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I contest that infinity != infinity-1. I base this on an assumption that there is more than one "infinity". Additionally, I will contest infinity != infinity ne

I contest that infinity != infinity-1. I base this on an assumption that there is more than one "infinity". Additionally, I will contest infinity != infinity necessarily. If we count on the number line from 1 to X, never stopping, that would be an infinite process. If we count on the number line from 1 to 2 by using fractional numbers that infinitely decrease, that is also an infinite process. However, both these infinite outcomes involve different components (different number values) and as such I consider them "different" infinities. I admit this is a fairly week argument, and much closer to "intuition" than my original comment. But, can you provide a mathematical proof that infinity = infinity-1?

jerf 16y ago
One infinity is not necessarily equal to another. You are correct. Equality is not a very useful concept for infinities, really, you need other concepts like one-to-one mappings. Take one away from your infinity and you haven't done anything to its cardinality. A one-to-one mapping between the old infinity and the new one is still trivially available. Words are useless. English is absolutely incapable of dealing with infinities in math, full stop. If you are arguing with English words, you're already wrong. That is why I said that I am presenting one useful way to think of them, and it is absolutely not the best or only way. What I showed was just a somewhat intuitive way to think of the infinity most commonly encountered in otherwise-conventional math and isn't totally correct there, either, it's just intuitively closer. Arguing about my intuitive shortcut misses the point entirely. You must go learn the math. I can't use the math terms to explain it to you here, either, because the correct understanding of the terminology would be tautological to understanding my point. Also, I am not trying to be mean here. It took mathematicians decades to work out consistent definitions of infinity, and if you are going to insist on banging on the definitions and exposing the inconsistencies, an admirable enterprise, the only definitions that will stand up to that scrutiny are the ones that have been honed over the decades, and they do not fit well into HN posts. (Even if they are posted, they are thoroughly based on other mathematics that they would be meaningless until you understand those other things.) There's no Royal Road to infinity.
- giberson 16y ago
  Very good points. Obviously, math is a language in itself and trying to make arguments from the trappings of an inexact translation to another language like English is folly. >I am not trying to be mean here No worries, no maliciousness interpreted. Thanks,
RiderOfGiraffes 16y ago
This is not the place to give a complete introduction to transfinite arithmetic. However, let me try to give you a sense of what's happening. You would agree that a collection of three cars is not the same as a collection of three cats, and yet you would agree that 3 equals 3. Just because we are counting different objects, that does not mean that the number of objects is necessarily different. Specifically, we regard two collections as being the same size if we can put them in one-to-one correspondence with nothing left over. When we are talking about the size of a collection, it doesn't matter what is in the collection. Similarly when we talk about the collection of positive even numbers and the collection of positive whole numbers. The collections are different, but it's possible to put each into one-to-one correspondence with the other. In this sense we feel that it's natural, right and appropriate to say that the two collections are the same size. Using the same reasoning we can show that the collection of positive whole numbers (all of them) is the same size as the collection of positive even numbers. We pair off each number with its double, and that gives us a one-to-one correspondence. Given that such a correspondence exists it would be perverse to claim that the collections are of different sizes. The size of each is the same "infinity", even though the objects themselves are different. And so we proceed. We can show that all sorts of collections are (in some cases surprisingly) the same size. Primes, even integers, rationals, algebraic numbers, all are collections that can be put into on-to-one correspondence. Having become comfortable with that, it then comes as a shock to discover that there are collections that have infinitely many objects, but which cannot be put into one-to-one correspondence. If you try to pair up the real numbers with the positive whole numbers you always have some reals left over. Always. The generally accepted conclusion is that the collection of real numbers is genuinely bigger. And so to your closing question. Suppose we (quite reasonably) define X-1 as "take a collection of size X, remove one object, and X-1 is the number of objects left." If you start with an infinite set (any infinity, and any example) and remove one element, then the result can be put into one-to-one correspondence with the original. This can be proved. It's not hard, but it's a bit icky, and I won't do it here. But under reasonable definitions of the terms, yes, I can provide a proof that infinity-1=infinity. I hope that helps.
- giberson 16y ago
  Thanks! I'd like to say !Ding, I get it--but that wouldn't quite be accurate. Suffice to say, I understand how you can use a one-to-one correspondence method to proof two collection's equality--despite not really understanding how you are generating those values (as its clearly above my math level). Therefore, from now on, I will at least not use the argument infinity!=infinity-1 when attempting to reason some conclusion. Thanks again, for taking the time to address my questions.

