Oh I see, you are talking of exclusive outcomes, not contradictions. Yes you are entirely right, exclusive outcomes work exactly that way. The probability of both Heads and Tails occuring at the same time on a coinflip is zero.
Contradictions in a system isomorphic with Propositional Logic do not and are an entirely separate mathematical object.
The Principle of explosion tells us that a if one supposes both truth and falsehood as base premises, one can derive any conclusion, by means of Propositional Logic theorems.
The intersection of a set with it’s complement is the empty set.
A consistent isomorphism of ZF Set theory and Propositional Logic is achieved by letting truth be a non-empty set and falshood be an empty set. Then intersection becomes logical conjunction and complement with respect to the truth set becomes logical inversion.
Now, the reason I can argue for such things is actually backed up by Godel’s incompleteness theorem. Propositional Logic and ZF Set Theory can both implement Peano Arithmetic, Bayesian Probability cannot. Thus Propositional Logic and ZF Set Theory are Complete but not Consistent, while Bayes is Consistent but not Complete.
argue for what things? I have no clue what the POE or Gtheorem have to do with komologrov provably assigning zero probability to contradictions.
What is the difference between exclusive outcomes and contradictions? How are they not the same mathematical object? If two exclusive outcomes end up resulting, you can also explode the universe.
For A to be exclusive with B, means that if A happened B did not.
So: If A and B, then ~B and B, then B or P, then ~B, so P.
The POE is not something unique to contradictions which exclusive outcomes lack.
Oh I see, you are talking of exclusive outcomes, not contradictions. Yes you are entirely right, exclusive outcomes work exactly that way. The probability of both Heads and Tails occuring at the same time on a coinflip is zero.
Contradictions in a system isomorphic with Propositional Logic do not and are an entirely separate mathematical object.
I would like to read more on that, because I believed them to be exactly equivalent.
A set’s intersection with its compliment
has a perfect isomorphism with
A propositions conjunction with its negation.
The Principle of explosion tells us that a if one supposes both truth and falsehood as base premises, one can derive any conclusion, by means of Propositional Logic theorems.
The intersection of a set with it’s complement is the empty set.
A consistent isomorphism of ZF Set theory and Propositional Logic is achieved by letting truth be a non-empty set and falshood be an empty set. Then intersection becomes logical conjunction and complement with respect to the truth set becomes logical inversion.
Now, the reason I can argue for such things is actually backed up by Godel’s incompleteness theorem. Propositional Logic and ZF Set Theory can both implement Peano Arithmetic, Bayesian Probability cannot. Thus Propositional Logic and ZF Set Theory are Complete but not Consistent, while Bayes is Consistent but not Complete.
argue for what things? I have no clue what the POE or Gtheorem have to do with komologrov provably assigning zero probability to contradictions.
What is the difference between exclusive outcomes and contradictions? How are they not the same mathematical object? If two exclusive outcomes end up resulting, you can also explode the universe.
For A to be exclusive with B, means that if A happened B did not. So: If A and B, then ~B and B, then B or P, then ~B, so P. The POE is not something unique to contradictions which exclusive outcomes lack.
This comment explains where our communication runs skew