Assuming an endless universe, there are as many Thursdays as there are days.
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Voting Style:  Open  Point System:  7 Point  
Started:  9/21/2008  Category:  Science  
Updated:  9 years ago  Status:  Voting Period  
Viewed:  2,645 times  Debate No:  5485 
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Georg Cantor established the idea bijection to establish the relationship between the sets of infinite numbers and establishing that they were the same size. He famously derived the counter intuitive proof that there as many unit squares as items on the number line in his 1874 Crelle paper.
http://en.wikipedia.org... Assuming the universe doesn't end and the word day still holds some meaning unto the infinite, then there are as many Thursdays as there are days in general. Today > Next Thursday 1 day from now > 1 week from next Thursday. 2 days from now > 2 weeks from next Thursday. 3 days from now > 3 weeks from next Thursday. 4 days from now > 4 weeks from next Thursday. 5 days from now > 5 weeks from next Thursday. 6 days from now > 6 weeks from next Thursday. 7 days from now > 7 weeks from next Thursday. 8 days from now > 8 weeks from next Thursday. ... For each day you site in the first set of days in general there is a 1 to 1 comparison to a Thursday in the second set.
I must negate. There are 1/7 as many Thursdays in an infinite amount of time as there are days in general. My opponent brings up bijection. Bijection applies when for every member in one set, there is exactly one member in the other set. This was indeed used to prove that there were just as many nit squares as whole numbers, since each whole number has its own unit square. how do we know that these two sets have the same number of members? Because each whole number must have its own square. The existence of 3 means it is necessary that 3^2 must also exist, since a number must be able to be squared. Now, let us look at what my opponent has attempted to do. He has made two sets, each of which is undeniably infinite given our assumptions [an infinite universe where days continue to hold their same meaning for all time]. Both sets are indeed infinite, but they are also unequal. You see, how do we know that there exists a corresponding Thursday for each day? True, we could do what my opponent has done, and simply write the first few down and make arrows corresponding them. However, this is begging the question. My opponent has simply ASSUMED that there is an equivalent number of members in each set. For the example of numbers and their squares, there was a reasoning behind them having the same number of members. the very existence of any number x necessitates the existence of x^2. To say that there were more numbers than squares would be to say that some numbers did not have corresponding squares, but every whole number has a square as a rule. However, it may be the case that every day does not have its own corresponding Thursday. This unproven point, which is basically the same point he is trying to prove, is something my opponent has just assumed. Now, I will prove that there are 7 times as many days as Thursdays. Both of our sets, number of days passed and number of Thursdays passed, depend on the variable time. I will denote time as t, which will be measured in days. Increasing t by 1 will always mean one more day. Increasing t by 7 will always mean one more Thursday. Let me write this in math notation. The number of days can be represented by the function D(t) = t The number of Thursdays can be represented by the function T(t) = t/7 Where t=1 is the first day we're counting, which in this case is a Friday. Already we can see a onetoseven correspondence. Now, the functions as we go into infinite. Number of days: lim t>(inf) (t) Number of Thursdays: lim t>(inf) (t/7) Each of these approaches infinity, but we already knew that. To get a ratio, we simply divide them. Since the units are the same, we can combine the two limits into one. Number of thursdays / Number of days: lim t>(inf) ((t/7)/(t)) This gets us inf/inf, which is indeterminate form. We could either cancel out the t's or use L'Hopital's rule, taking the derivatives of the top and bottom functions. That's what I'll do. Top: t/7 Derivative of top: 1/7 Bottom: t Derivative of bottom: 1 Now we are left with this: lim t>(inf) ((1/7)/1) = 1/7 Thus, the ratio of Thursdays to days is 1/7. That is because each Thursday has exactly seven days that correspond to it. Unit squares, however, have a onetoone correspondence, because the existence of any one whole number X necessitates the existence of it square, and the square of a number necessitates the existence of that number. Each Thursday, as I have shown, necessitates the existence of seven days, and each consecutive seven days necessitates the existence of one Thursday. So while both sets are indeed infinite, one contains exactly 7 times as many members as the other. Your turn. 

You're attempting to apply fractions to infinites in error. When dealing with infinite sets the only real question is the cardinality of the infinite set. The set of all days and the set of all Thursdays are both infinite sets of integers and as such are the same.
