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I can define a set as all possible numbers.

And then you cannot tell me there is an additional number outside the set.

Perhaps (I'd actually dispute that by asking you precisely what type of numbers are in your set, but I digress).

Regardless, let me be kind and assume that you've got a set of numbers that is uncountable; perhaps you're thinking of, say, the set of all real numbers. You're now quite pleased with yourself because you've got a set with an infinite number of elements.

However, I come along and claim that I can define a set with even more elements in it than yours. I can even be kind to you and say that I'll restrict myself to working with a set containing real numbers. The question therefore is: do you believe that I can construct a set of real numbers that contains more elements than your - already infinitely large - set of real numbers?

Because I can do so quite simply by taking my set to consist of all possible subsets of the real numbers. Both of our sets have infinitely many elements, but mine has more than yours.

BeamishBoy is arguing from the standard construction of numbers and set in modern mathematics which, for somewhat subtle reasons, disallows defining the set of all possible numbers. Due to the simplifying assumption that everything is a set (otherwise you could run out of numbers for talking about the size of sets which is similar to the argument BeamishBoy makes in his sibling post), numbers are defined in such a way that the set of all possible numbers must contain itself and sets containing themselves is disallowed due to https://en.wikipedia.org/wiki/Russell%27s_paradox .

This means that there is obvious way to define the largest infinity and leads to two separate kinds of numbers which act rather differently for infinities:

https://en.wikipedia.org/wiki/Ordinal_numbers for counting things where n != n+1 even n is infinite and https://en.wikipedia.org/wiki/Cardinal_numbers for measuring the size of sets where n == n+1 if n is infinite.

Member since:

2010-09-13

If I were to say to you, that in that I had this infinite set of whole numbers, plus I had the number 1.

I would still only have the infinite set of whole numbers, the number 1 is already included. I still fail to understand, how in the context of a child's game where you are attempting to say the highest number possibile and someone says infinity, how the number infinity plus 1 is larger.

Well, if you find it difficult to understand how this could be so in the context of the natural numbers, try thinking about it using a different set.

Think of the largest possible set you can (the number of elements in your set is something that a mathematician would call the

cardinalityof that set).Each time you come up with a set containing a huge number of elements, I can counter you by constructing a set containing

all of your elements, plus any other element that isn't already in your set. Thus I can construct a set containing an arbitrarily large number of elements; this is, in essence, one type of infinity.No. Infinity is not a number; it's a

concept.I actually think you suffer from an understanding of language.

I can define a set as all possible numbers.

And then you cannot tell me there is an additional number outside the set.

And infinity is a concept - I said that. You said that. You can pretend we are in disagreement, but we are not.