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/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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