The bijection between units and unit squared is exactly the same as the bijection between Thursdays and days. For every value that you can give in terms of, "X days from now", I can give a value in terms of, "X weeks from now". This is the exact same bijection as showing that anything you can give in terms of "X", I can give in terms of "X^2". Therefore, we find that both sets are of equal value. In fact, the squares increase so rapidly that by the time the units are at 1000 the unit squares are at 1,000,000 and as such there must be at least 999,000 numbers *NOT* part of the unit squares. This will get worse and worse as we continue ending up with an infinite number of numbers not in the set of unit squares (ironically there'd be as many numbers not in the unit squared set as there'd be in the unit squared set by a simply application of this same theorem). >>"My opponent has simply ASSUMED that there is an equivalent number of members in each set." I have done nothing of the sort. Given the set of days we note that for each and every member of the set we have a correlated member in the other; for day 39382 we have the Thursday of week 39382. Where have I lost any values? Give me a day for which I cannot have an equivalent Thursday within the bijection? Just as every whole number has a square, every week has a Thursday. So long as there are Thursdays each week then the existence of day X allows for the existence of Thursday X. You argue that the set of days is seven times larger than the set of Thursdays. Give me any value of X for which day X exists and Thursday X does not! That's the proof. For every value in set A there is a value in set B so set A is the same size as set B. I didn't just assume this. It was proven by Cantor in the 1800s. All infinite sets of integers can be placed into a bijection and though it's extremely counter intuitive every infinite integer set has the same number of elements: Infinity! My opponent then attempts to construct a faulty proof which much like dividing by zero leads to issues concludes that 1/7th of infinity isn't infinity. However, 1/7th of infinity IS infinity. Let's look as his same proof in context of the unit and unit squared proof Cantor proved. If we were to count each number within zed (the set of all integers) and place it into one set or the other, we would find that every number is in the set of units whereas there's going to only be about N^.5 (square root) as many numbers in the set of unit squares as there are in the units at any given point. So shouldn't there be infinity squarerooted unit squares?  Well, that's still infinity. The point of assembling the bijection of the set is to properly look for whether or not some value can or cannot be correlated to a value in the other set. For every day I can find you a corresponding Thursday within an infinite set. My opponent proceeds to show that the limit of the ratio between the sets as it approaches infinity is 1/7th. This demonstration is wrong because we aren't dealing with limits approaching infinity, we're dealing with infinity. We use limits because infinity itself is really really odd. Certainly as we move along the number line adding days to either set A or set B we will have about 7 times more in the set of days than the set of Thursdays. However, we aren't talking about doing that. We're talking about the infinite sets rather than the limit as it approaches.  Let us again look at the unit and unit squared example proven by Cantor (I'll provide the paper if you want) and show that that what you've done proves nothing. The set of units should be N at any point as we move along the number line (something we manifestly are not doing) and at any given point we will have N^.5 elements in the set of unit squared. Number of unit / Number of unit squares: lim t>(inf) ((t)/(t^.5)) This gets us inf/inf, which is indeterminate form. We could either cancel out the t's or use L'Hopital's rule, taking the derivatives of the top and bottom functions. That's what I'll do. Top: t Derivative of top: 1 Bottom: t^.5 Derivative of bottom: .5/t^.5 Now we are left with this: lim t>(inf) (1/.5/t^.5) = t^.5/.5 = 2*(t^.5) = inf. Therefore, we have infinity more units than unit squares. However, we know by Cantor that we actually have the same number? What happened here?  The procedure you carried out is proper to find out the ratio of the sets as it approaches infinity but worthless at infinity. You are doing something, which, though proves you've taken calculus and understood it has absolutely no bearing on the argument at hand.  Set of Days vs Set of Thursdays For every element of one set there is an element in the other set. If there exists a day X there must therefore exist a week X. The two sets are of equal value. There are as many Thursdays as there are days.
