Yes, my definition of the continuum is not adequate. Another poster gave a definition continuum close to mine but it is correct. I can search the thread and find the definition for you if you are interested. — MoK
Did not Godel and Cantor believe that once one sees Absolute Infinity — Gregory
I take 'one' to refer to humans, not to a god. But did Cantor or Godel say that any humans see absolute infinity? — TonesInDeepFreeze
if you search quotes by Cantor on the internet, there are these:
"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."
"A set is a Many that allows itself to be thought of as a One." — Gregory
If now is not the moment for AI, the near future will be. — keystone
Of course, one doesn't have to come within a million miles of a PhD just to learn basic mathematical logic. — TonesInDeepFreeze
I suppose it depends on what we're talking about. If we're talking about developing original ideas on continua I think more than a basic understanding of mathematical logic is required. — keystone
Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematical continua'. — keystone
If I recall correctly, another poster mentioned point-free geometry. — keystone
Do you accept Wikipedia as a reliable source? — Gregory
it seems unlikely that so msny sources are wrong to claim that Cantor believed Absolute Infinity was divinity and that the mathematics in our minds express a truth about truth itself, truth bring divinity. — Gregory
. I too find it unfitting that there be theorems in mathematics that can never be proven in any way. — Gregory
You are really going down the wrong road by resorting to AI for explanations. You are bound to take misinformation and confusion from it. — TonesInDeepFreeze
AI might become reliable in the future, or it could get a lot worse. A bot skims Internet articles that are themselves of dubious authority. The bots re-propagate the misinformation and even fabricate new misinformation. Then people re-propagate the misinformation re-propagated by the bots. A vicious sewage circle. — TonesInDeepFreeze
You were talking about learning basic mathematical logic. You said you are supplementing the book you are studying with AI. — TonesInDeepFreeze
Who is "the other poster"?
What standard mathematical definition of 'continua' are you referring to? — TonesInDeepFreeze
An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — TonesInDeepFreeze
You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?). — TonesInDeepFreeze
I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:
An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — keystone
point-based — keystone
You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?).
— TonesInDeepFreeze
I should not have used the word 'paradoxical' but rather logically impossible. — keystone
That makes no sense and is wrong: (1) By definition, a theorem is a statement that has a proof. (2) Incompleteness is not that there are statements that are unprovable "in any way". Rather, incompleteness is that if T is a consistent, formal theory that is sufficient for a certain amount of arithmetic, then there are statements in the language for T that are not provable in T. That does not preclude that statements not provable in T are provable in another theory — TonesInDeepFreeze
unprovable assumptions — Gregory
I think it is better to first get good at working in first order logic and then study the meta-theorems about first order logic. That's why I recommend this three-step sequence — TonesInDeepFreeze
So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone. — TonesInDeepFreeze
Df. the continuum = <R L> where R is the set of real numbers and L is the standard ordering on the set of real numbers.*
So, of course, there are points involved. — TonesInDeepFreeze
But the point of the argument by MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused. — TonesInDeepFreeze
Those are great logicians, great intellectual achievements. And a lot more (not necessarily in chronological order): Predecessors: Boole, De Morgan, Peirce, Cantor, Peano, Dedekind, Frege. Then Lowenheim, Skolem, Whitehead & Russell — TonesInDeepFreeze
Your picture of all of this is much too woozy — TonesInDeepFreeze
If Godel is widely misunderstood, the blame falls on those who explain it because i've seen many contradictory explanations of it — Gregory
Godel might have proven something about human conceptual thinking — Gregory
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