When remedies are past, the griefs are ended
By seeing the worst, which late on hopes depended.
To mourn a mischief that is past and gone
Is the next way to draw new mischief on.
What cannot be preserved when fortune takes
Patience her injury a mockery makes.
The robb'd that smiles steals something from the thief;
He robs himself that spends a bootless grief.
Friday 6 February 2009
Thursday 5 February 2009
The axiom of choice
The axiom of choice is incompatible with the axiom of determinacy.
The axiom of determinacy allows proof that ZF set theory is consistent.
The axiom of choice allows the creation of unmeasurable sets. There is no intuitive example of such a set, which is bad.
The axiom of choice allows the Banach–Tarski_paradox which is ridicululous until you remember that unmeasurable sets were ridiculous anyway.
The axiom of determinacy can be proved with infinite logic, which is cool.
So my vote would be against the axiom of choice. Sorry if that destroys your field but is there anything with a real application that requires the axiom of choice? Fourier analysis doesn't.
The joys of Wikipedia.
The axiom of determinacy allows proof that ZF set theory is consistent.
The axiom of choice allows the creation of unmeasurable sets. There is no intuitive example of such a set, which is bad.
The axiom of choice allows the Banach–Tarski_paradox which is ridicululous until you remember that unmeasurable sets were ridiculous anyway.
The axiom of determinacy can be proved with infinite logic, which is cool.
So my vote would be against the axiom of choice. Sorry if that destroys your field but is there anything with a real application that requires the axiom of choice? Fourier analysis doesn't.
The joys of Wikipedia.
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