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

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 ill-defined 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 one-to-one correspondence with these sets, but you can also make a one-to-seven 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.