Assuming an endless universe, there are as many Thursdays as there are days.

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 counter-intuitive 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