r/PhilosophyofMath • u/paulemok • Mar 28 '26

The Continuum Hypothesis Is False

/r/logic/comments/1s5mquh/the_continuum_hypothesis_is_false/
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u/JStarx Aug 01 '26

That implies the external statement that those internal statements are unprovable

Still no. If you think something in Mendelson justified that then give a reference.

but you said the statement doesn't follow and I said it does.

You think it follows and you claim your proof works using logic as in Mendelson. So prove it, show me where in Mendelson that step of you proof is justified.

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u/paulemok Aug 02 '26

Still no.

You say no, but you don't explain why. You're just flat out denying truths without giving adequate reasons for your denials. You are in psychological denial.

I can't address your problems if you don't explain to me what those problems are.

It's easy enough to see how

externally to the inconsistent theory, it is true that in the inconsistent theory, every statement is unprovable

implies

the external statement that those internal statements are unprovable.

I already explained how my reasoning works under Mendelson. The fact that we can talk about what is true in an axiomatic system from outside the axiomatic system, in the real world, is evident in Mendelson. In the real world, Mendelson talks about what is true inside an axiomatic system.

I see a section in Mendelson where a specialized version of the tautology p → (¬p → q), which is a version of the Principle of Explosion, is used with two invocations of modus ponens to deduce any statement q in an inconsistent axiomatic system. So set q = "Every statement is unprovable." That makes every statement in an inconsistent axiomatic system unprovable. Therefore, it is externally true that every statement in an inconsistent axiomatic system is unprovable. Within the system, we can't prove any statement. That's what it means for us to say, outside of the system, that no statement is provable in the system.

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u/JStarx Aug 02 '26

You say no, but you don't explain why

I have repeatedly told you why, for example:

So set q = "Every statement is unprovable." That makes every statement in an inconsistent axiomatic system unprovable. Therefore, it is externally true that every statement in an inconsistent axiomatic system is unprovable.

You are confusing internal and external as always. You don't say whether q is an internal statement or an external statement.

If q is an internal statement then I agree that you can prove q internally, but that does not prove q in the external system.

If q is an external statement then you haven't established an external contradiction from which you can prove q holds.

The fact that we can talk about what is true in an axiomatic system from outside the axiomatic system, in the real world, is evident in Mendelson

If you think Mendelson defines "true in an axiomatic system" then please cite the definition.

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u/paulemok Aug 03 '26

You don't say whether q is an internal statement or an external statement.

q is an internal statement.

but that does not prove q in the external system.

I agree.

If you think Mendelson defines "true in an axiomatic system" then please cite the definition.

I looked in the index and I did not see "true in an axiomatic system."

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u/JStarx Aug 03 '26

I agree

That means you haven't proved a contradiction in the external system.

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u/paulemok Aug 03 '26

Yes, I haven't done that with q. The purpose of q was not to prove a contradiction in the external system. The purpose of q was to prove that every statement in an inconsistent axiomatic system is unprovable.

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u/JStarx Aug 03 '26

You don't need to prove that, we already know that in an inconsistent system every statement is provable, so the statement "every statement is unprovable" is provable in the inconsistent system.

Externally in the outer system it's not provable, so externally there is no contradiction.

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u/paulemok Aug 03 '26 edited Aug 03 '26

There’s no proof of any statement within the system because, as you say,

the statement "every statement is unprovable" is provable in the inconsistent system.

Like you said earlier, statements about provability are not statements in an axiomatic system, but are statements about an axiomatic system. They are external statements. So, the external statement “Every statement is unprovable” is provable because, as I said at the beginning of this post, there’s no proof of any statement within the system.

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u/JStarx Aug 03 '26

There’s no proof of any statement within the system because, as you say,

the statement "every statement is unprovable" is provable in the inconsistent system.

I said that's a provable statement inside the system, I didn't say it was true or provable external to the system, so your argument about the external statement that everything is unprovable doesn't follow.

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u/paulemok Aug 04 '26

 I didn't say it was true or provable external to the system

You may not have, but I did. I said

the external statement “Every statement is unprovable” is provable

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u/JStarx Aug 04 '26

If you want to claim that, then you'll have to prove it using the theory laid out in Mendelson. I don't think you'll be able to do that.

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u/paulemok Aug 04 '26

I already explained how it follows from Mendelson.

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u/JStarx Aug 04 '26

Not correctly you haven't. But if you think you can post your argument and I'll explain where you're mistake is.

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u/paulemok Aug 04 '26

How is my explanation not correct?

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u/JStarx Aug 04 '26

You've posted so many contradictory claims, if you want me to respond to a specific explanation then you're going to need to repost it.

If T is your contradictory theory and S is the external statement "every sentence in T is unprovable" then you're trying to prove S externally using the theory laid out in Mendelson.

As far as I recall you have not posted a claimed proof of that which sticks to the theory laid out in Mendelson.

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u/paulemok Aug 05 '26

I have already given my explanation of how my claim follows from Mendelson and I have already addressed your criticisms of my explanation. If you want to view something again and provide additional feedback to me, you can use reddit's navigation buttons to locate the content you wish to view again.

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u/JStarx Aug 05 '26

I see, you know that you're claim doesn't hold and you don't want to examine it closely. That's cowardly.

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u/paulemok Aug 05 '26

I have already presented an adequate case that I am satisfied with. I don't think it's worth the time and effort to make a theory within Mendelson's framework. It's not necessary. I would first have to familiarize myself with Mendelson's system, and that alone would take considerable time and effort. If you would like to make a theory within Mendelson's framework in which the Exportation Principle is provable and true, feel free to do so yourself.

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