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AXIOM OF CHOICE: How do you CHOOSE from uncountable sets? (from only uncountable many choice functions)
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Graham Cooper
Jul 8, 2012, 11:54:46 PM7/8/12
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[AAUT]
I don't know about Russell, but the standard take is that the axiom
of
choice can be stated in (an interpreted language) L if there is a
sentence A in L such that we can prove (without appealing to choice)
that
A is true iff every set of non-empty sets has a choice function.
How many SETS are in ZFC?
How many FUNCTIONS are in ZFC?
HINT: use a fixed alphabet of 26 characters and count them.
****
None of all your F-A-N-T-A-S-T-I-C SET THEORY and NUMBER THEORY
holds up to the PIGEON HOLE PRINCIPLE!
You have MORE SETS THAN FORMULAS
A CHOICE FUNCTION for_each_set IS IMPOSSIBLE!
Herc
I don't know about Russell, but the standard take is that the axiom
of
choice can be stated in (an interpreted language) L if there is a
sentence A in L such that we can prove (without appealing to choice)
that
A is true iff every set of non-empty sets has a choice function.
How many SETS are in ZFC?
How many FUNCTIONS are in ZFC?
HINT: use a fixed alphabet of 26 characters and count them.
****
None of all your F-A-N-T-A-S-T-I-C SET THEORY and NUMBER THEORY
holds up to the PIGEON HOLE PRINCIPLE!
You have MORE SETS THAN FORMULAS
A CHOICE FUNCTION for_each_set IS IMPOSSIBLE!
Herc
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