 Okay, so here's this premise, if a runner completes a race course, then the runner has crossed a distance. If we reject this proposition, this premise, we are committed to the logical contradictory, which is a runner completes a race course, but the runner has not crossed a distance. I don't know how this could be anything other than a logical contradiction. Maybe if we really abuse the term race course to mean something other than a length a distance for people to run across, that's the only way this can't be a logical contradiction. But the minute that we say a race course is a distance for people to run across, well then, alright, just say that a runner has crossed a race course, but has not crossed a distance is a logical contradiction. Okay, so here's the premise, if the runner has crossed a distance, then the runner has crossed an infinite. If we reject this, we say the runner has crossed a distance, but the runner has not crossed an infinite. Well, so you might be very quick to reject this one. You know, you say, well, yeah, okay, let's say it's 100 meters, right? The race course is 100 meters. If the runner starts here and runs 100 meters over there, right, there's 100 meters, that's a finite distance. So it has not crossed an infinite, but you know, the mathematics here is sound, right? There is an infinite from the beginning to the end, right? What's halfway of 50 meters? I mean, you have to cross that halfway, well, 100 meters, well, that's 50 meters. And to cross halfway again, that's 75 meters. And to cross halfway again is what, 97 and a half, no, 87, 87 and a half meters, right? So here's the fraction, right? Here's how it works. You start, you know, at the starting point, and to get halfway across, you have to cross one half. To finish to the end, you have to cross three quarters. That's the next halfway point. Oh, sorry, yeah, three quarters. To finish to the end, you have to cross seven eighths. To finish to the end, you have to cross 15 sixteenths. To finish to the end, you have to cross 31, 32, 31, 30 seconds. To finish to the end, you have to cross 63, 64. To finish to the end, you have to cross 127 out of 128. The formula is actually pretty simple. You just take the denominator, you multiply it by two, right? So that tells you the next denominator, the next halfway point. And then for the numerator, you just simply take that denominator and minus one. Now you follow this formula and you can create an infinite set of fractions, right? Without end, there will be a one-to-one correspondence between each fraction and each number on the number line that constitutes an infinite. It actually gets worse than that, right? In order to get to the halfway point, you have to cross halfway. So you have to get past the quarter. To get to the quarter, you have to cross halfway. So you have to cross one eight. In order to cross one eighth, you have to cross one sixteenth. You have to cross one 32nd. You have to cross one 64. You have to cross 128. You have to cross 256. And it keeps going. not only do you to finish the race course you have to cross an infinite to get started you have to cross an infinite and this is just a geometry folks I'm not just making this up this is just a clever trick that Zeno thought of right this is geometry between any two points there's an infinite number of points it gets even worse than that take any two point right take any two points in the distance say one quarter to one half to get from the one quarter one half not only it to cross the one eighth right to get to the one it's across an infinite from the one quarter to the one eight there's an infinite number of fractions between that one too so yeah the mathematics here is sound it's not awesome you know it's not just making stuff up if you cross if the runner crosses a race course the runner has crossed an infinite you don't get to say the runners cross the race course and not an oops all right so this next premise it's impossible for a runner to cross an infinite now if we're gonna reject this we have to say it is possible that the runner has crossed an infinite and you say okay this is the way to go and Zeno goes oh come on you can't cross an infinite you don't have a you don't have enough time you have to keep going to the next step to keep going to the next step to keep going to the next step well here's a little clue folks here's where Zeno has made a mistake because that sense of infinite is different than what he was dealing with earlier right take any distance say a centimeter I can cross a hundred centimeters right so that thing you know I have crossed a hundred centimeters no problem but in order for me to cross say an infinite number of centimeters I have to keep going now in that case yeah it will never stop I will never reach the end of an infinite set of centimeters but that 100 centimeters are crossed that also has an infinite but it's a different kind of infinite the infinite that's impossible for you to cross is an infinite expanse that's where you have a fixed distance and an infinite number of them the distance the infinite that you cross when you cross a hundred centimeters that's not a fixed distance that's an infinite divisible so this is where Zeno has made a mistake and he actually has made a mistake here yeah you can cross an infinite divisible but you can't cross an infinite expanse right that you can't do so the error the Zeno has committed here and this is probably one of the old times over talk about an error in this course with philosophy the error the Zeno has committed here is equivocation that's where you have one term that means two different things and he uses the term infinite to talk about infinite expanse and an infinite divisible but these are two different kinds of infinite they are not equivalent to each other in fact if you start getting in mathematics the infinite expanse where we're dealing with a set of all counting numbers is smaller than the infinite of the divisible even though they're both infinite that's fun stuff isn't it so this is one of the only times as you know this is more than a few times I'll say that there's actually a mistake here right there's a real logical error Zeno has committed the logical error of equivocation when equivocates between an infinite expanse and an infinite divisible now I'm not going to talk about the other paradoxes that Zeno comes up with at least not here so you know if you think that Zeno has indeed made an error with the other paradoxes it's up to you to figure out where the error is and if he hasn't found an error well I mean I sorry if he hasn't committed an error with the other paradoxes well then he's found some deep problems with common sense and common sense probably doesn't you know give a good objection to parminities argument or if you say that it nevertheless does like okay well parminities argument results in absurdities common sense results in absurdities you're just siding with common sense but why if you don't have a why that's the abandonment of reason not the use of it