Indeed, I am attempting to apply fractions to infinite values, "in error." In error when we're talking about real numbers. However, infinity does not exist in the real numbers, and so it is clear that real numbers are not fit for our purposes, since we are talking about infinite values. Indeed, in the real numbers, adding a number to infinity does nothing. Take something we know, for example. We know that whole numbers, all integers and zero, contains exactly one more member than the natural numbers. However, since infinite is only a concept rather than a set of welldefined values in the real numbers, these sets have the same cardinality. That is to say, by my opponent's reasoning, they contain exactly the same number of members, even though we know that one of the sets is the exact same as the other set except with one more member. Indeed, the problem is rooted in the real number system as it pertains to infinite values. Even multiplying an infinite value by a scalar does not affect the value in the real number system. Since infinity is not a set of welldefined values in the real number system, we should seek another system where it is a meaningful concept, since our specific problem is dealing with the values of infinite numbers, which are not well defined in the reals. Let me be clear on something. Even though [in the reals] adding 1 to an infinite value doesn't change the infinite value, two infinite sets with a difference in members of one still has a different number of members. There is a reason Infinity  Infinity does not equal 0  because even though we cannot make the difference between the two infinities clear, there may still be some difference. Indeed, Infinity/Infinity may not always be 1. In the case we are given, the ratio of the infinities is onetoseven, meaning that each Thursday brings with it 7 days. Simply because we cannot differentiate these values in their infinite form does not mean that they are equal. If they were equal, Infinity/Infinity would indeed be 1, and Infinity  Infinity would be zero. So if we simply pretend that infinite numbers exist, and we graph the function I gave last round at some infinite value [meaning that some infinite number of days has passed], then we see that seven times as many days exist as Thursdays. The function indeed reduced itself to f(x)=1/7, meaning that the value is constant everywhere it is valid. And since we are now using a number system where infinite is valid, we see that it continues to be 1/7 at that point. For something to be equivalent [like the number of days and the number of Thursdays], there must be a difference of zero between them. However, it cannot be said that the difference between these two functions is ever zero, let alone at infinity. Indeed, the difference between the two functions, x and x/7, GROWS as we get to larger and larger numbers. At infinity, the difference between the two is infinite. And while that may not mean anything mathematically in the real numbers, it surely means something semantically in reference to this very discussion. The fact that there is any difference means we can negate. I apologize for so readily accepting the idea that the number of whole numbers and their squares is equivalent, since that only holds true mathematically using number systems with poorlydefined concepts of infinity. In the reals, if we define X as some infinite value, X + 1 equals X. We find that 7X = X. Thus, there can be any finite difference in the size of the two sets, or one set could be seven times as large as the other, yet my opponent would argue that since the real number system is so inept at handling infinite numbers, the two sets have the same number of members. Indeed, my opponent argues that 1/7 of infinity is infinity. This is true mathematically using reals, but we must also realize that one set has seven times as many members, since we are actively incrementing in days. At any time T, seven times as many days have passed as Thursdays. Thus, there are not 'as many' Thursdays as there are days. This will hold true at the end of every Thursday, even Thursdays of infinitelynumbered weeks. 

My opponent has seen the light. Perhaps having reviewed the evidence or begun to understand the cardinality established by a bijection. As the set of days is exactly equal to the set of Thursdays just as the unit set is equal to unit squared set because every element can be matched up exactly from the first set to second set on account of the nature of infinity.
Day X in the first set exactly matches Thursday X in the second set. There are no missing elements because for every element X you want in the first set we also have element X in the second set. And though the second set progresses to infinity faster than the first set they are both infinite sets of equal size. We aren't talking about the progression or rate of progression of the sets but rather their enumeration. Set theory is quite clear as proved by Cantor. There are as many Thursdays as there are days.  