Philosophy Now Forum

Truth is a property of something and as such is relative to this thing. Absolute truth would be then property of nothing. Absolute truth depends on nothing and describes nothing.

What I want to say is that difference between invention and discovery reflects the way common people see the world. From mathematical perspective the difference is much smaller if any.

Skepdick wrote: ↑Fri Mar 14, 2025 1:12 pm That's a really long-winded way to go about it.

All axioms are invented.
All consequences thereof (theorems) are discovered.

Invention is when from infinite enumerated sequence of axioms you pick some. Discovery is when you browse through infinite enumerated sequence of consequences.

With Halting problem Turing proved that you can't make halting procedure H that returns True if some other procedure halts, but you can make halting procedure that returns False is that other procedure halts. I don't see why is this so famous. It is more like obvious and trivial.

There is one thing I can't comprehend. If there is a sentence that is true and unprovable, how do you know that it is true? Did you prove it? If you dont prove it is true then it may be false and if you do prove it is true then it is not unprovable. Is there a rational solution of this dilemma?

If you replace provable(G) with G_is_provable you get the the formula G <=> not G_is_provable and this is just a simple propositional formula that does not and can not prove anything about provability in the first order logic. You are making a whole philosophy out of something trivial and dream that...

G is truth value of the sentence and G is false.
provable(G) is predicate and it's truth value is true.
G <=> not provable(G) is true.

Godel's first incompleteness theorem is expressed as: ∃G (G ⟺ ¬provable(⌜G⌝)) Interpretation: There exists a sentence G that is true but not provable or false and provable. Alternatively: If the theory at hand does not prove false sentences, then there exists a sentence that is true but not provabl...

If you create the following table: sentence1, proof1 sentence2, proof2 sentence3, proof3 ... In the table above, you end up with all the provable sentences in arithmetic along with their proofs. So, the question, "Is sentence S provable?" amounts to trying to look up S in the table above....