pairing function."

Making Sense of Nonsense: Panning The Gold From The Sand

What does make sense is the idea that the Hotel is a metaphor for the Universe. This is not science, by the way, now we have entered the realm of Metaphysics. We can see how this logic can be used to allow us to create computer software that will enable us to store information on our computers. We can see how we can keep adding more storage and how we can keep creating more files to store.

Math is Language

Set theory does have its uses. It is obvious that this reasoning is foundational to computer programming language.

Never Forget The 911: Math is About Money!

Newton was put in charge of the Royal Mint after all.

Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England's great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy comptroller of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life.[72][73] These appointments were intended as sinecures, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish clippers and counterfeiters.

Isaac Newton - Wikipedia

A Hotel Contains Rooms Not Other Hotels

SOURCE:

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Part Two: Twisted Logic Stripped

The next part in this series will look into things like the Moebius Strip and the fact that mathematical points are not real world things, but are ideas. Points do not occupy space. Points have no measurable dimensions. We will examine the religious like need to pretend Mathematics can reveal ultimate truths about Nature and existence without having to worry about the underlying logic, common sense and demonstrable principle. Mathematics is presented to us as some great Metaphysical truth in and of itself. Math is a tool and a language and is used like all languages, to model reality or ideas. And like with all language and other forms of human communication, math too can be wrong or can be used to tell us lies.