Forum:Did Taranovsky actually reach ZFC?

Remember this ordinal notation (by Dmytro Taranovsky) you tried hard to understand?

A few days ago, Taranovsky updated that page, writing that he found that his notation may go beyond second order arithmetic, and even ZFC. He found that the ordinal system in "Degrees of Reflection" is not identical to the on in the n=2 system in "Ordinal Notation System for Second Order Arithmetic", and it appears that the latter is much stronger.

Using the C function at the "Ordinal Notation System for Second Order Arithmetic" section, he said that if \(d=C(\Omega_2,C(\Omega_2 2))\), then \(C(\Omega_2+d,0)\) is the least recursively inaccessible ordinal, \(C(\Omega_2+d^2,0)\) is the least recursively Mahlo, and in the same way as the notation in "Degrees of Reflection".

\(C(\Omega_2 2,0)\) would be beyond all that, and using a certain working system(?), \(C(\Omega_2^2,0)\) corresponds to the first uncountable ordinal, and this is followed by ordinals with even larger cardinalities. And this way, this ordinal system may extend beyond ZFC, exceeding the strength of Loader's function and maybe even Friedman's finite promise games.

So what do you think about it? Do you think Taranovsky actually reached ZFC with his notation? -- ☁ I want more clouds! ⛅ 02:27, January 15, 2014 (UTC)