Recent content by jacquesb

Do you think with words? Do you feel the words follow the thought, or they make the thought? Do you hear them or see them mentally? Or do you think with pictures? Or with none of them, only pure thought?

I noticed a correspondence between Veblen functions and Madore's \psi collapsing function. \psi(\alpha) is defined as the least ordinal not in C(\alpha) , where C(\alpha) is the set of all ordinals constructible using only 0, 1, \omega , \Omega and addition, multiplication...

What is exactly a "naive method of extension" ? If Simmons ( http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/ordinal-notations.html ) really went much further than it is possible to go with my method, then the best would perhaps be to start where Simmons stopped and go further. I already...

I am fascinated by the topic of transfinite ordinals, which has the particularity of being the only mathematical domain that cannot be automated. In all other domains of mathematics, it is at least theoretically possible to deduce the theorems automatically from a formal system consisting of a...

Where did you see that his method goes "well-beyond" the Howard ordinal ? In http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/Fruitful.pdf , Simmons only speaks about "a method of producing ordinal notations `from below' (for countable ordinals up to the Howard ordinal)".

What I am trying to do now is first to understand what has already been done in the domain of ordinals construction, and then perhaps to go further. I began studying some tutorial presentations of ordinals, for example : -...

My two threads are not a repetition. My first thread was about a specific technical question concerning the fundamental sequences of \varphi_\alpha(\beta) . I consider it as solved for me since the definitions I read seem plausible to me now. My second thread is about the more general and vast...

The author wrote these formulas in page 10 in a slightly different but equivalent form : \phi_f 0 = f \phi_f(\alpha+1) = enumeration of fixed points of \phi_f \alpha \phi_f \lambda = enumeration of common fixed points of \phi_f \alpha for all \alpha < \lambda I rewrote these...

There is another way to express my construction, using Veblen functions indiced by the function used for \varphi_0 . As SSequence noticed in https://www.physicsforums.com/threads/fundamental-sequences-for-the-veblen-hierarchy-of-ordinals.933538/ , there are different conventions for...

I never saw the notation \Psi for the first uncountable ordinal, in everything I read it was denoted by \Omega . Finally I think it is more homogeneous to define : - \varphi'_0(\alpha) = \varphi(1_\alpha) - \varphi'_1(\alpha) = F(\alpha) - \varphi'_2(\alpha) = G(\alpha) -...

What is \Psi ?

Let me explain a bit more my idea. We start with the large Veblen ordinal (LVO) which is the least fixed point of the function \alpha \mapsto \varphi(1_\alpha) , \varphi(1_\alpha) representing the application of \varphi with transfinitely many variables with 1 at position \alpha and 0...

The binary Veblen function \varphi_\alpha(\beta) or \varphi(\alpha,\beta) (see https://www.physicsforums.com/threads/fundamental-sequences-for-the-veblen-hierarchy-of-ordinals.933538/) can be generalized to finitely many variables, for example \varphi(\alpha,\beta,\gamma) which can also be...

Is the condition 1 really necessary ? I mean, if f(0) = 0, couldn't we consider that the limit/least upper bound of 0, f(0), f(f(0)), ... is 0 which is the first fixed point of f ? At first sight it seems to me that the value at limit and the least upper bound are both equal to...

Thanks to SSequence for these explanations. (i) seems intuitively plausible to me, and also (2), the less evident is (1), but I have perhaps an idea of a possible proof. We have to prove that sup\{ \varphi _{{\beta [n]}}(\varphi _{{\beta }}(\gamma)+1) \,|\, n \in \omega\} is a common fixed...