My opponent however, realizing his mistake offers an argument which, if based on valid premises would win the debate. If we are comparing an infinite set of integers to an infinite set of reals then I would be wrong. My error would be palpable. However, under no possible interpretation can we accept that the set of days or the set of Thursdays are reals. He does however, mistake a few things when talking about "real" numbers. The "real" number line does not refer to down to earth concepts of numbers where infinity is impossible and we deal with common sense solutions to common sense problems. Rather, we are dealing with a number line which allows for infinite decimals. How many numbers are there between zero and one? On the integer number line we have 0 numbers between the two. There is no integer between zero and one. However, with the real number line we have .001, .01, .1, .002, .00000003, .5, .999999, .14159265358979323846 and literally an infinite number of numbers between the two! Cantor like his set theories for integers showed that among the real numbers that you can not only establish a bijection between the two infinite sets between integer sets but you can show that between zero and one there are an infinite number of numbers. And that that infinite number of numbers is exactly the same cardinality as the set of reals between negative infinity and infinity. Within the integers there are 0 numbers between 0 and 1. Within the reals there are as many numbers between 0 and 1 as there are on the entire number line. In fact, there's as many numbers between 0 and 0.00000000000000000000000001 as there are between 0 and 1 and the entire real number line. Now, an interesting point thing happens when you attempt to create a bijection between the integers and the reals; the reals win. You do not have item X in the set of integers for every item X in the set of reals because unlike the integers the real numbers are not countable. You can't create a 1:1 relation between the two. Now, this raises two problems for my opponent. In order for his argument to succeed he actually needs to show that the days are not discrete (they aren't countable) or that the Thursdays are not discrete while establishing that not both are reals (as if both are reals they are both infinite sets of reals rather than infinite sets of integers and again share the same cardinality). The set of Thursdays are countable! We have about 52 of them every year. The set of days are countable! We have about 356 of them every year. They are both integer sets and for every element X of the set of Thursday we have element X of the set of days. We can not only count them but we can count them at a 1 to 1 ratio. His argument is mistaken when it suggests that the concept of infinity doesn't work equally well with the set of real numbers. It is rather that not only are the set of real numbers infinitely big they are, unlike the set of integers, also infinitely small. We can show that some infinite functions within the reals advance faster towards infinity than other functions. Much as we would find if we compared the function, f(x) = x to the function, f(x) = x^2, in fact the latter function progresses towards infinity faster by a rate of 2X ( f2(x)' / f1(x)' ). However we already note the bijection proof by Cantor established that within the countable infinites that these two values are the same. So what's going on here?  We are dealing with integers.  >>For something to be equivalent [like the number of days and the number of Thursdays], there must be a difference of zero between them. However, it cannot be said that the difference between these two functions is ever zero, let alone at infinity. There is a difference of zero between the two. Within the set of days there exists a given element X. Within the set of Thursdays there is a given element X. In order to have a different number between the two there would need to exist some Thursday X where there does not exist a similar element day X or for some day X there needs to exist no acceptable Thursday X within the set. Allow me to show you why this works by ignoring the assumption of an endless universe. If, for example, the universe were to end in December of 2012 then we would be dealing with a finite length of about 4 years. Within those four years there would be about 1461 days, but only 208 or so Thursdays. So element 209 would exist within the set of days but there would not exist an element 209 within the set of Thursdays. We would not have a 1:1 relationship between the two. However, we aren't dealing with finite values where element X does not find a perfectly bijected element X in the other set. We are dealing with infinite numbers where the set of Thursdays have exactly as many elements as the set of days.  Cantor's proof only dealt with integer sets as we are dealing with here. My opponents attempt to obfuscate here is unwarranted. http://mathforum.org... Here is a quick explanation of the underlying mathematics in a very understandable explanation. Although, the notion seems counterintuitive it is actually remarkably easy to understand and a central idea within modern mathematics.  There are as many Thursdays as there are days. Count them! Give me any element X where there exists an element of day X or Thursday X and I'll give you the corresponding value in the other set. day X ==::== Thursday X
Indeed, one set grows seven times faster than the other set. This means that on average, for all times, the number of days will be seven times as large as the number of Thursdays. This holds true no matter how far we go, unto infinity. The only reason my opponent can claim that the sets are the same size is because infinity is so illdefined in our real number system. It's true that you'll never find a spot in one set that cannot match with a member of the next set. My opponent says that the number of Thursdays and the number of days are not real numbers, but integers. Ha ha, my friend, they are _both_. Perhaps ye learned sometime in the past, maybe in sixth grade or so, the different types of numbers. There are the natural numbers. These are 1, 2, 3, etc. Then there are the whole numbers. These are 0, 1, 2, 3, etc. All whole numbers are also natural numbers. Then there are the integers. These are ... 3, 2, 1, 0, 1, 2, 3 ..., or the positive and negative whole numbers. All whole numbers and all natural numbers are also integers. Then there are rational numbers. Rational numbers are any numbers that can be represented by fractions of integers. 