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text: '[summary: [3jz Collections of things] which are the same [4w5 size] as or smaller than the collection of all [-45h natural numbers] are called *countable*, while larger collections (like the set of all [-4bc real numbers]) are called *uncountable*.\n\nAll uncountable collections (and some countable collections) are [infinity infinite]. There is a meaningful and [5ss well-defined] way to compare the sizes of different infinite collections of things, and some infinite collections are larger than others.]\n\n[3jz Collections ] which have less than or the same [4w5 number of items] than the collection of all [-45h natural numbers] are called *countable*, while larger collections (like the set of all [-4bc real numbers]) are called *uncountable*.\n\nAll uncountable collections (and some countable collections) are [infinity infinite] and some infinite collections are larger than others %%note: At least, within mathematical systems which include the [69b], see the [4zp technical] page for details.%%. To demonstrate this, we'll explore a graphical demonstration with tiles and paths.\n\n[toc:]\n\n## Tiles and paths\n\n\n\nConsider, as shown above, a sidewalk that goes on forever in one direction, which is made up of equal-sized square tiles. The sidewalk is two squares across. Consider a person who walks forever on it, obeying the following rule: Each step the person takes must be to one of the two tiles immediately in front of that person; no going backwards, no skipping tiles, no going sideways, no standing in place forever. The following is the beginning of one possible path:\n\n\nNow let's ask two questions:\n\n 1. How many tiles are there?\n 2. How many possible paths are there?\n\nIn both cases, you could just say that there are infinitely many, and that would be correct. But now let's consider a third question:\n\n 3. Is the number of tiles the same as the number of possible paths?\n\nIt turns out that there is a meaningful and [5ss well-defined] way to compare the sizes of different infinite [3jz collections of things], and some infinite collections are larger than others. In particular, some infinite collections are *countable* (like the [3jz set] of all [-45h]s), while others are *uncountable* (like the set of all [-4bc]s). As we will see, it can be shown that the number of tiles on our infinite sidewalk is countable, but that the number of possible paths one could take, following the rules above, is uncountable. So there are in fact *more* possible paths than there are tiles.\n\nLet's dig into exactly what this means and why it's true.\n\n## Pairing off\n\nWe say that two collections of things are the "same size" if you can match the items together completely: you can pair each of the things in the first collection with exactly one of the things in the second collection, in such a way that there is nothing left unpaired. For example, given two sets of three things each, we may pair them. Here is an example of such a pairing:\n\n\n \nYou might think it obvious, then, that the number of paths our person can walk is bigger than the number of tiles. We can match each tile with the path that starts on a tile the same color as it, and changes to the other color after it hits this tile. For example, we would match the third red tile with the path\n\n\nIt is important to note, however, that it is not sufficient that we find some matching that leaves things left over. We must show that *every* matching leaves things left over. For example, an infinite sidewalk that is one tile across has just as many tiles as an infinite sidewalk that is two tiles across, as we can see from the picture below by matching the 1R on top with the 1R on bottom, the 1B on top with the 1B on bottom, the 2R on top with the 2R on bottom, and so on.\n\n \n \n\nIn fact, if we were only to require that *some* matching leave extra tiles, then the number of tiles in a sidewalk that is one tile wide would not be equal to itself, for we could match the first tile with 1B (in the bottom picture above), the second tile with 2B, and so on, and we would leave over half the tiles!\n\nIn fact, even if we had a *field* of tiles that is infinite in every direction, we would still have no more tiles than if we had only a sidewalk that is one tile across. The following matching shows this:\n\n\n\n\n## An unpairable path\n\nYou might wonder, given that there are so many different ways to match up infinitely many things, how we can know that there is no matching that catches everything. I will now prove that, no matter how you try to match paths (ways of walking) and tiles, you will miss some paths. Since we have already seen that the number of tiles in a sidewalk two tiles wide is the same as the number of tiles in a sidewalk one tile wide, I will show that any matching between paths and tiles in a sidewalk one tile wide misses some paths. I will do this by creating a path that does not match the path we have chosen for any tile %%note: This type of proof is known as a [-46z].%%.\n\nSuppose we are given a matching between tiles and paths. Since we have numbered the tiles in a sidewalk one tile wide ($\\fbox{1}\\fbox{2}\\fbox{3}\\fbox{4}\\fbox{5}\\fbox{6}\\fbox{7}\\fbox{8}\\overline{\\underline{\\vphantom{1234567890}\\cdots}}$), we also have a numbering of the paths in our matching. Consider a new path that differs from the [nth $n^\\text{th}$] path in our matching on the $n^\\text{th}$ tile, that is, the $n^\\text{th}$ step that you take. For example, if our first eight paths are\n\n\n\nthen our new path is\n\n\n\nClearly, this path is not any of the ones in the matching, because it differs from every single path at some point (in particular, it differs from the $n^\\text{th}$ path on the $n^\\text{th}$ tile, the $n^\\text{th}$ step you take, which is highlighted in yellow).\n \nBecause we can repeat this exercise no matter what matching we're given, that means *any* possible matching will always leave out at least one path. Thus, the number of paths a person can take must be strictly larger than the number of tiles in the sidewalk.\n\n## See also\n\nIf you enjoyed this explanation, consider exploring some of [3d Arbital's] other [6gg featured content]!\n\nArbital is made by people like you, if you think you can explain a mathematical concept then consider [-4d6]!',
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