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0 pointskazinator9y ago0 comments

So you're saying a countable infinity can be occupied, so that there is no room for one more.

You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.

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lisper9y ago

> So you're saying a countable infinity can be occupied, so that there is no room for one more.

No. I'm saying that a countable infinity doesn't have room for one more at the end. There is no end.

You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...

Twisol9y ago

You might be able to do something funny with a series whose terms are indexed by ordinals. The last `1` would then be indexed by `omega` (the first transfinite ordinal). I can't claim that the ordinary rules of arithmetic would make much sense, but it's certainly an idea that you can attempt to formalize.

lisper9y ago

It has been done:

https://en.wikipedia.org/wiki/Infinitesimal

nilkn9y ago

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended.

(This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rationals can be made into a sequence, but not with the same ordering.)

lisper9y ago

No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?

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lisper9y ago

> So you're saying a countable infinity can be occupied, so that there is no room for one more.

No. I'm saying that a countable infinity doesn't have room for one more at the end. There is no end.

You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...

Twisol9y ago

lisper9y ago

It has been done:

https://en.wikipedia.org/wiki/Infinitesimal

nilkn9y ago

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended.

lisper9y ago

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