Notesale: Turn your study into money

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The concept of infinity

By Jesper Meijerink
What is the highest number you can count? Is it one quadrillion, a one followed by 15 zero’s, or is
it a googolplex, a one to the power of ten to the power of one hundred
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But that’s incorrect, infinity is not a number
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For
example: you need infinite number to talk about amounts that are unending
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Before I can explain how you can count past infinity you need to know a few things, you might
already know this but when you don’t, this will be really complicated
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Natural numbers are zero, one, two, three and so on
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Cardinal numbers are number that refer to a quantity
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But how
many natural number are there? It can’t be some number in the naturals because there would
always be that number plus one
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Aleph is the first letter of the Hebrew
alphabet and aleph-null is the first and also smallest infinity there exists
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A set has cardinality aleph-null if and only if it is a countable infinite, so
there must be a bijection (one-to-one correspondence) between it and the natural numbers
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Because we can also count past Aleph-null
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But what if you add one line after the previously made unending string of lines
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You just start at the new line and then move back to the unending string of lines
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But what if we change the
way we counted
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What would that next line be called? It can’t be a natural number because we used
those in that unending string of lines
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In
order to reach that next line we need a new set of labels that extends beyond the naturals
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Ordinal numbers refer to numbers that indicate the position
or order of things
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Omega is the next label you’ll need after
using the infinite collection of every single counting number first
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The order type of a set is the first number not needed to name everything in the set
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So if you have 4 apples the
cardinality is 4 and the order type is also 4
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The
order type of all the naturals is omega but the cardinality is aleph-null
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But remember that these numbers refer to the same
amount of stuff just arranged differently, so omega plus one isn’t bigger than omega but it just
comes after
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Because statistics show that there are always infinities
bigger than aleph-null
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To find that out we use Cantor’s diagonal
argument, because Cantor’s diagonal argument shows us that the power set of aleph-null is
bigger than aleph-null itself
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For example, the set (1,2,3)
can be reconfigured into 8 different sets including the empty set
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So what would be the
power set of aleph-null
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You
will need to use diagonalization and even once you have used it and put the new subset back in
diagonalization can still be done
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But we have yet again hit another ceiling, because once we reach omega and go past it to omega
plus omega we can find another infinite set
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Will
we have to add a new symbol every time we describe something new? No, fortunately we can use
the scheme of replacement
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So
lets take the set of all the natural numbers and replace it by omega plus the natural number
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Using this we can make jumps of any size as long as we only use numbers we’ve already
reached
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We can go even further until
omega to the omega to the omega to the omega and so on
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And we can continue form here, but now there are all of these
ordinals, they are all well ordered so they have an order type, but after an infinite amount of
ordinals there is one ordinal that comes after all of them
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Because by definition omega one comes after every single
order type of aleph-null things so it must describe an arrangement of more things than the last
aleph
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But going back to the power set of aleph-null, its not in
between aleph-null and aleph-one, because there are no cardinals in between those two, it might
be the same as aleph-one but we just don’t know
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Reaching aleph-omega
and aleph-omega-omega-omega-etc
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There is no end
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Well, of course we can
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When you assume something is true in math it’s called axioms
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Mathematicians don’t fit there theories to some physical universe
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And all you have to do to achieve omega is to accept and say that omega
exists
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Beyond all the
natural numbers and reach aleph-null and omega and many more
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As stated above there is no end
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Such a number is called an
inaccessible cardinal, because you are unable to reach it
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But I disagree, because if something
really is an inaccessible cardinal it must also be uncountable, which is not the case with alephnull
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