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These kinds of infinity concepts are throughout math and physics. For example, the Universe is "infinite" yet it is also expanding. There is very little that is

These kinds of infinity concepts are throughout math and physics. For example, the Universe is "infinite" yet it is also expanding. There is very little that is logical about infinity itself and it has been the source of much philosophy throughout time. It took a couple thousand years just for mathematicians to theoretically prove that some infinities are larger than others, for instance.

adwn 11y ago
> There is very little that is logical about infinity itself [...] I assume that by "logical", you mean "intuitive"?
lutusp 11y ago
> There is very little that is logical about infinity itself ... On the contrary, infinity is a very logical notion -- much more logical than many other ideas we accept. For example, what value must "m" have for this statement to be true: sum(n:1 to m inclusive, 2^-n) = 1 (i.e. 1/2 + 1/4 + 1/8 ...) What value must "m" have, for this statement to be true: lim(n -> m), (1+1/n)^n = e (base of natural logarithms) For both these examples, for the described outcomes, m must equal infinity. Also, you say, "For example, the Universe is 'infinite' yet it is also expanding", but you also say, "It took a couple thousand years just for mathematicians to theoretically prove that some infinities are larger than others ..." If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding. Imagine a cube of space, in isolation, initially measuring one meter in any of its dimensions. If measurements reveal that this value is increasing over time, one may conclude that the cube is expanding. Now imagine that the cube is embedded in a space consisting of an infinity of such cubes in three spatial dimensions, all increasing in size in the same way. Does the fact that the cube is part of an infinity of such cubes prevent it from expanding? If this were true, it would be an easy test of the existence of an infinite spatial manifold -- but the idea of an expanding infinity is logically consistent.
- drdeca 11y ago
  I think that saying that m "equals infinity" is probably a bit of a simplification. One can say that some values are infinite, such as omega in the ordinal numbers and the surreal numbers, but neither of those values are what is meant in the examples you gave, which are both limits. If you plugged in omega (from the surreals) for those expressions instead of taking the limit, I think you would get a value which is off by around epsilon (i.e. 1/omega). I'm not sure of that but I think that's how it tends to work out. In the surreals one can indeed have a continuously increasing infinite value. But I'm not sure that that is a proper way to describe the "total volume of the universe"? Maybe it is, but idk.
  - lutusp 11y ago
    > I think that saying that m "equals infinity" is probably a bit of a simplification. Yes, it is, I agree. For the limit example, one might instead say, "As m approaches infinity, the result approaches 1." By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things. The other reason I agree is because infinity isn't a number. I'm guilty of compressing the lexical description of a limit to a ... ahem ... finite size. > ... what is meant in the examples you gave, which are both limits. One of my examples was an infinite sum, not a limit per se, but the same argument can be made, i.e. that infinity is addressed without being literally achieved.
    - drdeca 11y ago
      I believe that infinite sums are usually defined in terms of limits, which is why I said that both examples were limits. > By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things. I think I would say that a convergent sequence (most of the time that one cares) does not ever reach the target value (though, eventually constant sequences do, but those aren't interesting), but that the limit of the convergent sequence is equal to / is the target value. As opposed to saying that "the limit approaches". The limit isn't the thing that changes with another thing, but the limit of some changing?
- hellofunk 11y ago
  >If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding. True, and that is the point I was trying to make. My use of the word "logical" was perhaps a bit rushed, as another pointed out, more likely "intuitive" was a better word.
Oletros 11y ago
The Universe is not infinite

adwn 11y ago

> There is very little that is logical about infinity itself [...] I assume that by "logical", you mean "intuitive"?

lutusp 11y ago

> There is very little that is logical about infinity itself ... On the contrary, infinity is a very logical notion -- much more logical than many other ideas we accept. For example, what value must "m" have for this statement to be true: sum(n:1 to m inclusive, 2^-n) = 1 (i.e. 1/2 + 1/4 + 1/8 ...) What value must "m" have, for this statement to be true: lim(n -> m), (1+1/n)^n = e (base of natural logarithms) For both these examples, for the described outcomes, m must equal infinity. Also, you say, "For example, the Universe is 'infinite' yet it is also expanding", but you also say, "It took a couple thousand years just for mathematicians to theoretically prove that some infinities are larger than others ..." If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding. Imagine a cube of space, in isolation, initially measuring one meter in any of its dimensions. If measurements reveal that this value is increasing over time, one may conclude that the cube is expanding. Now imagine that the cube is embedded in a space consisting of an infinity of such cubes in three spatial dimensions, all increasing in size in the same way. Does the fact that the cube is part of an infinity of such cubes prevent it from expanding? If this were true, it would be an easy test of the existence of an infinite spatial manifold -- but the idea of an expanding infinity is logically consistent.

drdeca 11y ago
I think that saying that m "equals infinity" is probably a bit of a simplification. One can say that some values are infinite, such as omega in the ordinal numbers and the surreal numbers, but neither of those values are what is meant in the examples you gave, which are both limits. If you plugged in omega (from the surreals) for those expressions instead of taking the limit, I think you would get a value which is off by around epsilon (i.e. 1/omega). I'm not sure of that but I think that's how it tends to work out. In the surreals one can indeed have a continuously increasing infinite value. But I'm not sure that that is a proper way to describe the "total volume of the universe"? Maybe it is, but idk.
- lutusp 11y ago
  > I think that saying that m "equals infinity" is probably a bit of a simplification. Yes, it is, I agree. For the limit example, one might instead say, "As m approaches infinity, the result approaches 1." By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things. The other reason I agree is because infinity isn't a number. I'm guilty of compressing the lexical description of a limit to a ... ahem ... finite size. > ... what is meant in the examples you gave, which are both limits. One of my examples was an infinite sum, not a limit per se, but the same argument can be made, i.e. that infinity is addressed without being literally achieved.
  - drdeca 11y ago
    I believe that infinite sums are usually defined in terms of limits, which is why I said that both examples were limits. > By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things. I think I would say that a convergent sequence (most of the time that one cares) does not ever reach the target value (though, eventually constant sequences do, but those aren't interesting), but that the limit of the convergent sequence is equal to / is the target value. As opposed to saying that "the limit approaches". The limit isn't the thing that changes with another thing, but the limit of some changing?
hellofunk 11y ago
>If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding. True, and that is the point I was trying to make. My use of the word "logical" was perhaps a bit rushed, as another pointed out, more likely "intuitive" was a better word.

Oletros 11y ago

The Universe is not infinite