i always thought of zero as “the infinitely small” universe as in the other side or direction of “the infinitely large” universe, but then “thought” like time and space could also be a construct of the mind.

philosophically speaking i also wonder 😉 what the mind of the dolphin once they evolve to our current level in tech and science will think about this zero they invented for heir math… grin, thanks for all the fish

]]>Consider two sets, X and Y.

X contains (a b c d e)

Y contains (f g h i j)

Now perform a function in which you take all the members of X and pair them with members of Y.

a-f

b-g

c-h

d-i

e-j

If you remove all the members of X that have been successfully paired with a partner with Y, you find that X is now an empty set. You can map the contents of X onto Y, meaning that Y has at least as many members as X. Call this property A.

And you also find that if you perform the same operation on Y, pairing its members with X, this leaves Y an empty set, meaning X has at least as many members as Y. Call this property B.

If a pair of sets have both property A and property B, then they have the same cardinality. The sets have the same cardinality if and only if each one can be mapped onto the other without leaving a remainder.

http://en.wikipedia.org/wiki/Cardinality

http://en.wikipedia.org/wiki/Bijection

I got halfway through it by the time I thought of: “‘One’ is the condition by which something is in some set containing nothing else”.

It helps to recognise that when we say ‘one’, it is an abstract concept which stems from a reality in which the ‘thing’ has context. (One duck *in the room*).

Also, I find cardinality circular. “Three is the property defining the set ‘things in threes'”. This is the process that makes me find the rejection of axioms problematic.

On the whole, though, very well described ideas.

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