I made all ordinals up to $$\omega\cdot2$$ (Unfortunately it doesn't work with FGH), what should i do next and how would i do it?
Ordinal Library
Well, the good news is, you're not going to run out of ordinals. The bad news is, you are going to run out of computer programs, even if you live forever. So maybe you should get interested in something countable. For example, load all the world's award-winning novels into Snap!, and write the next one.
Or, you know, prove P=NP or something.
"countable", as in possible to count in general (like the extremely big numbers), or possible to count by a human that can live only 100 years or something?
Countable in the technical sense. Able to be put in one-to-one correspondence with the natural numbers or some subset thereof.
I'm also making set theory in it so not only does it support ordinals, but finite sets too.
Cool, finite sets are useful... I don't understand how that brings in set theory, which is about finite and infinite sets; you've been using it all along. All those (infinirte) ordinals you love come from set theory. For that matter, there are finite ordinals, which are the same as the finite cardinals but interpreted as "first, second, third..." instead of as "one, two, three..."
It's somewhat because the way to make larger ordinals is by using set theory symbols. For example, Buchholz's function uses set theory and symbols to map cardinals to ordinals. But madore's phi OCF (Ordinal Collapsing Function) is way simpler than buchholz, so i have a greater chance i might try to make it. Also, what OCFs do is map cardinals to ordinals, like omega and the bachmann howard ordinal.
Huh. Funny name. I would have guessed a function to collapse ordinals would map ordinals to cardinals! Oh well, shows what I (don't) know.
I meant from cardinals to large countable ordinals, to be used for functions that only allow countables, like Fast Growing Hierarchy, to make way larger numbers. For example, in Madore's Psi function:
$$\alpha>\zeta_0\rightarrow\psi(\alpha)=\zeta_{0}$$
$$\psi(\Omega) = \zeta_{0}$$
$$\psi(\Omega+1) = \varepsilon_{\zeta_{0}+1}$$
$$\psi(\Omega\cdot2) = \zeta_{1}$$
$$\psi(\Omega^{2}) = \eta_{0}$$
$$\psi(\Omega^{\omega}) = \varphi(\omega,0)$$
$$\psi(\Omega^{\Omega}) = \Gamma_{0}$$
$$\psi(\Omega^{\Omega^{\omega}}) = \textsf{Small Veblen Ordinal}$$
$$\psi(\Omega^{\Omega^{\Omega}}) = \textsf{Large Veblen Ordinal}$$
$$\psi(\varepsilon_{\Omega+1}) \text{(limit)} = \textsf{Bachmann-Howard Ordinal}$$
Okay.
The rules of the OCF:
Madore's Psi OCF
I'm not sure i will be able to apply the idea of set builder notation to it, since creating infinite lists is impossible. The only way i might be able to do it is by using the "keep items from" block, but what if i want to report a specific type of set, like $$\left\lbrace a+bi|a\in\mathbb{R},b\in\mathbb{R}\right\rbrace$$?
Could you use streams?
How would I create a predicate that reports if an element is contained in a infinite stream, instead of just a finite sample of it, and how would i visualize Ω as a list instead of reporting a ring with the block Ω in it (since that was the first thing i came up with)?
Well obviously you can't check every element of an infinite stream in finite time. :~)
Can you order the items? If so, as soon as you see an item bigger than the one you're looking for, you can stop.
But if this is a collection of all the ordinals, then it's not countable, so it won't fit even in a stream.
I have a lot of things to make in the Ordinal Library.
First is the Ordinal comparison system. What if i needed to compare $$\omega \text{ vs } \omega\cdot2$$? Now obviously $$\omega*2$$ is bigger, but what if the iterations for $$\omega\cdot2$$ was less than that of $$\omega$$? The computer would think $$\omega$$ is bigger, since it has more things in it, so I need a comparison system and a "is a successor" predicate so that all ordinals in form $$\omega\cdot\alpha$$ are limit. I'll continue later.