jerf 16y ago

One infinity is not necessarily equal to another. You are correct. Equality is not a very useful concept for infinities, really, you need other concepts like one-to-one mappings. Take one away from your infinity and you haven't done anything to its cardinality. A one-to-one mapping between the old infinity and the new one is still trivially available. Words are useless. English is absolutely incapable of dealing with infinities in math, full stop. If you are arguing with English words, you're already wrong. That is why I said that I am presenting one useful way to think of them, and it is absolutely not the best or only way. What I showed was just a somewhat intuitive way to think of the infinity most commonly encountered in otherwise-conventional math and isn't totally correct there, either, it's just intuitively closer. Arguing about my intuitive shortcut misses the point entirely. You must go learn the math. I can't use the math terms to explain it to you here, either, because the correct understanding of the terminology would be tautological to understanding my point. Also, I am not trying to be mean here. It took mathematicians decades to work out consistent definitions of infinity, and if you are going to insist on banging on the definitions and exposing the inconsistencies, an admirable enterprise, the only definitions that will stand up to that scrutiny are the ones that have been honed over the decades, and they do not fit well into HN posts. (Even if they are posted, they are thoroughly based on other mathematics that they would be meaningless until you understand those other things.) There's no Royal Road to infinity.

giberson 16y ago
Very good points. Obviously, math is a language in itself and trying to make arguments from the trappings of an inexact translation to another language like English is folly. >I am not trying to be mean here No worries, no maliciousness interpreted. Thanks,

RiderOfGiraffes 16y ago

This is not the place to give a complete introduction to transfinite arithmetic. However, let me try to give you a sense of what's happening. You would agree that a collection of three cars is not the same as a collection of three cats, and yet you would agree that 3 equals 3. Just because we are counting different objects, that does not mean that the number of objects is necessarily different. Specifically, we regard two collections as being the same size if we can put them in one-to-one correspondence with nothing left over. When we are talking about the size of a collection, it doesn't matter what is in the collection. Similarly when we talk about the collection of positive even numbers and the collection of positive whole numbers. The collections are different, but it's possible to put each into one-to-one correspondence with the other. In this sense we feel that it's natural, right and appropriate to say that the two collections are the same size. Using the same reasoning we can show that the collection of positive whole numbers (all of them) is the same size as the collection of positive even numbers. We pair off each number with its double, and that gives us a one-to-one correspondence. Given that such a correspondence exists it would be perverse to claim that the collections are of different sizes. The size of each is the same "infinity", even though the objects themselves are different. And so we proceed. We can show that all sorts of collections are (in some cases surprisingly) the same size. Primes, even integers, rationals, algebraic numbers, all are collections that can be put into on-to-one correspondence. Having become comfortable with that, it then comes as a shock to discover that there are collections that have infinitely many objects, but which cannot be put into one-to-one correspondence. If you try to pair up the real numbers with the positive whole numbers you always have some reals left over. Always. The generally accepted conclusion is that the collection of real numbers is genuinely bigger. And so to your closing question. Suppose we (quite reasonably) define X-1 as "take a collection of size X, remove one object, and X-1 is the number of objects left." If you start with an infinite set (any infinity, and any example) and remove one element, then the result can be put into one-to-one correspondence with the original. This can be proved. It's not hard, but it's a bit icky, and I won't do it here. But under reasonable definitions of the terms, yes, I can provide a proof that infinity-1=infinity. I hope that helps.

giberson 16y ago
Thanks! I'd like to say !Ding, I get it--but that wouldn't quite be accurate. Suffice to say, I understand how you can use a one-to-one correspondence method to proof two collection's equality--despite not really understanding how you are generating those values (as its clearly above my math level). Therefore, from now on, I will at least not use the argument infinity!=infinity-1 when attempting to reason some conclusion. Thanks again, for taking the time to address my questions.