I am going to prove what infinity divided by infinity … Suppose we have two infinite sets and we show that there cannot possibly be a bijective function between the two, thus showing that their cardinalities are different. Indeed, there is no sense in which this is even a totally reasonable question to ask. So I guess I’m essentially rephrasing what was said in Lesson 5: the bijectiveness implies equality as that’s the only time when something is neither lesser nor greater than the other thing (in ordered domains or w/e). For weird kinds of multiplication, read on into the later lessons when we learn about groups And as for the amount of intuition we use, think of it as follows. Assign the next even number, which is 8, to this square. Definition of infinity in the Definitions.net dictionary. Two halves of the symbol can signify two sides inside one person. Infinity symbol tattoo is attractive for its plain design and also for its limitless possibilities to make it look unique. Since we haven’t assigned anything to 3 yet, we assign the next even number, which is 10, to 3. Any pointers for these? We use intuition to motivate our definitions, because we can make whatever definitions we want. TBH, I found quantumlotus’s explanation a bit hard to follow (not much experience with math, but loving the lessons so far! Secondly, we recall that and and and (etc.) In this sense, FRAC appears to have “infinity times infinity” elements. The problem with FRAC is that there is no meaningful order to them. So, we use bijection to show that the seemingly larger set of fractions is of the same (infinite) size as the set of integers. After all, any number divided by itself is equal to one, however infinity is not a real or rational number. It was first used by John Wallis in 1655. Namely, the negative numbers in B have a kind of order, as do the positive numbers of B. The real numbers therefore have the “larger” cardinality because we can define a surjective function from the real numbers to the natural numbers, but we can’t define a surjective function from the natural numbers to the real numbers (if we could, then we’d simply “get rid of the duplicates” and therefore have a bijective function, which we just showed was impossible). This Latin word itself originated from the Greek word “apeiros”, meaning “endless”. It can symbolize joining of the two halves of a person into one body, in a “complete” person. Clearly figure 1 only shows a small portion of the entire infinite chart, primarily because it would take me too long to write out the whole thing! What does infinity mean? Zomboss sends the player to Infinity in an attempt to defeat the Gnomes and their leader, Gnomus the Gnome King. Actually, what we’ll show, is that “infinity type 1 times 2 is infinity type 1”. > Note also that this also applies to finite sets: if the cardinalities are different, we can only define a surjective function from one to the other, and not vice versa, but if their cardinalities are the same (so that we have a bijective function between the two), then there is a surjective function both ways. Now head diagonally down to the left, and land on the square containing . Information and translations of infinity in the most comprehensive … Then to prove things about those definitions we need completely rigorous logic. 1 2. superjake3001. That said, the reason we think this definition is most meaningful is that it aligns perfectly with how we’re used to counting finite sets, intuitively. If we call this function from A to FRAC “F”, then we see that and similarly (mapping to the negative fractions). Moreover, since we’re skipping duplicates, this function is perfectly injective. Thanks for adding your explanation though, as you never know which explanation will stick for which readers. In the function from to , we were able to exploit the fact that B has a sort of order to it (as does A). > Indeed, there are some fascinating proposals for strange types of numbers and However, similar symbol was used by Romans to express “large number”. Implicit in this statement is the fact that any whole number is itself a fraction merely by putting “1” downstairs. Namely, if we first know that two sets are not equal, then we know that one is strictly “larger” than the other (as in cardinality), in the same way as for numbers, say. Double infinity symbol tattoos are famous as well.Double infinity symbol is created by entwining two infinity symbols on top of each other. Infinity times infinity is infinity. Then consider injectiveness to imply the property of the domain being ‘lesser of equal than’ the co-domain (in terms of cardinality); as for each element of the domain, there is a distinct element of the co-domain and possibly more to follow. Moreover, the initial two fractions that we chose in this process could have been any of the infinitely fractions that exist! The farther into the chart we go, the more this function “jumps around”. The point is, though, that once one makes the definitions that we have made, the results and the structure that we have found regarding infinity truly is there, and there’s no room to maneuver. Thanks a lot. Note, for example, that 1 shows up on every diagonal entry because and so on. Change ), You are commenting using your Twitter account. Two sets of circles, on top of each other show perfect balance between the partners where they can be together side by side, independently as well as intertwined. )ness’; and that’s true as all the elements of the co-domain are mapped to by one or more elements of the domain. Then the resulting function would be surjective, but not injective, implying that the cardinality of fractions is less than the cardinality of natural numbers. whereas e^(-infinity) can always be written as (1/e)^infinity and for numbers between (0,1) , raised to infinity is always 0 . At first, you may think that infinity divided by infinity equals one. In fact, this chart contains more than every positive fraction, since it contains every fraction as well as all of its duplicates! This game mode also reveals interesting facts about the Gnomes' job in controlling time. Let us now construct this function. Given two distinct fractions we are of course able to say which is bigger, but suppose that you give me a fraction and ask me what the “next biggest” fraction is. e^(infinity) is always gonna be infinity !!!! Thanks for this great explanation, but the chart about fractions seems a little paradoxical. This lesson will be devoted to showing that this vague idea of “infinity times infinity” is in fact the same as infinity type 1. Now, we could indeed define a SURJECTIVE but not INJECTIVE function from the even numbers to the whole numbers, simply by sending, say, 2, 4, 6, 8, and 10 to 1 (thus making it not injective), and then sending 12 to 2, 14 to 3, 16 to 4, 18 to 5, and so on. It’s the same argument only from a sort of backwards viewpoint, but using essentially the same facts. It shows that every single individual is infinite in its nature and the combinations of the individuals without any predetermined ending makes the world infinite. Now take a step to the right. We then call the positive whole numbers across the top “numerators” and those going down the left “denominators”. It also stood for the unity of a man and a woman. Recall that in lesson 11 we saw how the set of fractions (which we denoted by FRAC) is so extremely infinite. It is unlocked in the last quest of the single player mode of each side. I get it now: My statement would be valid only if there was no bijective function between them… this is brilliant. The infinity symbol shows the variety of the possible versions of every single day in our life and what could have happened even if one of the decisions we made would have been different. In order to get a bijective function from natural numbers to fractions, we skip over the “duplicates” such as 2/2. ( Log Out / But wait! It is often our intuition that helps guide us through the proofs, but at the end of the day it is rigorous logic that we need for the proofs. In mathematics, infinity is the conceptual expression of such a "numberless" number. If the cardinalities are different, only one of the two sets can have a surjective function defined from it to the other one. Firstly, we recall that every fraction is nothing but “some whole number upstairs, and some (non-zero) whole number downstairs”. So enough with the fluffy philosophy—let’s do some math. Namely, our definition of what it means for a set to have a certain number (finite or infinite all the same) of elements in it is precisely that—a definition.