Recent content by BDV

I agree! Thank you for the clarification, it reinforces that intuition derived from examining finite phenomena and collections is a weak guide when dealing with infinities.

To review an old topic. Still cannot wrap my head around the induction hypothesis making P(0) true from vacuously true vacuous predecessors. OTOH, once I use the negative of the induction hypothesis things become crystal clear (to me). That is, when the induction hypothesis is TRUE, for P(α)...

I meant more exotic than the principal ultrafilters and complements of finite set subalgebras. I realized last evening that non-principal ultrafilters may also contain objects with complements greater than finite sets. So there may be some richness right there. I just wanted to get a good...

Shouldn't the solution be t=ω? Possibly t=ω+1 ?!

Hello, With the axiom of choice, we are left with two options for ultrafilters: a) principal ultrafilters, built from a singleton {x}. b) nonpricipal filters of which all contain the cofinite filter, ergo complements of finite sets subalgebras. Isn't this kind of flimsy? To get to more...

I apologize, it is the space of linear functions from E to F.

Hello, I have seen (in H Cartan's differential calculus) a proof that if F is a Banch space, L(E,F) where E is some vector space, is also a Banach space. One of the main points of the proof is based on the behaviour of a function being "proper" (continuous) on a ball of arbitrary radius "n"...

But the procedure is very clear that at each step the number of balls increases by 9. N0 steps of 9 balls each. However, the RL procedure ensures that the resulting set is not effectively denumerable (cf. Sierpinski's usage of the word, e.g in "Cardinal and Ordinals"). Actually we cannot...

Ergo, having removed all named balls (ie balls covered under the natural notation) you are left with N0 of unnamed ones (not covered under natural notation). One cannot take the intuition from the completion of a finite number of steps and apply it to the totality of the infinite procedure...

Why should I be able to name even one of the remaining elements? The elimination procedure in the RL paradox simply ensures I have removed the entirety of the vocabulary for naming elements, so the remaining elements are unnamed. The RL pradox procedure yields a completely unorderd...

Ross-Littlewood vase filling paradox (from Wikipedia): To complete an infinite number of steps, it is assumed that the vase is empty at one minute before noon, and that the following steps are performed: The first step is performed at 30 seconds before noon. The second step is performed...

So I read, maybe there I chanced on a glimmer of understanding. P(0) is semantically included in the inductive hypothesis. In practice, though, one would have to prove P(0) separately. Now, if that is not correct, my problem continues to be computing properties of existing objects based...

Thank you for the replies and the links. Yossell, I had thought that your statement: "And in general, for any P, if P(0) is false, then the inductive hypothesis fails - for, vacuously, every predecessor of 0 has P, but P(0) is false. In this sense, for the inductive hypothesis to hold...

Sorry to be a thick skull on this topic but : P(a) for a<0 is also trivially (vacuously) false. It would make sense to me that by the formalism/formulation of the inductive hypothesis, proving the inductive hypothesis includes having to prove P(0). I am talking about a specific P.

Many authors, in online notes and in published books (e.g. Suppes in his Axiomatic Set Theory - 2nd ed. page 195-196, don't bother with P(0). So is P(0) - true needed as a precondition (my reading of wolfram link) or not? The only way out of this confusion (for me) is iff the inductive...