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

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 square-rooted 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.