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Martin Sleziak
Martin Sleziak
7:22 AM
There seems to be discussion going on related to foundations on metamathematics:
These topics are not my strong suit. (In fact, I prefer to avoid methamatematics.) But I thought it is worth mentioning, just in case somebody else is interested in the discussion there.)
It seems that it started as a follow-up discussion to this question:
1
Q: Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?

Thomas KlimpelI heard that the P vs NP question is equivalent to a $\Pi_2^0$ sentence, and that the Riemann hypothesis is equivalent to a $\Pi_1^0$ sentence. Many known mathematical theorems state that some specific $\Pi_2^0$ sentence is actually true, take Goodstein's theorem for example. The related Paris-Ha...

logic computability ordinals
 
 
2 hours later…
Jim
Jim
9:15 AM
0
Q: Finding similar permutation.

Jim$A$ is a nonsymmetric matrix. $\beta_k, \gamma_k$ are sets of permutations. Each permutation acts on column of $A$. $\beta_k, \gamma_k$ have same permutations but different labeling is used to label columns of $A$. $\beta_k$ used one kind of labeling, $ \gamma_k$ used a different labeling. Con...

group-theory graph-theory finite-groups algebraic-graph-theory spectral-graph-theory
 
 
1 hour later…
Martin Sleziak
Martin Sleziak
10:29 AM
in Mathematics, 8 hours ago, by Leaky Nun
Is $\ln\omega$ defined?
in Mathematics, 10 secs ago, by Martin Sleziak
@LeakyNun I have seen logarithms of cardinals. But I have not encountered logarithms of ordinals.
 
 
1 hour later…
Martin Sleziak
Martin Sleziak
11:51 AM
in Mathematics, 3 mins ago, by Martin Sleziak
@LeakyNun Maybe Cantor normal form could give you something like logarithm with the base $\omega$? I am not sure. (I did not encounter situation where such things would be needed.)
in Mathematics, 35 secs ago, by Martin Sleziak
@LeakyNun I simply tried to search in Google Books and I found this in Enderton's Set Theory.
in Mathematics, 40 secs ago, by Martin Sleziak
Logarithm Theorem. Assume that $\alpha$ and $\beta$ are ordinal numbers with $\alpha\ne0$ and $\beta$ (the base) greater than $1$. Then there are unique ordinal numbers $\gamma$, $\delta$ nad $\rho$ (the logarithm, the coefficient, and the remainder) such that $$\alpha=\beta^\gamma\cdot\delta+\rho \land 0\ne\delta\in\beta \land \rho\in\beta^\gamma$$
 

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