Basic Transfinite Arithmetic Explained
Submitted by on Mon, 11/02/2009 at 5:01pm.
Well to say basic isn't exactly fair. But this is cool and important and not too hard! So George Cantor, the Father of Set Theory (an informal title) was the first to show that there are different "levels" of actual infinities. What he did was prove this by the all-powerful diagonalization reductio proof. The actual infinities that result are named Alephs, which are Hebrew letters, and they have different subscripts starting with 0 that show their ordinality.
1.What G.F.L.P Cantor did first was to assume that there was an actual 1-1 correspondence. This means that every member of one infinite set should be able to correspond to another member of another infinite set.
2.Then he began constructing numbers that differed according to each successive decimal place in an ordinary fashion. These were paired to a corresponding number in a different set (obviously).
3.But by doing this, he was able to construct a number that different from all other numbers in the set in every decimal place by the above. This is diagonalization!
4.Therefore, 1-1 correspondence failed between sets. Which in turn means that our starting assumption must be false. Which means there are different levels of infinity.
Not too hard, right? So the infinity of natural and rational numbers is aleph-null (an aleph with a subscript of 0). The next level is aleph-1, and so on. We can actually manipulate these actual infinities ;).
The big problem became: does aleph-1=C? That is to say, C is the cardinality of all the real numbers. This is the CONTINUUM HYPOTHESIS: is there no set of cardinality larger than aleph-0 and smaller than that of the reals?
Well, the answer is not so simple. It could be true, it could be false, it could be neither. Paul Cohen and Kurt Godel proved that it was either true or false depending on a certain starting scheme that we use. Famous mathematicians were of divided opinion.
Thanks for viewing and good synapses,
strangequark
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