Again, retracting the law of excluded middle does not provide contradictions. Intuitionist mathematics eschews the law of excluded middle. If classica...
Godel's 2nd incompleteness theorem is not that certain systems can't be proven consistent, but rather that if they are consistent then they can't be p...
I don't know all the mathematics for engineering, but I don't imagine that reliably building bridges or other common practical endeavors depend on set...
Retracting excluded middle wouldn't allow contradictions. The logic is monotonic: we don't get additional theorems from subsets of a consistent axiom ...
It's probably fair to say that the import of foundations for the mathematics for the sciences is mostly theoretical as opposed to practical. But some ...
One my choose to hold that a proposition does not exist until it is has been expressed. But even if we restrict to the set of propositions that have b...
SEP is clear that 'Kp' means "We know that p is true" and that the proof uses sentences. And Wikipedia is even more explicit (I don't automatically tr...
The distinction between propositions and sentences is an involved subject in philosophy and logic. But no matter, it is not the case that the intended...
As I said, the general topic regards propostions, but the formal portion of the argument uses sentences. One should read the expositions. And, again, ...
I am not sure that all discussants here understand: (1) 'Kq' stands for "q is known to be true" and it does not stand for "q is known to be a sentence...
Who stated it? To be clear, Fitch does not hold that p -> Kp. What specific quotation or reference is given by anyone (other than a flagrantly errant ...
The topic is about propositions, but more formally about sentences. Yet it doesn't matter toward the point that 'Kq' does not stand for 'We know of th...
"We don't know that the earth is round" and "We believe that the earth is flat"? The differences are so easy to point out that I don't see the sense i...
'known' in this context doesn't mean 'we know that the sentence itself exists'. 'known' in this context means 'we know that the sentence is true'. If ...
There is no statement in the expositions I've seen of Fitch that 'p' stands for a true sentence. One may go back and read the exact expositions to see...
Contrarian, I would think, to the preponderance of philosophers and to everyday common sense. It is an extraordinarily outlandish view that every trut...
Fitch does not claim that all truths are known. That is ridiculous a misunderstanding of him. What he shows is that If all truths are knowable then al...
To believe For all q, we have q -> Kq is extraordinarily contrarian. It should not be overlooked that 'Kq' does not stand for 'q is knowable' but rath...
Formally, it's about sentences, regarded in terms of two primitive modal operators, whether true or false, whether known to be true or false or not. T...
What does it mean to say that falsehoods are or are not in the scope of Fitch's paradox? What does being "in the scope" of a paradox exactly mean in t...
I'm not inclined to go through the earlier posts. But, from what I do see, I don't know exactly what people are claiming about truth in this context. ...
If anyone is claiming there is an incorrect step, then I'd like to know where it is here: Axiom schemata: (a) Kq -> q (b) K(q & r) -> (Kq & Kr) (c) q ...
(1) Just to be clear, Fitch does not hold that for all p we have p -> Kp. Rather, the import is that if for all p we have p -> LKp, then for all p we ...
Suppose some Sn is true. So Sn+1 is false. So there is some k > n+1 such that Sk is true. But Sn is true and k > n, so Sk is false. So Sn is false. So...
"There is a set that is a member of itself" is not in and of itself contradictory. The famous contradictory statement is "There is a set of all sets t...
The first video (I didn't watch the second video) is stupid nonsense and disinformation. In this context, infinite summation is defined only for conve...
A conjunction of a statement and its negation is of the form P & ~P where P is any statement. A conjunction of a statement and its negation in the lan...
Perhaps in some of the articles cited, it is mentioned that paraconsistent logic is suited for such situations as contradictory data entries. I am not...
None of the axioms of set theory mention 'set'. So objection to any axioms on the basis that they mention 'set' are ill-founded. / 'set' is not a prim...
An objection was made that the axiom of extensionality does not "distinguish" between sufficiency and necessity. The axiom is: Axy(Az(zex <-> zey) -> ...
About the schema of separation, if we say there is one axiom for each predicate, we need to be careful what 'predicate' means. There is one axiom for ...
Z is axiomatized by: extensionality schema of separation pairing union power infinity regularity It is uncontroversial that AC, ZL, and WO are equival...
They are equivalent in Z, so, a fortiori, they are equivalent in ZF. But they are not logically equivalent. Z |- AC <-> ZL & ZL <-> WO & AC <-> WO But...
Footnote 1 of the SEP article says: "Talk of ‘first’ and ‘last’ members here is just a matter of convention. We could just as well have said that an i...
Another poster made it appear as if I hold that words (such as 'least') or symbols don't have explicit definitions, and that ambiguity results. My poi...
I mentioned finding a recursive definition of 'prior'. It's simple. We define the set of symbols prior to a symbol s: If s is primitive, prior(s) = 0 ...
Comments