TonesInDeepFreeze

I said that my commentary is based only on the clip posted at the top of this thread. Yesterday I got hold of McGee's paper. It turns out that his arg...

PPS Two iconic books that handle forcing in detail are: 'Set Theory' - Jech 'Set Theory: An introduction To Independence Proofs' - Kunen Jech's book i...

PS. We don't need definitions of 'cardinality' an 'ordinal' to state CH and GCH. We can state it equivalently: GCH. If S is infinite, then is no set X...

I'm not a set theorist, but I have some thoughts. I haven't seen articles before that give a layman's explanation of forcing and of axioms for proving...

I like the idea of structuralism, and my own personal understanding of set theory is that we may think of the axioms as specifying structural relation...

"Replace" might not be a good way of putting. A better way of putting it might be that sets "play the role" of numbers, or something like that. And wh...

That's not the case in set theory.

Or maybe we can think of vacuity and unity as the bases. Then we have 0 and 1, the binary. But in set theory, with the pairing operation, we can defin...

https://thephilosophyforum.com/discussion/comment/568807 With a Hilbert style system, the axiom we use to derive modus tollens is given in the intuiti...

https://plato.stanford.edu/entries/structuralism-mathematics/

I think you meant 'transitive set well ordered by ?'.

My mistake.

My mistake. My proof does not use RAA. I see that what I left tacit is not RAA, but just modus tollens: Let f:X -> PX Let S = {y e X | ~y e f(y)} S e ...

I don't know whether this bears on anything here, but just in case, there is huge difference between: (1) ~~P RAA premise ... contradiction ... infer ...

I did bother. I read your post three times but couldn't figure it out.

I don't know what you're saying. You lost me.

I can see it both ways. Starting with an RAA premise provides a clear structure. Not starting with RAA, but instead talking about an arbitrary functio...

There's a double negative in what you're saying. RAA premise would not need to deny ~P. Rather, in this case, the premise is P. Anyway, that's not the...

In the beginning, I didn't deny any claim claim whatsoever. And, of course, I wouldn't even think of denying the claim that S is not in the range of f...

Of course the premise "A Republican wins" restricts. The impression that there is not good reason to believe "If Reagan doesn't win then Anderson wins...

That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity."

For some mathematicians its a stylistic preference.(I'm not sure, but I think maybe my version is Cantor's version.)* The proofs prove the exact same ...

The definition of 'precedes' ('less than') had been given by fishfry many posts ago. To have missed it is to have not paid attention to the posts. Df....

df: S is equinumerous with T <-> there is a bijection between S and T theorem: For every S, there is a unique T such that T is an ordinal & S and T ar...

'inherent' has not been given a mathematical definition. Dispute about it can go on and on and on, and in circles, for as long as people have the oppo...

https://thephilosophyforum.com/discussion/comment/568451 We would continue to prove that the uncountability of Pw implies the uncountability of R: It ...

k = the-least-ordinal_x such that Px <-> (k is an ordinal & Pk & Ah(h e k -> ~Ph))

We don't need to suppose toward contradiction that there is a surjection. Let f:X -> PX Let S = {y e X | ~y e f(y)} S e PX S e ran(f) -> EyeX f(y) = S...

https://sites.duke.edu/wsa/papers/files/2011/05/wsa-defenseofmodusponens1986.pdf I just now read that. My argument is basically the same as theirs.

From Donkeys and Elephants to Lungfish and Porpoises. McGee has another supposed impeachment of MP. https://sites.duke.edu/wsa/papers/files/2011/05/ws...

You are terribly confused. You asked me to prove there is not bijection between a countable set and an uncountable set. And I told you where to find t...

By definition, there is no bijection between a countable set and an uncountable set. By theorem, there is no bijection between N and R. The proof has ...

Impressionistic descriptions are fine for stoking creativity in mathematics and sometimes for making certain mathematical concepts intuitive. But they...

Please. If you have a rigorous definition of "infinite more" different from set theoretic "greater cardinality" then fine, state your definition, and ...

Beating a dead horse about donkeys, elephants, and red herrings. Directly responding to the clip and its one sentence intro: MP is valid. McGee does n...

I just realized I made a really rookie mistake in some of my attempts several posts back. prob(x) is not presumably the poll rating of x. For example,...

I said that I only have the clip from the article to reference plus the locution 'strictly valid'. Anything I say is in that context alone. If there i...

Not by the premises of the argument. The point is not to challenge the premises, but rather to show that the conclusion has as great a reason for beli...

Here's the new puzzle for me: Based on the level of McGee's research in logic and his associations, he must be extremely intelligent and knowledgeable...

My analysis doesn't need to say anything about Carter. All I need to point out is that the conclusion of the example has strength of reason to belief ...

'incomplete' is not part of my analysis. MP is valid. McGee claims MP is not "strictly valid" which I can only take to mean that MP does not preserve ...

I agree. "false -> P" is true. And again, it's not even about MP: S v H is equivalent to ~S -> H.

Everything is immaterial vs Some things are material Everything is material vs Some things are immaterial Everything is immaterial vs Some things are ...

Or if Idealism and Materialism can't be shoehorned into strict Pro and Con that way, then there could be just one particular representative propositio...

Usually a debate has Pro and Con such that Con is the negation of Pro. Is that the framework with Idealism vs Materialism here? What is the definitive...

That article is good because it's hard to find layman's terms explanations of forcing and the proposed axioms. But a couple of points: "Cantor realize...

E -> P implies ~P -> ~E. CORRECT ~(~E -> ~P) is inconsistent with ~P -> ~E. INCORRECT

Ironic that in a thread about an alternative to head to head debating we have a head to head debate.

Suppose there are additional facts and logic that would improve your opponent's argument. Then you might easily win the Steelman by bringing in the ad...

MODUS PONENS HOCUS POCUS I think fdrake and Andrew M had the right idea, but it needed a follow-through. I think sime had the solution in a general fo...