How could you possibly go beyond infinity? I am certain you would be able to tell me that there are an infinite number of values which I can substitute in for a.
You could also say that this is an infinite set of numbers. So the second set is smaller than the first, yes? How is infinity bigger than infinity? How is infinity smaller than infinity? Now I am going to introduce a concept which might help make a little more sense of what I have just said: countable and uncountable sets. A countable set is one for which I can say there is a definite first, second, third, fourth… element. I could say 0. Or 1 x 10 ? I could move the tiniest imaginable bit higher up the number line than zero and there would still be a number that lies between this new position and my original starting point.
If instead we look at set 2. Then the third element is 4, the fourth is 6, etc. This is evidently a countable set. Log in Sign Up. Save Word. Definition of infinity. Synonyms for infinity Synonyms eternity , everlasting , foreverness , perpetuity Visit the Thesaurus for More. Examples of infinity in a Sentence The view tapers off into infinity. First Known Use of infinity 14th century, in the meaning defined at sense 1a. Learn More About infinity. Time Traveler for infinity The first known use of infinity was in the 14th century See more words from the same century.
Style: MLA. English Language Learners Definition of infinity. Kids Definition of infinity. I'm talking of course about the infamous catchphrase of Buzz Lightyear in Toy Story or, as Imdb puts it:. It seems a throwaway line, but hidden there is a deep philosophical meaning. It pops up here and there in pop culture and even in random discussions with friends. In this sentence helped father and son to survive a shipwreck.
And let's not forget infinity plus one! And the lovey-dovey couples smooching "I love you infinity times and more". There are probably infinite other examples and more! It sounds paradoxical: how can you go beyond the infinity? It's not possible to reach the end of infinity because it is infinite , so how can one overtake it? One needs to change completely point of view: from potential infinity to actual infinity.
Let me go back to the example of Achilles and the tortoise , just like this blog started. You have the tortoise and Achilles, say, 10 meters apart, Achilles in the back. They run. Achilles is much faster than the tortoise, so when Achilles reached the point where the tortoise was, the tortoise just made one meter. Achilles run that meter, and the tortoise is 10 cm ahead. And so on, and so on. Will Achilles ever reach the tortoise? If you have the mentality above, it will not: you have this infinite succession of states: 10 m distance, 1 m distance, 10 cm distance, This is the so-called potential infinite, potential because you never realize it in full, you just see parts of it.
But we know that Achilles reaches the tortoise, right? We're not that stupid. So what? Well, this is where Cantor shines! He managed to treat in a formal way the actual infinite. You pick all the infinity, you put it in a sack, and you treat it as a whole, completed object.
So you can imagine what happens after it, in this case Achilles overtake the tortoise, turns and makes very funny faces. Think of the potential infinity as an infinity you are walking inside.
Like, just go out now and try to reach the horizon, it will always go on and on and you will never reach it. But the actual infinity is like seeing everything from above, so instead of walking pick a drone and go up, up, up, and you can embrace everything with a single view.
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