1/3 is an example. All integers, whole numbers, and natural numbers are rational numbers. Next. there are real numbers, which include rational and irrational numbers. Irrational numbers are numbers that cannot be represented as fractions of integers, like pi. All the number types I have talked about so far are also members of the real numbers. So we see that these sets are indeed sets of real numbers. Both of these sets are sets of integers as well as sets of reals. That considered, take a look at this statement from my opponent: "If we are comparing an infinite set of integers to an infinite set of reals then I would be wrong." Well, we are! Therefore, my opponent has accepted defeat. I will continue this charade nonetheless. I have already accepted that these sets have the same cardinality per Cantor's set theory. However, this can only establish that there is an equivalent _number_ of members in a finite set. In infinite sets, while it does describe just how infinite the set is, it does not have the power to show that the number of members is equivalent. We are dealing with number systems where 7 times infinity equals the same infinity, due to the sheer inability of the system to handle infinities. I have established that things can only be equivalent if their difference is 0. However, I can prove that the difference in number of members of these sets is not zero. By both my opponent's logic and my own, the number of Thursdays IS equal to the number of, say, Sundays. Which is also equal to the number of Mondays, and the number of Tuesdays, and the number of Wednesdays, and Fridays, and Saturdays. So now, we have 7 infinite sets, each with equivalent numbers of members. Thur = number of Thursdays FriWed = number of all other days combined. This is also an infinite value. Days = number of all days. For the number of days and the number of thursdays to be equivalent, Days  0 = Thur. However, this is false. It is undeniably true, by both my opponent's logic and my own, that Days  FriWed = Thur That is to say, # of days  # of all days that aren't Thursdays = number of Thursdays Thus, we see that Days 0 = Thur must be false. I just showed, as I did last round, that the difference between the two is infinite. The difference cannot be both infinite and zero at the same time, and so they are not equal. My opponent stresses that simply because we can correspond the two, they must have equivalent numbers of members. I can't give an example of a # for which there is a day but not a Thursday, so they must be the same size, right? This is only valid logic for finite numbers. To ask me to produce a # where there is a day but not a Thursday is nonsensical, since as we know, there are infinite values in both groups. Since I am only capable of naming finite values, due to limitations of math in dealing with infinities, of course I cannot name an element number where there is a day but not a Thursday. However, I will do the best I am able: Let us define a certain infinity as w. We will say that this is the infinite length of time for which the universe will exist. After w days, w/7 Thursdays will have passed, also an infinite number. There is no 'w'th Thursday, since there are only w/7 Thursdays. I hope that made sense, but if it did not, fault our crippled ability to work with infinities. However, to say that since infinities seem impossible to define well that we should simply break math completely on them is a travesty. Sure, it's what we have to do with real numbers [and all the subgroups thereof], but that is only because infinity does not exist in these number systems, which is proven by the Archimedean principle. I have already stated that infinity doesn't exist as a real number though, and that went unaddressed by my opponent. It's true that you 'can' make a onetoone correspondence with these sets, but you can also make a onetoseven correspondence with them, which happens to be the way reality works with Thursdays and days. Every seventh day, we can match one day with one Thursday, just like reality. The other days would correspond not to more Thursdays, but to other days of the week. I hope I have sufficiently argued my point, and I thank my opponent for this fascinating debate. 
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I fully accepted that two sets of aleph0 items have the same cardinality namely aleph0, even if combined. While it may be mathematically a moot point to speak of coefficients and one cold likewise argue for any (rational) coefficient (rational in that it can be stated as a fraction unlike pi or e), because you could argue for that particular bijection.
You could argue that in an endless universe, there are twice as many Thursdays as there are days, in general. The issue is that because they are all true, they are all pointless and the rather worthless. Every set of aleph0 values has aleph0 values regardless the nature of the set or whether one set can be said to be a subset of the other. I can easily argue for the truth of the proposition and be correct, even if it doesn't matter much because the stated proposition doesn't mean much of anything mathematically.
I freely concede that a logically sound argument that there are 7 times as many days as Thursdays, could be made along the same lines as my argument, but that doesn't mean my argument is incorrect, it means that the importance of my claim is nonexistent.
In your first post there, you stated:
... = 1/3d = 1/2d = t = d = 2d = 3d = 4d = 5d = ...
Note the middle section "... t = d ....", that's the proposition and you're noting that it's true. Therefore, Pro is correct.
I freely concede the topic of the debate is rather irrelevant with regard to what we mean by infinity, and that cardinality shows that such points are rather without relevancy to anything. But, I didn't argue the topic was very important, I argued that it was true And it is true, even if that doesn't mean much.
