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The Last Alpha Number

This is my candidate for a large named number. It is called The Last Alpha Number and is calculated using The Alpha Function. It is approximately \(f_{svo}(10000)\) and therefore it is not big (by this website standard).


The Alpha Function

The Alpha Function calculates an ordinal for a given real number input. For example:

\(\alpha(0.00) = 0\)

\(\alpha(1.00) = 1\)

\(\alpha(2.00) = 8\)

\(\alpha(e) = 9\)

\(\alpha(3.00) = 10\)

\(\alpha(\pi) = 11\)

\(\alpha(4.00) = f_{\omega}(8)\)

The function accepts real numbers up to 10,000 at which point it asymptotically goes to infinity. The Alpha Function is based on my work on the S Function (substitution function). The limit of the Alpha Function is represented as follows:

\(\alpha(10000) = S(2,T^{\omega}(0),1) = \omega\) equals infinity by definition


Last (calculated) Alpha Number

My candidate for one of the largest named numbers is the Last Alpha Number. By this I mean the last Alpha Number that can be calculated given the limits of computer processing time and number precision.

Using my desktop PC and program code written on Microsoft Excel VBA, the last calculated Alpha Number I can get is:

\(\alpha(9999.99999999998) = S(2,T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T\)

\((T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(T(0))))))))))))))))\)

\()))))))))))))))))))))))))))))))))))),1)\)

or

\(\alpha(9999.99999999998) = S(2,T^{52}(0),1)\)

This is obviously arbitrary and would be easy to beat with programming code changes to increase the number precision used in the calculations.

Therefore my candidate for large named number is the Last (designated) Alpha Number.


Last (designated) Alpha Number

My candidate for one of the largest named numbers is:

\(\alpha(d) = S(2,T^{10000}(0),1)\) where \(d\) is a real number very close to 10,000

The necessary precision required to write \(d\) would be a lot. Simply explained it is 9999.9999 recurring for many many digits. The value of d is not important.

The Last Alpha Number is not really that big but I calculate \(\alpha(d) = S(2,T^{10000}(0),1) = f_{svo}(10000)\).


Further References

Further references to relevant blogs can be found here: User:B1mb0w

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