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Originally Posted by ecriture d'adulte
My system does NOT have the statement Con(PA) being true anywhere in it. Godel's 2nd specifically tells us that it cannot.
Edit: I see it now. It's a misunderstanding: "Your system" = The system created by the statements provided in the specific post, not "Your system L."
"(P and not(Q)) and Q" is a logical contradiction, yes? This has nothing in particular to do with Godel. This is just a basic truth table. The statement contains both Q and not(Q), and so gives a logical contradiction. This is the construction that I am considering the in bolded statement in question.
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I don't know why you are switching to "logical contradiction" instead of just inconsistent.
It highlights the somewhat obvious distinction that you are refusing to make.
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If PA is consistent, do you think ~Con(PA) by itself is a logical contradiction?
No. I'm saying "if Con(PA) and not(Con(PA)), then contradiction."
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That's really the only way to save what you said, but it really tortures the meaning of contradiction into just a synonym for false.
The error your making is confusing basic consistency with soundness or intuitively true or a higher level concept like w consistency.
A contradiction is a statement. False is a truth value. I don't really see how they're synonyms for each other if they aren't even the same type of object. I'll lay out all of my definitions:
Math:
* A logical contradiction is a statement that is false regardless of the truth values of the propositions contained within that statement. For example, P and not(P) is a contradiction.
* A set of statements S is consistent if there are no derivable statements P (from S) such that both P and not(P) are derivable.
* An argument is sound if the assumptions are true and the deductive logic is valid.
* An axiom is a statement that is accepted to be true without proof.
Edit: To be absolutely clear....
--- False is a truth value of a statement: "The statement P is false."
--- A contradiction is a statement: "The statement 'P and not(P)' is a contradiction because 'P and not(P)' is false for all possible truth values of P."
--- Consistent is a property of a collection of statements: "The collection of statements S is consistent because you cannot derive a contradiction."
Philosophy:
* A statement is a contingent truth if there exist possible universes in which the statement is false.
* A statement is a necessary truth if there do not exist possible universes in which the statement is false.
You may or may not accept the philosophy definitions. But do you accept the mathematical definitions? Because if you do:
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Originally Posted by wiki
Now, assuming PA is really consistent, it follows that PA + ¬Con(PA) is also consistent
This is correct.
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Originally Posted by me
I think PA + not(Con(PA)) [leads to] a logical contradiction if "PA is consistent" is true [in addition].
This is correct as well, with the language cleaned up to be tighter.
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Originally Posted by you
The easiest example would be the set of axioms 1&2 taken together called L:
L:
1. All the axioms of Peano Arithmetic
2. The axioms of Peano Arithmetic are inconsistent.
We are taking these statements to be true. And if that's the case, then if you further assume that it's true that the axioms of Peano Arithmetic are consistent ("really consistent"), then that statement combined with PA + not(Con(PA)) gives a logical contradiction.
But this does not imply that PA + not(Con(PA)) is inconsistent. And I don't think I ever claimed that anywhere.
Last edited by Aaron W.; 06-11-2020 at 02:01 AM.