The reason being that by the logic of this statement, there are also as many Thursdays as months, years, decades, centuries, millennia, hours, seconds, etc. This result gives us no useful notion of the size of an infinite set. Consequently, it can only make sense to use cardinalities to determine whether an infinite set is countable or not. In other words, we can only draw the conclusion that the set of Thursdays and and the set of days each have the same cardinality, and they are countably finite. If you were to read deeper into Set Theory and Cantor's work this is what you will find.
The notion of cardinality changes as we move to infinite sets. There is no sense to the statement that two infinite sets have the same number of elements. Whereas with two finite sets of the same size we can form a bijection between them to determine that they have the same number of elements, we cannot do the same for infinite sets (as shown in my previous comment).
So when we look at infinite sets we use the notion of cardinality in a different sense. Whereas previously it was used to measure the number of elements in a set, it is now used to determine what type of infinity that a set is. This may seem a strange notion, but there are in fact several different sizes of infinity.
The two simplest types are, countable infinity and uncountable infinity. A countably infinite set is one in which the elements can be listed in a line, and an example is the set of days.
Set of days = {day 1, day2, day 3, ...}
An uncountably infinite set is an infinite set which can not be listed in this way, and an example is the set of real numbers. (see Cantor's diagonalization argument for a proof that the real numbers are uncountable.)
So we have that countable infinity gives us one type of infinity, and is itself a "size" or cardinality. We also have that the test for showing two infinite sets have the same cardinality is to show that there exists a bijection between the two sets. Consequently, if such a bijection exist then there the two sets are of the same cardinality.
It should be clear that the set of Thursdays and the set of days both have the same cardinality, they are both countably infinite sets. I hope it is also clear that this is the only way that we can sensibly compare the size of these tow sets.
In conclusion, my answer, and the mathematically correct one, is that the statement "Assuming an endless universe, there are as many Thursdays as there are days" is incorrect. The re
The message from Pro is that if two infinite sets have the same cardinality then they contain the same number of elements. In formulating your hypothesis that the number of Thursdays is equal to the number of days in an infinite timeline, you are suggesting that should we divide the number of Thursdays by the number of days we get 1. I am sure you can see that this is a silly proposition as no sense can be made of dividing two infinities. I hope that you also recognise that in dividing these numbers and getting 1 is the same as Con dividing the two sets and getting 1/7.
The above is a consequence of the fact that there is ambiguity in choosing a bijection between the set of Thursdays and the set of days. Pro, you have chosen to associate Thursday of week x to day x, yet there is no reason for doing this as one could also associate Thursday of week x to day day 2x. This would give the result that there are twice as many Thursdays as days. In fact, you could choose to have as many Thursdays to days as you wish.
As a result you have the following as true:
If number of Thursdays = t,
and number of days = d,
Then,
... = 1/3d = 1/2d = t = d = 2d = 3d = 4d = 5d = ...
One should be able to see that this result is absurd, and mathematically useless. The consequence is that there is no meaning to the notion that there are as many Thursdays as days. But Pro, please take comfort in that the same could be said of the number of Thursdays being 1/7 of the number of days.
I will now explain a briefly what Cantor meant by two infinite sets having the same cardinality, and what exactly cardinality is.
Cardinality gives the notion of the size of a set. When dealing with finite sets, cardinality is the same as the number of elements in each set, if they are the same then they have the same cardinality.
However, most of the rest of your comment is quite accurate.
I would go Pro for the reasons below.
Con's problem is that he attempts to look at infinity in a finite fashion.
"after w days (where w is infinity) have passed".... that is flawed. Since w is infinity there will never be an "after w days".
Con has conceded in the last paragraph of the debate that you can make a 1:1 cardinality for these two sets. Therefore, in his very conclusion to the argument he has in fact endorsed Pro's proposition.
One way to deal with this is the following:
1. Take out all the thursdays from now to infinity and put them in a separate set called T {1,2,3,4,5....}
2. Keep the remaining set of days from now to infinity as it is and name it D{1,2,3,4,5.....}
3. Place the two sets side by side.
4. You will see that for each T there is one and only one D.
Yes, it's hard to put your head around it. That's because us humans aren't really used to infinities ;)
I can't vote yet but if I could I'd go Pro.