On Fri, May 27, 2011 at 12:49 PM, Lonnie Olson <[email protected]> wrote:
> On Fri, May 27, 2011 at 10:13 AM, Daniel C. <[email protected]> wrote:
>> Some infinities are larger than others. Consider the set of all
>> numbers between 0 and 1 - there are an infinite number of them. (0.1,
>> 0.01, etc.) The set of all numbers between 0 and 2 (or, for that
>> matter, 0 and 1.1) is also infinite, but it's a larger set than the
>> set of numbers between 0 and 1. It is in fact possible for something
>> that is infinite to become larger, while still being infinite.
>
> In your example, your use of the term infinite is in relation to the
> count of numbers, not the sums or values.
> This count is in fact infinite, no upper bound. There are no more
> numbers between 0 and 1 than there are between 0 and 2. This upper
> bound is limitless. You cannot multiply infinity by two, since
> infinity is already fully encompassing that result. There is no value
> greater than infinity.
To expand on this, the words 'infinity' and 'infinite' are used to
describe a number of different, although similar, concepts. When you
get into mathematics, you get precise but different usages of the
term. You get into 'different sized infinities' when dealing with
Cantor's cardinality of infinite sets. The baseline infinite set
would be the size of the set of natural numbers, which he calls
countably infinite. This has cardinality of aleph-naught. Any set
that has more members than can be mapped one-to-one onto the natural
numbers is then of higher cardinality than aleph-naught. The real
numbers, for example, contain all of the natural numbers but also an
infinite number of others between each natural number. They are given
the cardinality aleph-one, and are called 'uncountably infinite'.
The set theoretic notion of infinite sets is not quite the same thing
as the notion of infinity and infinitesimal values in calculus, which
were originally rather fuzzy concepts that they kept around because
they worked so well rather than because they had a strong logical
argument for them. Later efforts were able to put them on strong
theoretical footing through math that I don't really understand.
Anyway, my point is that the question 'is the universe infinite?' is
too vague to give a yes or no answer to, because it's not really clear
what is being asked. Do you mean 'does the universe have an edge?'
Or maybe 'is there a bound to the volume of the universe?' Or perhaps
'Is there a bounded amount of matter/energy in the universe?'. Or
even, 'will the universe ever cease to exist?' The answers to these
questions and the certainty of those answers varies. The link I sent
earlier regarding cosmology has a lot of information to help answer
them.
--Levi
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